Relations And Functions Questions

We provide relations and functions practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on relations and functions skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

Relations And Functions Questions

List of relations and functions Questions

Question NoQuestionsClass
1If function ( boldsymbol{f}(boldsymbol{x})=frac{1}{2}- )
( tan left(frac{pi x}{2}right) ;(-1<x<1) ) and ( g(x)= )
( sqrt{3+4 x-4 x^{2}}, ) then the domain of
( (g o f) ) is
12
2The range of the function ( y= ) ( frac{1}{(2-sin 3 x)} ) is
( ^{mathrm{A}} cdotleft(frac{1}{3}, 1right) )
в. ( left[frac{1}{3}, 1right) )
c. ( left[frac{1}{3}, 1right] )
D. None of these
12
3An example for a function which is a
relation, (Domain-R, Codomain-R) is:
This question has multiple correct options
A ( cdot y=x )
В. ( y=x-1 )
c. ( y=x^{2} )
D. none of these
12
4The domain of the function ( f(x)= )
( frac{sqrt{5-x}}{(x+1)(x-2)(x-3)}+ )
( a sin ^{-1}left[frac{2 x-5}{3}right] ; a neq 0 ) and ( [cdot] ) denotes
greatest integer function, is:
A ( cdot[1,5]-{2,3} )
3
B . [1,2)( cup(3,5] )
c. ( left[1, frac{11}{2}right) )
D・ [1,5]
11
5if ( boldsymbol{f} ) and ( boldsymbol{g} ) are real function defined by
( boldsymbol{f}(boldsymbol{x})=mathbf{2} boldsymbol{x}-mathbf{1} ) and ( boldsymbol{g}(boldsymbol{x})=boldsymbol{x}^{2}, ) then
A. ( (3 f-2 g)(1)=1 )
В. ( (f g)(2)=10 )
C ( cdot g^{3}(2)=128 )
D ( cdotleft(frac{sqrt{f}}{g}right)(2)=frac{sqrt{3}}{2} )
12
6let ( f(x)=sin x, g(x)=[x+1] ) and ( g(f(x))=h(x) )
then ( h^{prime}left(frac{pi}{2}right) )
12
7( f(x)=log frac{(1+x)}{1-x} ) then ( f(log (x))=? )
A ( cdot log (1+log x)-log (1-log x) )
B. ( log (1+log x)+log (1-log x) )
c. ( log (1-log x)-log (1-log x) )
D. None of these
11
8( boldsymbol{f}: boldsymbol{Q} rightarrow boldsymbol{Q} ) is defined by ( boldsymbol{f}(boldsymbol{x})=mathbf{1 5} boldsymbol{x}+mathbf{7} )
is
A . injective only
B. surjective only
c. bijective
D. neither injective nor surjective
12
9The domain of function ( y=log _{3}(5+ )
( left.4 x-x^{2}right) ) is
A ( .(0,2] )
B . ( (-infty,-1) cup(5, infty) )
c. (0,9]
D. (-1,5)
12
10Determine whether the operation ( ^{prime} *^{prime} ) on
( N ) defined by ( a * b=a+b-2 ) for al
( a, b in N ) is a binary operation or not:
12
11Classify the following function ( boldsymbol{f}(boldsymbol{x}) )
defined in ( R rightarrow R ) as injective,
surjective, both or none
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+boldsymbol{6} boldsymbol{x}^{2}+boldsymbol{1} mathbf{1} boldsymbol{x}+boldsymbol{6} )
A. Surjective but not injective
B. Injective but not surjective
c. Neither injective nor surjective
D. Both injective and surjective
12
12Let ( boldsymbol{A}={1,2,3,4} ; B={3,5,7,9} )
( C={7,23,47,79} ) and ( f: A rightarrow B, g: )
( B rightarrow C ) be defined as ( f(x)=2 x+1 )
and ( g(x)=x^{2}+x . ) Express ( (g circ f)^{-1} )
and ( f^{-1} circ g^{-1} ) as the sets of ordered
pairs and verify and ( (boldsymbol{g} circ boldsymbol{f})^{-1}=boldsymbol{f}^{-1} circ )
( boldsymbol{g}^{-1} )
12
13Let ( boldsymbol{f}(boldsymbol{x})=frac{(boldsymbol{x}-boldsymbol{a})(boldsymbol{x}-boldsymbol{b})}{(boldsymbol{x}-boldsymbol{c})} ) Which of the
following are TRUE?
This question has multiple correct options
A. Range of ( f(x) ) is ( R, b leq c leq a )
B. Range of ( f(x) ) is ( R, a leq c leq b )
C. Range of ( f(x) ) is proper subset of ( R ),for ( a leq b leq c )
D. Range of ( f(x) ) is proper subset of ( R ),for ( c leq b leq a )
11
14The graph of the equation ( x y=k, ) where
( k<0, ) lies in which two of the
quadrants shown above?
A . I and II
B. I and III
c. II and III
D. II and IV
E . III and IV
12
15Is ( g={(1,1),(2,3),(3,5),(4,7)} )
function? If this is described by the formula.
( g(x)=alpha x+beta, ) then what values should be assigned to ( alpha ) and ( beta ? )
12
16Vikas and Vasu punt a ball into the air.
The equation ( boldsymbol{h}=-mathbf{1 6} boldsymbol{t}^{2}+boldsymbol{6 0} boldsymbol{t} )
represents the height of ball in feet, seconds after it was punted for Vikas’s ball. Which of the following can be Akanksha’s ball height equation if her ball goes higher?
A ( cdot h=-16left(t^{2}-3 tright) )
B . ( h=-8 tleft(2 t^{2}-9right) )
c. ( h=-4left(2 t^{2}-5right)^{2}+48 )
D. ( h=-4left(2 t^{2}-6right)^{2}+52 )
12
17Give a relation ( mathrm{R}={(1,2),(2,3)} ) on the set
of natural numbers, add a minimum number of ordered pairs.
This question has multiple correct options
A. enlarged relation is symmetric
B. enlarged relation is transitive
c. enlarged relation is reflexive.
D. enlarged relation is equivalence relation
12
18If ( y ) is second entry and ( x ) is first entry then its ordered pair will be
A. ( (x, y) )
(i) ( x )
в. ( (y, y) )
c. ( (y, x) )
D. None of the above
12
19Let ( R ) a relation on the set ( N ) be defined
by ( {(x, y) mid x, y in N, 2 x+y=41} . ) Then
( boldsymbol{R} ) is
A. Reflexive
B. Symmetric
c. Transitive
D. None of these
12
20Show that function ( f: N rightarrow N ) given by
( f(x)=3 x ) is one-one but not onto.
12
218.
The number of real solutions of the equation
1×2-3x +2=0 is
(1982 – 2 Marks)
(a) 4 a. (b) 1 (C) 3 6 (d) 2
12
22The range of the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}-[boldsymbol{x}] )
where ( [x] ) represents the greatest
integer less than or equal to ( x ) is
( A cdot{0} )
в. [0,1]
c. (0,1)
()
D. [0,1]
12
23f ( M={x: x in N, 1<x leq 4} ) and
( N={y: y in W, y<3} ; ) find ( : N times M )
Also find the number of such ordered
pairs.
12
24Let ( boldsymbol{S}={1,2,3, ldots, 100} . ) The number
of non-empty subsets ( boldsymbol{A} ) of ( boldsymbol{S} ) such
that the product of elements in ( boldsymbol{A} ) is even is :-
A ( cdot 2^{50}left(2^{50}-1right) )
B . ( 2^{100}-1 )
c. ( 2^{50}-1 )
D. ( 2^{50}+1 )
12
25If ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{B} ) is surjective then
A. no two elements of ( A ) have the same image in ( B )
B. every element of ( A ) has an image in ( B )
C. every element of ( B ) has at least one pre-image in ( A )
D. ( A ) and ( B ) are finite non empty sets
12
26If the set ( A ) has 3 elements and the set
( B={3,4,5} ) then find the number of elements in ( (boldsymbol{A} times boldsymbol{B}) ) ?
12
27Suppose ( f ) is the collection of all
ordered pairs of real numbers and ( x=6 ) is the first element of some ordered pair
in
f. Suppose the vertical line through ( x ) ( =6 ) intersects the graph of ( f ) twice. Is ( f ) a function? Why or why not?
12
28The relation ( R ) and ( R^{prime} ) are symmetric in
the set ( A ), then show that ( R cup R^{prime} ) and
( boldsymbol{R} cap boldsymbol{R}^{prime} ) are symmetric.
12
29Identify the first component of an ordered pair ( (mathbf{0},-mathbf{1}) )
( mathbf{A} cdot mathbf{0} )
B. –
( c cdot 2 )
( D )
12
30Let ( R ) be the equivalence relation on the
( operatorname{set} Z ) of integers given by ( boldsymbol{R}= )
( {(a, b): 2 text { divides } a-b} . ) Write the
equivalence class ( [mathbf{0}] )
12
31Given ( boldsymbol{f}: boldsymbol{x} rightarrow mathbf{2} boldsymbol{x}+boldsymbol{p} ) and ( boldsymbol{f}^{-1}: boldsymbol{x} rightarrow )
( boldsymbol{m}(boldsymbol{4} boldsymbol{x}+boldsymbol{3}), ) where ( boldsymbol{p} ) and ( boldsymbol{m} ) are
constants. Find the value of ( p ) and value
of ( boldsymbol{m} )
12
32The value of the function ( boldsymbol{f}(boldsymbol{x})= )
( frac{x^{2}-3 x+2}{x^{2}+x-6} ) lies in the interval
( ^{mathbf{A}} cdot(-infty, infty) /left{frac{1}{5}, 1right} )
B ( cdot(-infty, infty) )
( mathrm{c} cdot(-infty, infty) / 1 )
D. None of these
12
33Let ( X ) be the set of all citizens of India.
Elements ( x, y ) in ( X ) are said to be related if the difference of their age is 5 years.
Which one of the following is correct?
A. The relation is an equivalence relation on ( x ).
B. The relation is symmetric but neither reflexive nor transitive
c. The relation is reflexive but neither symmetric nor transitive
D. None of the above
12
34Make h the subject of the formula given
above.
A ( quad h=frac{s+2 pi r^{2}}{2 pi r} )
B. ( h=frac{s-2 pi r^{2}}{2 pi r} )
c. ( h=frac{s-2 pi r^{2}}{4 pi r} )
D. none of the above
11
35If ( R ) is a relation from a set ( A ) to a set ( B )
and ( S ) is a relation from ( B ) to a set ( C )
then the relation ( S O R )
A. is from ( A ) to ( C )
B. is from ( C ) to ( A )
c. does not exist
D. none of these
12
36If ( boldsymbol{R} ) is a symmetric relation on a set
( A={1,2,3}, ) then write the relation
between ( R ) and ( R^{-1} )
12
37The figure shows a relationship
between the sets ( P ) and ( Q . ) Write this
relation in
(i) in set-builder form (ii) roster form
12
38( f(x) ) and ( g(x) ) are linear function such that for all ( x, f(g(x)) ) and ( g(f(x)) ) are Identity functions. If ( f(0)=4 ) and ( g(5)= )
17, compute ( f(2006) )
11
39Let ( boldsymbol{f}: boldsymbol{x}, boldsymbol{y}, boldsymbol{z} rightarrow(boldsymbol{a}, boldsymbol{b}, boldsymbol{c}) ) be a one-one
function. It is known that only one of the following statements is true:
(i) ( boldsymbol{f}(boldsymbol{x}) neq boldsymbol{b} )
(ii) ( f(y)=b )
(iii) ( f(z) neq a )
A ( . f={(x, a),(y, b),(z, c)} )
B. ( f={(x, b),(y, a),(z, c)} )
c. ( f={(x, b),(y, c),(z, c)} )
D. ( f={(x, b),(y, c),(z, a)} )
12
40The smallest positive root of the equation, tan x – x = 0 lies
(1987-2 Marks)
(e) None of these
12
41The domain of the function ( boldsymbol{f}(boldsymbol{x})= )
( log _{3+x}left(x^{2}-1right) ) is
A ( cdot(-3,-1) cup(1, infty) )
B . [-3,-1)( cup[1, infty) )
c. (-3,-2)( cup(-2,-1) cup(1, infty) )
D ( cdot[-3,-2) cup(-2,-1) cup[1, infty) )
12
42The distinct linear functions which
maps from [-1,1] onto [0,2] are
A. ( x+1,-x+1 )
в. ( x-1, x+1 )
c. ( -x+1 )
D. ( -x+2 )
12
4318. How many real solutions does the equation
x7 + 14×5 + 16×3 + 30x – 560 = 0 have?
[2008]
(a) 7. (b) (c) 3 (d) 5
10
Lot
(.)
.
12
44Prove that the relation ( R ) on ( Z ) defined
by ( (a, b) in R Leftrightarrow 5 ) divides ( a-b, ) is an
equivalence relation on ( Z )
12
45Let ( R=left{left(a, a^{3}right): a ) is a prime number right.
less than ( 10} ). Find dom(R).
12
46If ( f(x)=cos u x+cot x, ) find ( f^{prime}left[frac{pi}{4}right] )12
47For the function ( f ) graphed in the ( x y- )
plane above, if ( boldsymbol{f}(-mathbf{2 . 5})=boldsymbol{k}, ) then find
( boldsymbol{f}(mathbf{2 k}) ? )
A . 0
B.
c. 1.5
D. 2
E . 0.5
12
48( f(x)=x^{2}+5, ) Find ( f(5) )12
49Let ( L ) be the set of all lines in a plane
and ( R ) be the relation in ( L ) defined as
( boldsymbol{R}=left{left(boldsymbol{L}_{1}, boldsymbol{L}_{2}right): boldsymbol{L}_{1} perp boldsymbol{L}_{2}right} . ) Show that ( boldsymbol{R} ) is
symmetric but neither reflexive nor transitive.
12
50If ( boldsymbol{f}: boldsymbol{R}-{-1,1} rightarrow boldsymbol{R} ) is defined by
( f(x)=frac{x}{x^{2}-1}, ) verify whether ( f ) is one
to-one or not.
12
51Let ( S ) be the set of all points in a plane. Let ( R ) be a relation on ( S ) such that for any
two points a and ( b, a R b ) iff ( b ) is within 1
centimetre from a. Check R for
reflexivity, symmetry and transitivity.
12
52The domain of the function ( mathbf{f}(boldsymbol{x})=sqrt{frac{boldsymbol{x}-mathbf{2}}{boldsymbol{x}+mathbf{2}}}+sqrt{frac{mathbf{1}-boldsymbol{x}}{mathbf{1}+boldsymbol{x}}} ) is
A. ( R )
B. [-2,2]
c. [-1,1]
D.
12
53( boldsymbol{f}(boldsymbol{x}+mathbf{1})=(-1)^{x+1} boldsymbol{x}-boldsymbol{2} boldsymbol{f}(boldsymbol{x}) ) for ( boldsymbol{x} in mathbb{N} )
and ( f(1)=f(1986) . ) Then sum of digits
of ( (boldsymbol{f}(mathbf{1})+boldsymbol{f}(mathbf{2})+ldots .+boldsymbol{f}(mathbf{1 9 8 5}))= )
( mathbf{A} cdot mathbf{4} )
B. 3
( c cdot 7 )
D. 1
11
54Write the domain of the relation ( boldsymbol{R} )
defined on the set ( mathbb{Z} ) of integers as
follows:
( (a, b) in R Leftrightarrow a^{2}+b^{2}=25 )
12
55Illustration 2.17 A function f(x) is defined as f(x) = x +3.
Find f(O), F(1), f(x?), f(x + 1) and f(f(1)).
12
56Given ( boldsymbol{A}={mathbf{5}, mathbf{6}, mathbf{7}} ) and ( boldsymbol{B}={mathbf{3}, mathbf{4}} )
Form all possible ordered pairs and write the total number of ordered pairs formed so that the first component is from set ( A ) and second component is
from set B.
What is the total number of such pairs?
12
5711. Let f(x) be a function satisfying the condition f(-x)=f(x)
for all real x. If f'() exists, find its value. (1987- 2 Marks)
12
58Area of the region bounded by the curve ( boldsymbol{y}=boldsymbol{x}^{2} ) and ( boldsymbol{y}=boldsymbol{s e c}^{-1}left[boldsymbol{s i n}^{2} boldsymbol{x}right] ) (where ( [.] )
denotes the greatest integer function) is
A ( cdot frac{pi}{3} sqrt{pi} )
В. ( frac{2 pi sqrt{pi}}{3} )
c. ( frac{4 pi sqrt{pi}}{3} )
D. ( frac{6 pi sqrt{pi}}{3} )
E ( cdot frac{3 pi sqrt{pi}}{2} )
11
59f ( A={a, b} ) and ( B={1,2,3} ) find
( (boldsymbol{A} times boldsymbol{B}) cap(boldsymbol{B} times boldsymbol{A}) )
12
60Find range of ( f(x)=frac{x^{2}+2 x+3}{x}, ) the
range is ( R-(a, b), ) find ab?(If the
answer is ( mathrm{m} ). Find ( -mathrm{m} ) )
12
61If relation ( mathrm{R}= )
( {(x, x+2): x in N, 1 leq x<4} ) then ( R )
is
в. {(2,3),(4,2),(5,3)}
c. {(1,3),(2,4),(3,5),(4,6)}
D. none of these
12
62Let ( boldsymbol{X}={1,2,3,4,5} . ) fr the relation ( boldsymbol{g}= )
( {(1,2),(2,3),(3,4),(4,5),(5,1),} ) on
( X ) is a function from ( X ) to ( X ).then find
( boldsymbol{g}(boldsymbol{g}(boldsymbol{g}(boldsymbol{4}))) )
11
63If ( f(x)=ln left(frac{1+x}{1-x}right), ) then ( fleft(frac{2 x}{1+x^{2}}right) )
equals.
( mathbf{A} cdot[f(x)]^{2} )
B. ( f(x)^{3} )
c. ( 2 f(x) )
D. ( 3 f(x) )
12
64Let ( boldsymbol{f} ) be an injective map with domain ( {x, y, z} ) and range {1,2,3} such that exactly one of the following statements is correct and the remaining are false:
( boldsymbol{f}(boldsymbol{x})=mathbf{1}, boldsymbol{f}(boldsymbol{y}) sqrt{mathbf{1}}, boldsymbol{f}(z) sqrt{mathbf{2}} . ) The value of
( boldsymbol{f}^{-1}(1) ) is
( A )
B.
( c cdot z )
D. none of these
11
65The function ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R}, boldsymbol{f}(boldsymbol{x})= )
( x^{2}, forall x in R ) is
A. Injection but not surjection
B. Surjection but not injection
c. Injection as well as surjection
D. Neither injection nor surjection
12
66Two functions ( f(x) ) and ( g(x) ) are defined as ( f(x)=log _{10}left|frac{x-2}{x^{2}-10 x+24}right| ) and
( boldsymbol{g}(boldsymbol{x})=sin ^{-1}left(frac{mathbf{2}[boldsymbol{x}]-mathbf{3}}{mathbf{1 5}}right), ) where
denotes greatest integer function, then the number of even integers for which
( f(x)+g(x) ) is defined, is
12
67If ( g(f(x))=|sin x| ) and ( f(g(x))= )
( (sin sqrt{x})^{2}, ) then
A ( cdot f(x)=sin ^{2} x cdot g(x)=sqrt{x} )
B. ( f(x)=sin x, g(x)=|x| )
C. ( f(x)=x^{2}, g(x)=sin sqrt{x} )
D. f and g can not be determined
12
68The function ( mathbf{f}: mathbf{R} rightarrow mathbf{R} ) defined by
( mathbf{f}(mathbf{x})=sin mathbf{x} ) is
This question has multiple correct options
A. neither one one nor onto
B. onto
c. one-one
D. many one
12
69If the function ( f(x)=x^{4}-62 x^{2}+a x+ )
9 is maximum at ( x=1 ), then the value
of a is
12
70( operatorname{Let} f(x)=frac{x^{2}-4}{x^{2}+4} ) for ( |x|>2, ) then the
function ( boldsymbol{f}:(-infty,-2] cup[2, infty) rightarrow )
(-1,1) is
A. One-one into
B. One-one onto
c. Many one into
D. Many one onto
12
713.
If f(x) = cos[rº]x + cos[-T?]x, where [x] stands for the
greatest integer function, then
(1991 – 2 Marks)
(a) =-1 (b) f(n)=1
(©) f(-1) = 0
(d) r
=1
12
7211. Let S (x)=sinsin(sinx) for all x e R and g(x) =
sin x for all x E R. Let (fog)(x) denote f(g(x)) and (gof)(x)
denote g(x)). Then which of the following is (are) true?
(JEE Adv. 2015)
(2) Range of fis [2]
(6) Range of fog is [28]
©
lim ()_
( *-+0 g(x) 6
(d) There is an xeR such that (gof)(x) = 1
12
73The domain of the function ( boldsymbol{f}(boldsymbol{x})= )
( log _{x}(cos x) ) is
A ( cdotleft(-frac{pi}{2}, frac{pi}{2}right)-{1} )
в. ( left(frac{pi}{2}, frac{pi}{2}right)-{1} )
c. ( left(-frac{pi}{2}, frac{pi}{2}right) )
D. ( left(0, frac{pi}{2}right)-{1} )
12
74If ( boldsymbol{x}, boldsymbol{y} in[mathbf{0}, mathbf{2} boldsymbol{pi}], ) then find the total
number of ordered pairs ( (boldsymbol{x}, boldsymbol{y}) )
satisfying the equation ( sin x cos y=1 )
12
75If ( boldsymbol{A}={boldsymbol{a}, boldsymbol{b}, boldsymbol{c}}, ) then the relation ( boldsymbol{R}= )
( {(b, c)} ) on ( A ) is
A . reflexive only
B. symmetric only
c. transitive only
D. reflexive and transitive only
12
76Let ( f ) and ( g ) be real-valued functions
such that
( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})+boldsymbol{f}(boldsymbol{x}-boldsymbol{y})=2 boldsymbol{f}(boldsymbol{x}) )
( boldsymbol{g}(boldsymbol{y}) forall boldsymbol{x}, boldsymbol{y} boldsymbol{epsilon} boldsymbol{R} )
if ( f ) is not identically zero and ( f|(x)| leq )
( mathbf{1}, forall boldsymbol{x} epsilon boldsymbol{R}, ) then ( |boldsymbol{g}(boldsymbol{y})| leq mathbf{1}, forall boldsymbol{y} boldsymbol{epsilon} boldsymbol{R} )
If true enter 1 else enter 0
A. True
B. False
12
77If ( R ) is a relation from a finite ( operatorname{set} A )
having ( m ) elements to a finite set ( B )
having ( n ) elements, then the number of relations from ( A ) to ( B ) is:
( A cdot 2^{m n} )
B . ( 2^{m n}-1 )
( c cdot 2 m n )
D. ( m^{n} )
12
7865. If f(x)+ 2
) = 3x,x+ 0 and
[JEE M 2016]
S= {xI R: f(x)=f(-x)}; then S:
(a) contains exactly two elements.
(b) contains more than two elements.
(c) is an empty set.
(d) contains exactly one element.
11
799.
Let f(x) = (x + 1)2 – 1, x 2-1. Then the set
{x: f(x) = f'(x)} is
Jo-1 -3+iV3 -3- iv3 |
2
-2
(a) 70, -1,
(b) {0, 1,-1}
(c) {0,-1}
(d) empty
12
80Show that subtraction are not binary operation on natural number ( N )12
81If ( boldsymbol{A}=left{boldsymbol{x}: boldsymbol{x}^{2}-boldsymbol{3} boldsymbol{x}+boldsymbol{2}=boldsymbol{0}right} ) and ( boldsymbol{B}= )
( left{x: x^{2}+4 x-5=0right} ) then the value of
( A-B ) is
( mathbf{A} cdot{1,2} )
B . {2}
( c cdot{1} )
D ( cdot{5,2} )
12
82f ( A={a, b} ) and ( B={1,2,3} ) find
( A times A ) and ( B times B )
12
83( f(x)=frac{k^{x}}{k^{x}+sqrt{k}}(k>0) ) then
( sum_{r=1}^{2 n-1} 2 fleft(frac{r}{2 n}right)=a n+b ) where ( a-b )
is equal to
11
84ff ( f(x+1)=3 x-9 ) then find the value
of ( boldsymbol{f}left(boldsymbol{x}^{2}-mathbf{1}right) )
12
85Find domain and range ( . ) If ( boldsymbol{y}=sqrt{boldsymbol{x}-mathbf{1}} )12
86Let ( x ) be non-empty set. ( P(x) ) be its power set. Let ( ^{*} ) be an operation defined on element of ( boldsymbol{P}(boldsymbol{x}) boldsymbol{b} boldsymbol{y}, boldsymbol{A} * boldsymbol{B}= )
( boldsymbol{A} cap boldsymbol{B} forall boldsymbol{A}, boldsymbol{B} in boldsymbol{P}(boldsymbol{x}) . ) Then
(i) Prove that ( ^{*} ) is a binary operation in ( boldsymbol{P}(boldsymbol{x}) )
(ii) is ( ^{*} ) associative?
(iii) Is ( ^{*} ) commutative?
12
87( f(x)=frac{log frac{1}{2}+log (2+4 x)}{x} ) for ( x neq 0 ) at ( x=2 )12
88Solve the following inequalities.
( log _{1 / 2}(x+4)<2 )
11
89Calculate the number of roots of ( boldsymbol{f}(|boldsymbol{x}|) ) if ( f(x)=(x-2)(x+3)(x-4) )
( A cdot 3 )
B. 6
( c cdot 0 )
D. 4
11
90Find the number of ordered pairs in
( boldsymbol{R}^{-1} boldsymbol{o} boldsymbol{R} )
12
91If ordered pair ( (a, b) ) is given as (-2,0)
then ( a= )
A . -2
B. 0
( c cdot 2 )
D. None of the above
12
92If ( boldsymbol{A}={boldsymbol{x}:-mathbf{1} leq boldsymbol{x} leq mathbf{1}}=boldsymbol{B} . ) Discuss
the following function w.r.t one-one onto bijective and write their characteristics ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}}{boldsymbol{2}} )
A. one one, into
B. one one, onto
c. many one, into
D. many one, onto
12
93The relation ‘is a factor of on the set of
natural numbers is
not
A. reflective
B. symmetric
c. anti symmetric
D. transitive
12
94If ( f(x)=sin ^{2} x ) and the composite
function ( g(f(x))=|sin x|, ) then ( g(x) ) is
equal to
A ( cdot sqrt{x-1} )
B. ( sqrt{x} )
c. ( sqrt{x+1} )
D. ( -sqrt{x} )
12
95The function ( f(x) ) is defined on the interval ( [0,1] . ) Find the domain of the function:
( boldsymbol{f}(boldsymbol{2} boldsymbol{x}+boldsymbol{3}) )
( ^{A} cdot-frac{3}{2} leq x leq-1 )
в. ( -frac{3}{2} leq x leq frac{3}{2} )
( ^{mathrm{c}}-1 leq x leq frac{3}{2} )
D. ( -1 leq x leq 1 )
12
96Define a relation ( R ) on the set ( N ) of
natural numbers by ( boldsymbol{R}={(boldsymbol{x}, boldsymbol{y}): boldsymbol{y}= )
( x+5, x ) is a natural number less than
( mathbf{4} ; boldsymbol{x}, boldsymbol{y} in boldsymbol{N}} . ) Depict this relationship
using roster form. Write down the domain and the range.
12
97Let ( A ) and ( B ) be two sets such that ( A times )
( boldsymbol{B}= )
( {(a, 1),(b, 3),(a, 3),(b, 1),(a, 2),(b, 2) )
then
A ( cdot A={1,2,3} ) and ( B={a, b} )
B . ( A={a, b} ) and ( B={1,2,3} )
( mathbf{c} cdot A={1,2,3} ) and ( B subset{a, b} )
D. ( A subset{a, b} ) and ( B subset{1,2,3} )
12
98Total number of equivalence relation defined in the set ( s={a, b, c} ) is
A . 5
B. 31
( c cdot 2^{3} )
D. ( 3^{3} )
12
99The relation ( R ) defined on ( operatorname{set} A= )
( {boldsymbol{x}:|boldsymbol{x}|<mathbf{3}, boldsymbol{x} boldsymbol{epsilon} boldsymbol{I}} ) by ( boldsymbol{R}= )
( {(x, y): y=|x|} ) is
A ( cdot{(-2,2),(-1,1),(0,0),(1,1),(2,2)} )
B – {(-2,2),(-2,2),(-1,1),(0,0),(1,-2),(2,-1),(2,-2)}
c. {(0,0),(1,1),(2,2)}
D. None of the above
12
100If ( f(x)=cos (log x) ) then ( f(x) f(y)- )
( frac{1}{2}left[fleft(frac{x}{y}right)+f(x y)right] ) has value
A . -1
B. 2
( c .-2 )
( D )
11
101What type of a relation is “Less than” in the set of real numbers?
A. only symmertric
B. only transitive
c. only reflexive
D. equivalence relation
12
102Let ( boldsymbol{f}(boldsymbol{x})=mathbf{4} boldsymbol{x}-mathbf{3} . ) If ( boldsymbol{f}(boldsymbol{a})=mathbf{9} ) and
( f(b)=5, ) then calculate ( f(a+b) )
( mathbf{A} cdot mathbf{5} )
B. 7
c. 14
D. 16
E . 17
12
103If ( f ) is even function and ( g ) is an odd
function, then ( f_{o} g ) is ( ldots ). What fun.
A. Even
B. Odd
C. Neither even nor odd
D. Either even
11
104Let ( boldsymbol{f}(boldsymbol{x}) ) be defined on ( [-2,2], ) and is
given by ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{l}-1,-2 leq x leq 0 \ x-1,0 leq x leq 2end{array} text { and } g(x)=right. )
( boldsymbol{f}(|boldsymbol{x}|)+|boldsymbol{f}(boldsymbol{x})|, ) then find ( boldsymbol{g}(boldsymbol{x}) )
11
105The domain of the function ( f(x)= ) ( frac{sin ^{-1}(x-3)}{sqrt{9-x^{2}}} ) is
A ( cdot[1,2] )
в. [2,3]
c. [2,3]
D. [1,2)
12
106Which of the following is always true
A ( cdot(p Longrightarrow q) equiv sim q Longrightarrow sim p )
в. ( (p Longrightarrow q) equiv p wedge sim q )
( mathbf{c} . sim(p vee q) equiv vee p vee sim q )
D. ( (p vee q) equiv sim p wedge sim q )
12
107Number of solutions of the equation ( [y+ ) ( [y]]=2 cos x ) is
(where ( y=(1 / 3)[sin x+[sin x+[sin x]]] )
and []( =text { greatest integer function }) )
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
( D cdot infty )
11
108The domain of ( boldsymbol{f}(boldsymbol{x})=frac{mathbf{1}}{sqrt{boldsymbol{x}^{2}-boldsymbol{4}}} ) is
( mathbf{A} cdot(-infty, 4) )
в. (-2,3)
( c cdot(4, infty) )
D. ( (-infty,-2) cup(2, infty) )
12
109( y=frac{1}{sqrt{(4+3 cos x)}} ) Is the function one-
one ? Explain
12
110( f ) is a linear function. Values of ( x ) and
( f(x) ) are given in the table; complete the table
begin{tabular}{cc}
( boldsymbol{x} ) & ( boldsymbol{f}(boldsymbol{x}) ) \
-3 & 17 \
0 & ( – ) \
( – ) & 1 \
4 & -18 \
hline \
( – ) & -30
end{tabular}
12
111If ( f(x)=frac{x+1}{x-1} ; ) then ( fleft(frac{1}{2}right)= )
A ( cdot frac{1}{2} )
B. ( -frac{1}{2} )
( c cdot frac{3}{2} )
D. –
11
112Find the domain of the following
functions:
( boldsymbol{f}(boldsymbol{x})=sqrt{mathbf{3}-mathbf{2}^{boldsymbol{x}}-mathbf{2}^{mathbf{1 – x}}} )
A . [0,1]
B . [0,4]
c. [0,2]
D. None of these
12
113If ( f(x)=3 x-7, ) then what is ( f )
chickpea ( ) ? )
A. ( 3 times ) chickpea -7
B. chickpea –
c. chickpea
D. chickpea +7
12
114If ( z_{1} ) and ( z_{2} ) both satisfy the relation ( z+ ) ( bar{z}=2|z-1| ) and ( arg left(z_{1}-z_{2}right)=frac{pi}{4} )
then the imaginary part of ( left(z_{1}+z_{2}right) ) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. None of these
11
115The number of real solutions of
( cos ^{-1} x+cos ^{-1} 2 x=frac{pi}{4} ) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. infinitely many
12
11630.
Let f(x) = –
– for n
2 and
(1+xnn
g(x) = (fofo.of) (x). Then [xn-2 g(x)dx equals.
f occurs n times
(2007 -3 marks)
1-
1
(a)
nx”) n +K (b)
(1 + nx”)
n +K
n(n-1)
n-1
(e)
(+ rux”)* + K (a) 1 (1+x)**+K
1+-
-(1+nx”) n +K
1+ nx”)
n +K
(d)
n(n+1)
n+1
12
117If ( f(A) ) is a proper subset of ( B ), then ( f ) ( A rightarrow B ) is called a/an ( ldots ldots . . ) function
A . into
B. onto
c. one – one
D. indentity
12
118Determine the domain and the range of the relation ( R ) defined by ( R= )
( {(x+1, quad x+5): x in )
( {0, quad 1, quad 2, quad 3, quad 4, quad 5}} )
12
119Let ( R=left{left(a, a^{3}right): a ) is a prime number right.
less than ( 10} ). Find range (R).
12
120Let ( ^{*} ) be a binary operation on the set ( boldsymbol{Q} )
of rational numbers as follows:
( boldsymbol{a} * boldsymbol{b}=boldsymbol{a}^{2}+boldsymbol{b}^{2} )
Find which of the binary operations are commutative and which are
associative.
12
121The binary operation ( *: boldsymbol{R} times boldsymbol{R} rightarrow boldsymbol{R} ) is
defined as ( a * b=2 a+b . ) Find ( (2 * 3) * )
4
12
122Assertion
Let ( boldsymbol{f}:(boldsymbol{e}, infty) rightarrow boldsymbol{R} ) defined by ( boldsymbol{f}(boldsymbol{x})= )
( log (log (log x)) ) is invertible.
Reason
( f ) is both one-one and onto.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
123( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R}, boldsymbol{f}(boldsymbol{x})=boldsymbol{3} boldsymbol{x}+boldsymbol{2} )
( boldsymbol{g}: boldsymbol{R} rightarrow boldsymbol{R}, boldsymbol{g}(boldsymbol{x})=boldsymbol{6} boldsymbol{x}+mathbf{5} )
for the given functions ( left(g o f^{-1}right)(10)= )
A . 21
B . 29
( c cdot 7 )
D.
12
12418. Lety(***)-16); “) for all real x and y. If s”0)
exists and equals – 1 and f(0)=1, find f(2). (1995 – 5 Marks)
12
125ff ( [x] ) is the integral part of a real number x. Then, solve ( [mathbf{2 x}]-[boldsymbol{x}+mathbf{1}]= )
( 2 x )
11
126If
( * ) is defined on the set ( R ) of all real
numbers by ( *: boldsymbol{a} * boldsymbol{b}=sqrt{boldsymbol{a}^{2}+boldsymbol{b}^{2}}, ) find
the identify element, if it exists in ( boldsymbol{R} )
with respect to ( * )
12
127Let ( * ) be a binary operation on the set of all non zero real numbers, given by ( a * )
( boldsymbol{b}=frac{boldsymbol{a} boldsymbol{b}}{mathbf{5}} forall boldsymbol{a}, boldsymbol{b} in boldsymbol{R}-{mathbf{0}}, ) find the value of
( x, ) given that ( 2 *(x * 5)=10 )
12
128f ( n(A)=4 ) and ( n(B)=6, ) then the
number of surjections from ( boldsymbol{A} ) to ( boldsymbol{B} ) is
A ( cdot 4^{circ} )
B. ( 6^{text {4 }} )
c. 0
D. 24
12
129Let z satisfy ( |z-3|=4, ) then find max
and min. value of ( |z-5-7 i| )
12
130( operatorname{Let} boldsymbol{f}(boldsymbol{x}):left{begin{array}{llll}boldsymbol{x}, & boldsymbol{x} & boldsymbol{i} boldsymbol{s} & boldsymbol{r} boldsymbol{a} boldsymbol{t} boldsymbol{i} boldsymbol{o} boldsymbol{n} boldsymbol{a} boldsymbol{l} \ boldsymbol{0}, & boldsymbol{x} & boldsymbol{i} boldsymbol{s} & boldsymbol{i} boldsymbol{r} boldsymbol{r} boldsymbol{a} boldsymbol{t} boldsymbol{i} boldsymbol{o} boldsymbol{n} boldsymbol{a} boldsymbol{l}end{array}right. )
and
( boldsymbol{g}(boldsymbol{x}):left{begin{array}{llll}mathbf{0}, & boldsymbol{x} & boldsymbol{i} boldsymbol{s} & boldsymbol{r} boldsymbol{a} boldsymbol{t} boldsymbol{i} boldsymbol{o} boldsymbol{n} boldsymbol{a} boldsymbol{l} \ boldsymbol{x}, & boldsymbol{x} & boldsymbol{i} boldsymbol{s} & boldsymbol{i} boldsymbol{r} boldsymbol{r} boldsymbol{a} boldsymbol{t} boldsymbol{i} boldsymbol{o} boldsymbol{n} boldsymbol{a} boldsymbol{l} boldsymbol{l}end{array}right. )
ff ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) and ( boldsymbol{g}: boldsymbol{R} rightarrow boldsymbol{R}, ) then ( (boldsymbol{f}- )
( boldsymbol{g}) ) is
A. one-one and into
B. neither one-one nor onto
c. many-one and onto
D. one-one and onto
12
131The equation ( x^{2}-2=[sin x], ) where ( [.] )
denotes the greatest integer function, has
This question has multiple correct options
A. infinity many roots
B. exactly one integer root
c. exactly one irrational root
D. exactly two roots
11
132Left ( boldsymbol{f} ) be an odd function defined on the
set of real numbers such that for ( x geq )
( mathbf{0}, boldsymbol{f}(boldsymbol{x})=mathbf{3} sin boldsymbol{x}+mathbf{4} cos boldsymbol{x} )
Then ( f(x) ) at ( x=-frac{11 pi}{6} ) is equal to
A ( cdot-frac{3}{2}-2 sqrt{3} )
B. ( frac{3}{2}-2 sqrt{3} )
c. ( frac{3}{2}+2 sqrt{3} )
D. ( -frac{3}{2}+2 sqrt{3} )
12
133Let ( A={1,2,3} . ) The total number of
distinct relations that can be defined
over ( A ) is:
A ( cdot 2^{9} )
B. 6
( c .8 )
D. None of the above
12
134( operatorname{Let} boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}mathbf{1}+|boldsymbol{x}|, quad boldsymbol{x}<-mathbf{1} \ {[boldsymbol{x}], quad boldsymbol{x} geq-mathbf{1}}end{array} ) where right.
denotes the greatest integer function,
Find ( boldsymbol{f}{boldsymbol{f}(-mathbf{2 . 3})} )
11
135The domain of the function ( f(x)=log _{10} )
( 4-x^{2} mid ) is.
12
136Determine whether or not each of the
definition of ( * ) given below gives a
binary operation. In the event that ( * ) is not a binary operation, given justification for this
(i) ( operatorname{On} Z^{+}, ) define ( * ) by ( a * b=a-b )
(ii) ( operatorname{On} Z^{+}, ) define ( * ) by ( a * b=a b )
(iii) On ( R, ) define ( * ) by ( a * b=a b^{2} )
(iv) On ( Z^{+}, ) define ( * ) by ( a * b=|a-b| )
On ( Z^{+}, ) define ( * ) by ( a * b=a )
12
137( * ) is said to be commutative in ( boldsymbol{A} ) for all
( boldsymbol{a}, boldsymbol{b} in boldsymbol{A} )
A ( . a+b=b+a )
B. ( a * b=b * a )
c. ( a-b=b-a )
( a )
D. ( a * b neq b * a )
12
138In the set of all positive integers show
that the relation ‘>’ is not an
equivalence relation.
12
139If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) and ( boldsymbol{g}: boldsymbol{R} rightarrow boldsymbol{R} ) are defined
by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}-[boldsymbol{x}] ) and ( boldsymbol{g}(boldsymbol{x})=[boldsymbol{x}] ) for
( boldsymbol{x} in boldsymbol{R}, ) where ( [boldsymbol{x}] ) is the greatest integer
not exceeding ( x, ) then for every ( x in ) ( boldsymbol{R}, boldsymbol{f}(boldsymbol{g}(boldsymbol{x}))= )
A . ( x )
в. 0
c. ( f(x) )
D. ( g(x) )
12
140( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R}, boldsymbol{g}: boldsymbol{R} rightarrow boldsymbol{R} ) and ( boldsymbol{f}(boldsymbol{x})=sin boldsymbol{x} )
( boldsymbol{g}(boldsymbol{x})=boldsymbol{x}^{2} ) then ( boldsymbol{f o g}(boldsymbol{x})= )
A ( cdot x^{2}+sin x )
B. ( x^{2} sin x )
( mathbf{c} cdot sin ^{2} x )
( D cdot sin x^{2} )
12
141Show that the relation ( R ) on the ( operatorname{set} A= )
( {x in Z ; 0 leq x leq 12}, ) given by ( R= )
( {(a, b): a=b}, ) is an equivalence
relation.
12
142In each of the following cases, state whether the function is one-one, onto or
bijective. Justify your answer.
(i) ( f: R rightarrow R ) defined by ( f(x)=3-4 x )
(ii) ( f: R rightarrow R ) defined by ( f(x)=1+x^{2} )
12
143Let ( A=Q times Q ) and let ( ^{star} ) be a binary
operation on A defined by ( (a, b) * )
( (c, d)= )
( (a c, b+a d) ) for ( (a, b),(c, d) epsilon A )
Determine, whether ( ^{*} ) is commutative and associative. Then, with respect to
on ( A )
Find the invertible elements of ( A )
12
144Solution of ( |4 x+3|+|3 x-4|=12 ) are
A ( cdot x=-frac{7}{3}, frac{3}{7} )
в. ( x=-frac{5}{2}, frac{2}{5} )
c. ( _{x=-frac{11}{7}}, frac{13}{7} )
D. ( x=-frac{13}{7}, frac{11}{7} )
11
145The period of the function ( f(x)= ) ( sin 3 x cos [3 x]-cos 3 x sin [3 x], ) where ( square )
denotes the greatest integer function is
( mathbf{A} cdot mathbf{6} )
B. 3
c. ( 1 / 3 )
D. 1/6
11
146( operatorname{Let} g(x)=1+x-[x] quad ) and ( f(x)= )
( left{begin{array}{lll}-1 & text { if } & boldsymbol{x}mathbf{0}end{array}right. )
equals
( A )
B.
c. ( f(x) )
D. ( g(x) )
12
147If functions ( f, g: R rightarrow R ) are defined as
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+mathbf{1}, boldsymbol{g}(boldsymbol{x})=mathbf{2} boldsymbol{x}-mathbf{3}, ) then find
( f o g(x), g o f(x) ) and ( g o g(3) )
12
148Find, if possible, ( (boldsymbol{f} circ boldsymbol{g})(boldsymbol{0}) )
( f(x)=frac{1}{(x+1)}, g(x)=frac{1}{(x-1)} )
12
149Function ( f(x) ) is defined as ( f(x)= ) ( left{begin{array}{l}frac{x-1}{2 x^{2}-7 x+5}, quad x neq 1 \ -frac{1}{3}, quad x=1end{array}right. )
Is ( f(x) ) differentiable at ( x=1 ) if yes find
( boldsymbol{f}^{prime}(mathbf{1}) )
12
150Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}left(frac{1}{x-1}+frac{1}{x}+frac{1}{x+1}right), boldsymbol{x}>1 )
Then
A. ( f(x) leq 1 )
B. ( 1<f(x) leq 2 )
c. ( 23 )
11
151If ( A={1,2,3} ) and ( B={4,5} ) then the
number of function ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{B} ) which is
not onto is
( A cdot 2 )
B. 6
c. 8
D. 4
12
152Let ( A={1,2} ) and ( B={2,3,4} )
If ( boldsymbol{B} times boldsymbol{B}= )
{(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)
Please enter 1 or else 0
12
153If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R}, boldsymbol{g}: boldsymbol{R} rightarrow boldsymbol{R} ) are defined by
( boldsymbol{f}(boldsymbol{x})=mathbf{5} boldsymbol{x}-mathbf{3}, boldsymbol{g}(boldsymbol{x})=boldsymbol{x}^{2}+mathbf{3}, ) then
( left(g o f^{-1}right)(3)= )
A ( cdot frac{25}{3} )
в. ( frac{111}{25} )
c. ( frac{9}{25} )
D. ( frac{25}{111} )
12
154If ( f(x)=frac{sin ([x] pi)}{x^{2}+x+1}, ) where ( [x] ) denotes
the integral part of ( x . ) Show that ( f(x) ) is
a constant function.
11
155Find the solution set of the inequality
( |boldsymbol{y}-mathbf{5}| leq mathbf{1} )
A. ( 4 leq y leq 6 )
в. ( 4 leq y geq-6 )
c. ( 4 geq y leq 6 )
D. ( -4 leq y leq 6 )
11
156If ( boldsymbol{f}(boldsymbol{x})=mathbf{3} boldsymbol{x}-boldsymbol{2} ) and ( boldsymbol{g}(boldsymbol{x})=frac{boldsymbol{x}+mathbf{1}}{mathbf{2}} )
then ( boldsymbol{g} boldsymbol{o} boldsymbol{f}(boldsymbol{3})= )
12
157The velocity ( n m / s ) of a particle is
proportional to the cube of the time. If the velocity after ( 2 s ) is ( 4 m / s, ) then ( n ) is
equal to.
A ( cdot t^{3} )
в. ( frac{t^{3}}{2} )
c. ( frac{t^{3}}{3} )
D. ( frac{t^{3}}{4} )
12
158( operatorname{Let} boldsymbol{E}=left[frac{mathbf{1}}{mathbf{3}}+frac{mathbf{1}}{mathbf{5 0}}right]+left[frac{mathbf{1}}{mathbf{3}}+frac{mathbf{2}}{mathbf{5 0}}right]+ldots+ )
up to 50 terms, then which of the
following is true? Here ( square ) denotes the greatest integer function.
This question has multiple correct options
A. ( E ) is divisible by exactly 2 primes
B. ( E ) is prime
c. ( E leq 30 )
D. ( E geq 35 )
11
159The domain of the function ( boldsymbol{f}= )
( left((x, y): x, y in R, 3^{x}+3^{y}=3right) ) is
( A cdot(0,1] )
B. [0, 1]
( c cdot(-infty, 0] )
( D cdot(-infty, 1) )
12
160If ( f(x)=x^{2} quad ) and ( g(x)=2 x+1 ) be two
real valued function. Find ( boldsymbol{f}(boldsymbol{g}(boldsymbol{x})) ) and
( frac{f(x)}{g(x)} )
12
161Find the number of all onto functions
from the set ( A={1,2,3, ldots n} ) to itself
12
16231. Suppose p(x) = a, + a,x+ a2x2 +……. + ax”. If
P(x)) set-1 – 1 for all x 20, prove that
| a1 +2a2 + ….. + nan|si.
(2000 – 5 Marks)
12
163On every Friday, a veggie store owner offers a discount of ( 30 % ) on potatoes,
which he sell for ( $ 0.90 ) per pound on other days. The store also sells carrots for ( $ 3.50 ) per pound. If a customer buys
2 pound of carrots and ( p ) pounds of
potatoes then the total cost, ( c ), customer
has to pay is
A ( c=0.63 p+7 )
7
B. ( c=0.9 p+7 )
c. ( c=0.3 p+3.5 )
D. ( c=0.9 p+3.5 )
12
164A real valued function ( f(x) ) satisfies the functional equation ( boldsymbol{f}(boldsymbol{x}-boldsymbol{y})= )
( boldsymbol{f}(boldsymbol{x}) boldsymbol{f}(boldsymbol{y})-boldsymbol{f}(boldsymbol{a}-boldsymbol{x}) boldsymbol{f}(boldsymbol{a}+boldsymbol{y}), ) where ( boldsymbol{a} )
is a given constant and ( f(0)=1, ) then
A. ( f(2 a-x)=f(x), f(x) ) is symmetric about ( x=a )
B. ( f(2 a-x)=0=f(x), f(x) ) is symmetric about ( x=a )
c. ( f(2 a-x)+f(x)=0, f(x) ) is not symmetric about ( x= )
D. ( f(x)=0, f(x) ) is symmetric about ( x=a )
11
165Classify the following function as injection, surjection or bijection:
( boldsymbol{f}: boldsymbol{Q} rightarrow boldsymbol{Q}, ) defined by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+mathbf{1} )
12
166( boldsymbol{f}(boldsymbol{x})=boldsymbol{4}-(boldsymbol{6}-boldsymbol{x})^{2 / 3} ) in ( [mathbf{5}, boldsymbol{7}] )
A. Lagrange’s theorem is applicable
B. Rolle’s theorem is applicable
C. Lagrange’s theorem is applicable but not the Roll’s theorem
D. Both theorem are not applicable
11
167Let ( mathrm{R} ) be a relation from ( mathrm{N} ) to ( mathrm{N} ) defined
by ( R=left{(a, b): a, b in N text { and } a=b^{2}right} . ) Is
the following statement is true for ( (boldsymbol{a}, boldsymbol{a}) in boldsymbol{R} forall boldsymbol{a} in boldsymbol{N} )
A. True
B. False
12
168Find the domain and range of the
relation
( boldsymbol{R}={(-mathbf{1}, mathbf{1}),(mathbf{1}, mathbf{1}),(-mathbf{2}, mathbf{4}),(mathbf{2}, mathbf{4})} )
12
169If ( n(A)=m ) and ( n(B)=n ; ) then
( boldsymbol{n}(boldsymbol{A} times boldsymbol{B})=boldsymbol{m} boldsymbol{n} )
What is the value of ( n(A times B)=m n ). if
( boldsymbol{m}=boldsymbol{6} ) and ( boldsymbol{n}=boldsymbol{8} )
12
170Let ( boldsymbol{f}, boldsymbol{g} ) and ( boldsymbol{h} ) be functions from ( boldsymbol{R} ) to ( boldsymbol{R} )
Show that
(i) ( (boldsymbol{f}+boldsymbol{g}) boldsymbol{o} boldsymbol{h}=boldsymbol{f} boldsymbol{o} boldsymbol{h}+boldsymbol{g} boldsymbol{o} boldsymbol{h} )
(ii) ( (boldsymbol{f} cdot boldsymbol{g}) boldsymbol{o} boldsymbol{h}=(boldsymbol{f} boldsymbol{o} boldsymbol{h}) cdot(boldsymbol{g} boldsymbol{o} boldsymbol{h}) )
12
171The set of all real numbers satisfying ( e^{left(frac{1}{x}-1right)}<1 )
A. ( (0, infty) )
)
B . ( (-infty, 0) cup(1, infty) )
( c cdot(-infty, infty) )
D. (0,1)
12
172How many non-negative integral values of ( x ) satisfy the equation ( left[frac{x}{5}right]=left[frac{x}{7}right] ? )
(Here ( [x] ) denotes the greatest integer
less than or equal to ( x ). For example ( [3.4]=3 ) and ( [-2.3=-3) )
11
173Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be defined by ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{cc}x+2 & (x leq-1) \ x^{2} & (-1<x<1) text { then the value } \ 2-x & (x geq 1)end{array}right. )
of ( boldsymbol{f}(-1)+boldsymbol{f}(mathbf{0})+boldsymbol{f}(1) ) is
( A cdot 0 )
B.
( c cdot 2 )
D. –
12
174f ( n(A times B)=36 ) then ( n(A) ) can
possibly be
( A cdot 7 )
B. 8
( c .9 )
D. 10
12
17512. Letf(x) = x² and g(x) = sin x for all x € R. Then the set of
all x satisfying (fogogof) (x) = (gogof)(x), where (fog)(x)
= f(g(x)), is
a. I vnt, ne {0, 1, 2,…}
b. Vnt , ne {1, 2, …}
1 + 2nt, ne {…, -2, – 1,0, 1, 2…}
2
d. 2nt, ne {…, -2,-1,0, 1, 2, …} (IIT-JEE 2011)
12
176The complete range of values of ( ^{prime} a^{prime} ) such that ( left(frac{1}{2}right)^{|x|}=x^{2}-a ) is satisfied for maximum number of values of ( boldsymbol{x} ) is
B ( cdot(-infty, infty) )
c. (-1,1)
D. ( (-1, infty) )
11
177Determine the value of the constant ( k )
so that the function
( boldsymbol{f}(boldsymbol{x})=left{begin{aligned} frac{sin 2 x}{mathbf{5} boldsymbol{x}}, & text { if } boldsymbol{x} neq mathbf{0} \ boldsymbol{k}, text { if } boldsymbol{x} &=mathbf{0} end{aligned}right. )
continuous at ( boldsymbol{x}=mathbf{0} )
12
178If ( boldsymbol{f}(boldsymbol{x})+mathbf{2} boldsymbol{f}(mathbf{1}-boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{2} ; forall boldsymbol{x} in boldsymbol{R} )
then ( f(x) ) is given by
A ( cdot frac{(x-1)^{2}}{3} )
B. ( -frac{(x-2)^{2}}{3} )
c. ( x^{2}-1 )
D. ( x^{2}-2 )
11
179If ( f o g=|sin x| ) and ( g o f=sin ^{2} sqrt{x}, ) then
( f(x) ) and ( g(x) ) are
A ( cdot f(x)=sqrt{sin x}, g(x)=x^{2} )
B. ( f(x)=|x|, g(x)=sin x )
c. ( f(x)=sqrt{x}, g(x)=sin ^{2} x )
D. ( f(x)=sin sqrt{x}, g(x)=x^{2} )
12
180Find all solution of the equation. ( frac{(mathbf{2}|boldsymbol{x}|-mathbf{3})^{2}-|boldsymbol{x}|-mathbf{6}}{mathbf{4} boldsymbol{x}+mathbf{1}}=mathbf{0}, ) which
belong to the domain of definition of the
function ( boldsymbol{y}=(2 boldsymbol{x}+mathbf{1}) /left(boldsymbol{x}^{2}-boldsymbol{3} boldsymbol{6}right) )
12
181How many ordered pairs of ( (mathrm{m}, mathrm{n}) ) integers satisfy ( frac{m}{12}=frac{12}{n} ? )
A . 30
B. 15
c. 12
D. 10
12
182Let
( {x} &[x] ) denotes the fraction and integral part of a real number ( x ) respectively, then match the column
12
183A constant function is a periodic function.
A. True
B. False
11
184If ( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}]+left[boldsymbol{x}+frac{1}{3}right]+left[boldsymbol{x}+frac{2}{3}right], ) then
([.] denotes the greatest integer
function)
A. ( f(x) ) is discontinuous at ( x=1,10,15 )
в. ( lim _{x rightarrow frac{2}{3}} f(x)=2 )
( ^{mathbf{c}} cdot int_{0}^{2 / 3} f(x) d x=frac{1}{3} )
D. ( f(x) ) is continuous at ( x=frac{n}{3} ), where ( n ) is any intege
11
185( (f circ f)(x) )12
186Find the correct co-related number.
( mathbf{5}: mathbf{3 6}:: mathbf{6}: ? )
A . 48
B. 50
c. 49
D. 56
12
187If a function from ( R rightarrow R ) such that
( f(x)=x^{2} forall x epsilon R ) then show that ( f ) is
not one-one.
12
188Find the left hand and right hand limits of the greatest integer function ( f(x)= ) ( [x]= ) greatest integer less than or equal
to ( x, ) at ( x=k, ) where ( k ) is an integer.
Also, show that ( lim _{x rightarrow k} f(x) ) does not exist.
11
189( boldsymbol{f}(boldsymbol{theta})=sin theta(sin theta+sin 3 boldsymbol{theta}) operatorname{then} boldsymbol{f}(boldsymbol{theta}) ? )
( A cdot geq 0 ) only when ( theta geq 0 )
B. ( leq 0 ) for all real ( theta )
( mathbf{c} . geq 0 ) for all real ( theta )
D. ( leq 0 ) only when ( theta leq 0 )
11
190The equation ||( x-1|+a|=4 ) can have
real solutions for ( x ) if ( a ) belongs to the
interval
This question has multiple correct options
( A cdot(-infty, 4] )
В. ( (-infty,-4] )
( c cdot(4,+infty) )
D. [-4,4]
11
191The domain of ( boldsymbol{f}(boldsymbol{x})=sqrt{mathbf{2 5}-boldsymbol{x}^{2}} ) is
( mathbf{A} cdot(-infty,-5) )
B. ( (5, infty) )
c . [-5,5]
D. ( [-infty, infty] )
12
192If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R}, boldsymbol{g}: boldsymbol{R} rightarrow boldsymbol{R} ) are defined by
( boldsymbol{f}(boldsymbol{x})=mathbf{5} boldsymbol{x}-boldsymbol{3}, boldsymbol{g}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{3}, ) then
( left(g o f^{-} 1right)(3)= )
A ( cdot frac{25}{7} )
в. ( frac{111}{25} )
c. ( frac{9}{25} )
D. ( frac{25}{111} )
12
193Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a function defined by
( f(x)=frac{x^{2}-8}{x^{2}+2} . ) Then, ( f ) is
A. one-one but not onto
B. one-one and onto
c. onto but not one-one
D. neither one-one nor onto
12
194If ( f ) is a real function defined by ( f(x)= ) ( frac{x-1}{x+1}, ) then prove that ( f(2 x)=frac{3 f(x)+1}{f(x)+3} )11
195A relation ( R ) is defined from a ( operatorname{set} A= )
[2,3,4,5] to a ( operatorname{set} B=[3,6,7,10] ) as
follows:
( (x, y) in R Leftrightarrow x ) is relatively prime to ( y )
Express ( R ) as a set of ordered pairs and determine its domain and range.
12
196Let ( T ) be the set of all triangles in a plane with ( boldsymbol{R} ) a relation in ( boldsymbol{T} ) given by
( left.boldsymbol{R}=left{left(boldsymbol{T}_{1}, boldsymbol{T}_{2}right)right}: boldsymbol{T}_{1} text { congruent to } boldsymbol{T}_{2}right} )
Show that ( R ) is an equivalence relation.
12
1977.
Letf:(-1,1)→ IR be such that S(cos40) =
2 2 for
04(0.7) 69). Then the value (9) of 5(1) is (are)
@ 1-V 6) 1+ / 01- a) + V
11
12
198Find all real values of ( x ) such that
( boldsymbol{f}(boldsymbol{x})=boldsymbol{g}(boldsymbol{x}) ) where ( boldsymbol{f} ) and ( boldsymbol{g} ) are
functions given by
( boldsymbol{f}(boldsymbol{x})=mathbf{3} boldsymbol{x}+sqrt{boldsymbol{x}} ) and ( boldsymbol{g}(boldsymbol{x})=boldsymbol{2} boldsymbol{x}+boldsymbol{6} )
This question has multiple correct options
A .4
B. 9
( c .3 )
D. 6
12
199Assertion
Domain of ( f(x) ) is singleton.
Reason

Range of ( boldsymbol{f}(boldsymbol{x}) ) is singleton.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect

12
200If ( f(x)=x^{2}-3 x+4 ), then find the
value of ( 3left(x_{1}+x_{2}right) ) if ( x_{1}, x_{2} ) satisfying
( boldsymbol{f}(boldsymbol{x})=boldsymbol{f}(boldsymbol{2} boldsymbol{x}+mathbf{1}) )
12
201A binary operation ( ^{*} ) on the set {0,1,2,3,4,5} is defined as ( boldsymbol{a} * boldsymbol{b}=left{begin{array}{c}boldsymbol{a}+boldsymbol{b}, text { if } mathbf{a}+mathbf{b}<mathbf{6} \ mathbf{a}+mathbf{b}-mathbf{6} text { if } mathbf{a}+mathbf{b} geq mathbf{6}end{array}right} ) show
that zero is the identity element of this
operational each element 'a' of the set is invertible with 6 -a being the inverse of
( mathbf{a}^{prime} )
12
202If ( boldsymbol{A}={boldsymbol{a}, boldsymbol{b}, boldsymbol{c}}, boldsymbol{B}={boldsymbol{c}, boldsymbol{d}, boldsymbol{e}}, boldsymbol{C}= )
( {a, d, f}, ) then ( A times(B cup C) ) is
A ( cdot{(a, d),(a, e),(a, c)} )
в. ( {(a, d),(b, d),(c, d)} )
c. ( {(d, a),(d, b),(d, c)} )
D. none of these
12
203The range of ( boldsymbol{f}(boldsymbol{x})= )
( -3 cos sqrt{3+x+x^{2}} ) is
A. [-1,1]
в. [-2,2]
c. [-3,3]
D. [-4,4]
12
204Which of the function defined below are
one-one function(s)?
This question has multiple correct options
A. ( f(x)=x+1,(x geq-1) )
B. ( g(x)=x+frac{1}{x},(x geq 0) )
C ( cdot h(x)=x^{2}+4 x-5,(x>0) )
D. ( f(x)=e^{-x},(x geq 0) )
12
205For two sets ( A ) and ( B, A times B=B times A )
A. True
B. False
12
206Given ( boldsymbol{A}={mathbf{5}, mathbf{6}, mathbf{7}} ) and ( boldsymbol{B}={mathbf{3}, mathbf{4}} )
Form all possible ordered pairs and write the total number of ordered pairs formed. So that the first component is from ( mathrm{B} ) and second is from ( mathrm{A} )

Find the total number of such pairs.

12
207Show that each of the relation ( R ) in the
( operatorname{set} A={x in Z: 0 leq x leq 12}, ) given by
(i) ( boldsymbol{R}= )
( {(a, b):|a-b| text { is a multiple of } 4} )
(ii) ( boldsymbol{R}={(boldsymbol{a}, boldsymbol{b}): boldsymbol{a}=boldsymbol{b}} )
is an equivalence relation. Find the set of all elements related to 1 in each case.
12
208Assertion
( f ) is a function defined on the interval
[-1,1] such that ( f(sin 2 x)=sin x+ )
( cos x )
Statement I: If ( boldsymbol{x} inleft[-frac{boldsymbol{pi}}{boldsymbol{4}}, frac{boldsymbol{pi}}{boldsymbol{4}}right], ) then
( fleft(tan ^{2} xright)=sec x )
Reason
Statement II: ( boldsymbol{f}(boldsymbol{x})=sqrt{mathbf{1}+boldsymbol{x}}, forall boldsymbol{x} in )
[-1,1]
A. Statement I is true, Statement II is also true:
Statement II is the correct explanation of Statement
B. Statement I is true, Statement II is also true Statement II is not the correct explanation of Statement
c. statement lis true, statement II is false
D. statement lis false, statement II is true
12
209( operatorname{Let} g(x)=f(log x)+f(2-log x) ) and
( boldsymbol{f}^{prime prime}(boldsymbol{x})<mathbf{0} forall boldsymbol{x} in(mathbf{0}, boldsymbol{3}) . ) Then find the
interval in which ( g(x) ) increases.
( ^{mathrm{A}} cdotleft(0, frac{1}{e}right) )
B. ( left(frac{1}{e}, eright) )
( c cdot(0, e) )
D. None of these
11
210Find the domain of definition of the
following function. ( boldsymbol{y}=frac{sqrt{mathbf{3} boldsymbol{x}-mathbf{7}}}{sqrt[6]{boldsymbol{x}+mathbf{1}}} )
12
211The function ( f(x)= )
( left{begin{array}{l}frac{cos 3 x-cos 4 x}{x^{2}}, text { for } x neq 0 \ frac{7}{2}, text { for } x=0end{array} ) at right.
( boldsymbol{x}=mathbf{0} ) is
A. right continuous only
B. discontinuous
c. left continuous only
D. continuous
12
212If ( f ) and ( g ) are functions such that fog is onto then
A. ( f ) is onto
B. ( g ) is onto
c. ( g o f ) is onto
D. Neither ( f ) nor ( g ) is onto
12
213ff ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{B} ) is bijective function such
( operatorname{that} n(A)=10, operatorname{then} n(B) )
12
214Life expectancy is defined by the formula ( frac{2 S B}{G}, ) where ( S= ) shoe size ( boldsymbol{B}= ) average monthly electric bill in dollars, and ( G=mathrm{GMAT} ) score. If Melvin’s
GMAT score is twice his monthly electric bill, an his life expectancy is 50 what is his shoe size?
A . 40
B. 50
c. 60
D. 80
12
215Let ( A={1,2,3,4} ) and ( R={(2,2),(3,3),(4, )
4), ( (1,2)} ) be a relation on A. Then R is
A. Reflexive
B. Symmetric
c. Transitive
D. None of these
12
21634. Using the relation 2(1 – cos x) < x2, x = 0 or otherwise
(211
prove that sin (tan x) 2x, V x 6 0,
03 – 4 Marks)
12
217If ( n geq 1 ) is any integer, ( d(n) ) denotes
the number of positive factors of ( n, ) then
for any prime number ( mathbf{p}, mathbf{d}left(mathbf{d}left(mathbf{d}left(mathbf{p}^{7}right)right)right)= )
A .
B . 2
( c cdot 3 )
D. 4
12
218(a) 0< x <1
(c) –0 < x < 0
e domain of definition of the function f(x) given by the
equation 2x + 2y = 2 is
(2000S)
(b) 0 <x<1
(d) –00<x<1
12
219Let ( R ) be the equivalence relation in the
( operatorname{set} A=0,1,2,3,4,5 ) given by ( R= )
( (a, b): 2 ) divides ( (a-b) . ) Write the
equivalence class [0].
12
220If ( f(x)=x+3, ) then find the value of
the function ( boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(-boldsymbol{x}) )
12
221Let ( boldsymbol{f}= )
{(1,1),(2,3),(0,-1),(-1,-3)} be a
function described by the formula ( f(x)=a x+b ) for some integers ( a, b )
Determine ( a, b )
A ( . a=2, b=-1 )
В. ( a=1, b=0 )
c. ( a=0, b=-1 )
D. None of these
12
222Show that each of the relation ( R ) in the
( operatorname{set} A={x in Z: 0 leq x leq 12}, ) given by
is an equivalence relation. Find the set of all elements related to 1 in each case
( boldsymbol{R}={(boldsymbol{a}, boldsymbol{b}): boldsymbol{a}=boldsymbol{b}} )
12
223( operatorname{Let} n(A)=n . ) Then the number of all
relations on ( boldsymbol{A} ) is
( mathbf{A} cdot 2^{n} )
B ( cdot 2^{(n)} )
c. ( 2^{n^{2}} )
D. none
12
224Let ( boldsymbol{R}={(boldsymbol{x}, boldsymbol{y}): boldsymbol{x}+boldsymbol{2} boldsymbol{y}=boldsymbol{8}} ) be a
relation on N. Write the range of R.
12
225Verify ( boldsymbol{A}={mathbf{1}, mathbf{2}} )
( boldsymbol{B}={1,2,3,4} C={5,6} D= )
( {5,6,7,8} . ) Prove ( A times(B cap C)=(A times )
( boldsymbol{B}) cap(boldsymbol{A} times boldsymbol{C}) )
12
226If ( boldsymbol{A}={mathbf{2}, mathbf{3}, mathbf{5}}, boldsymbol{B}={mathbf{2}, mathbf{5}, boldsymbol{6}}, ) then
( (boldsymbol{A}-boldsymbol{B}) times(boldsymbol{A} cap boldsymbol{B}) ) is
A ( cdot{(3,2),(3,3),(3,5)} )
B – {(3,2),(3,5),(3,6)}
c. {(3,2),(3,5)}
D. none of these
12
227Let ( boldsymbol{A}={mathbf{1},-mathbf{1}} . ) Then find ( boldsymbol{A} times boldsymbol{A} )12
228The minimum value of ( boldsymbol{f}(boldsymbol{x})= )
( |boldsymbol{x}-mathbf{1}|+|boldsymbol{x}-mathbf{2}|+|boldsymbol{x}-mathbf{3}| ) is:
( mathbf{A} cdot mathbf{1} )
B. 2
( c cdot 3 )
D. None of the above
12
2295.
x), then
(1998 – 2 Marks)
Ifg(x)) = sinx and f(g(x))=(sin
(a) f(x)=sinºx, g(x)= x
b) f(x)=sin x, g(x)= |x|
(c) Sx)= x2, g(x)=sin vx
(d) fand g cannot be determined.
12
230If the function ( boldsymbol{f}(boldsymbol{x})=log left(frac{1+boldsymbol{x}}{1-boldsymbol{x}}right) )
then the value of ( fleft(frac{2 x}{1+x^{2}}right) ) is equal to
A ( . f(x) )
B. ( 2 f(x) )
c. ( 3 f(x) )
D. ( 4 f(x) )
11
231If ( boldsymbol{f}(mathbf{1})=mathbf{1}, boldsymbol{f}(boldsymbol{n}+mathbf{1})=mathbf{2} boldsymbol{f}(boldsymbol{n})+mathbf{1}, boldsymbol{n} geq )
1, then ( f(n) ) is:
( mathbf{A} cdot 2^{n+1} )
в. ( 2^{text {г }} )
c. ( 2^{n}-1 )
D. ( 2^{n-1}-1 )
11
232Let ( boldsymbol{f}:{boldsymbol{x}, boldsymbol{y}, boldsymbol{z}} rightarrow{1,2,3} ) be a one-one
mapping such that only one of the following three statements and remaining two are false : ( boldsymbol{f}(boldsymbol{x}) neq ) ( mathbf{2}, boldsymbol{f}(boldsymbol{y})=mathbf{2}, boldsymbol{f}(boldsymbol{z}) neq 1, ) then
A ( . f(x)>f(y)>f(z) )
в. ( f(x)<f(y)<f(z) )
c. ( f(y)<f(y)<f(z) )
D. ( f(y)<f(z)<f(x) )
12
233Determine the domain and range of the following relation.
( boldsymbol{R}={(boldsymbol{a}, boldsymbol{b}): boldsymbol{a} in boldsymbol{N}, boldsymbol{a}<mathbf{5}, boldsymbol{b}=mathbf{4}} )
12
234Find the number of reflexive relations
from set ( A ) to ( A ), defined as ( A=a, b, c )
12
235( f(x)=left{begin{array}{ll}1 & x>0 \ 0 & x=0 text { and } \ -1 & x<0end{array}right. )
( boldsymbol{g}: boldsymbol{R} rightarrow boldsymbol{R}, boldsymbol{g}(boldsymbol{x})=[boldsymbol{x}], ) then ( (boldsymbol{f} circ boldsymbol{g})(boldsymbol{pi}) )
is:
( A )
B.
( c )
D. –
12
236Show that the relative ( R ) in the ( operatorname{set} A= )
{1,2,3,4,5} given by ( R= )
( {(a, b):|a-b| text { is even }}, ) is an
equivalence relation. Show that all the
elements of {1,3,5} are related to each
other and all the elements of {2,4} are
related to each other. But no element of
{1,3,5} is related to any element of
{2,4}
12
237If ( f(x)=3 x+1, g(x)=x^{3}+2, ) then
( frac{boldsymbol{f}+boldsymbol{g}}{boldsymbol{f} boldsymbol{g}}(mathbf{0})= )
A . ( x )
B. 1
( c cdot 3 )
( D cdot frac{3}{2} )
11
238Find the intervals in which the following function is (a) increasing (b)
decreasing ( f(x)=x^{4}-8 x^{3}+22 x^{2}-24 x+21 )
12
239The largest interval lying in ( left(frac{-pi}{2}, frac{pi}{2}right) ) for which the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{4}^{-boldsymbol{x}^{2}}+ )
( cos ^{-1}left(frac{x}{2}-1right)+log (cos x) ) is defined
A ( cdotleft[-frac{pi}{4}, frac{pi}{2}right) )
в. ( left(0, frac{pi}{2}right) )
( mathbf{c} cdot[0, pi] )
D. ( left(-frac{pi}{2}, frac{pi}{2}right) )
12
240Determine the domain and range of the
relation R defined by ( boldsymbol{R}=left{left(boldsymbol{x}, boldsymbol{x}^{3}right): boldsymbol{x} ) is right.
a prime number less than ( 10} )
12
241The inequality ( |2 x-3|<1 ) is valid
when x lies in the interval
( mathbf{A} cdot(3,4) )
B ( cdot(1,2) )
( c cdot(-1,2) )
D ( cdot(-4,3) )
11
242If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) is defined by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{2} boldsymbol{x} )
( mathbf{3} forall boldsymbol{x} boldsymbol{epsilon} boldsymbol{R}, ) which of the following is / are
correct?
( mathbf{A} cdot f ) is a function
B. ( f ) is an injection
c. ( f ) is surjection
D. ( f ) is not onto
12
243If the range of ( 5 cos theta+3 cos left(theta+frac{pi}{3}right)+ )
( mathbf{3} ) is ( [boldsymbol{a}, boldsymbol{b}], forall boldsymbol{theta} in mathbf{R} ) then ( boldsymbol{a}+boldsymbol{b} ) is equal to
( mathbf{A} cdot mathbf{6} )
B. 2
c. 8
D. 4
12
244Which one of the following relations on
( R ) is equivalence relation
A. ( x R_{1} y Leftrightarrow|x|=mid y )
В. ( x R_{2} y Leftrightarrow x geq y )
c. ( x R_{3} y Leftrightarrow x mid y )
D. ( x R_{4} y Leftrightarrow x<y )
12
245Show that the function ( f ) in ( A=R=frac{2}{3} ) defined by ( f(x)=frac{3 x+2}{5 x+3}, neq frac{-3}{5} ) is
one-one and onto.Hence find ( boldsymbol{f}^{-1}(boldsymbol{x}) )
12
246If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) is defined by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}- )
( [boldsymbol{x}]-frac{1}{2} forall boldsymbol{x} in boldsymbol{R}[boldsymbol{x}] ) is the greatest
integer not exceeding ( x ) then ( {x in R: ) ( left.boldsymbol{f}(boldsymbol{x})=frac{1}{2}right}= )
A. the set of all integers
B. N, the set of all natural numbers
c. ( phi ), the empty set
( D )
11
247Solve: ( 2 x-frac{1}{3}=frac{1}{5}-x )12
248If ( f: R rightarrow R ) be a differentiable function and ( f(0)=0 ) and ( f^{prime}(0)=1 ) then
( lim _{x rightarrow 0} frac{1}{x}left[f(x)+fleft(frac{x}{2}right)+ldots+fleft(frac{x}{100}right)right] )
equals
A . 1
B cdot ( 1+frac{1}{2}+frac{1}{3}+ldots+frac{1}{100} )
D. Does not exist
12
249( operatorname{Let} boldsymbol{F}(boldsymbol{k})= )
( left(1+sin frac{pi}{2 k}right)left(1+sin (k-1) frac{pi}{2 k}right)(1+sin )
The value of ( boldsymbol{F}(mathbf{1})+boldsymbol{F}(mathbf{2})+boldsymbol{F}(mathbf{3}) ) is
equal to
A ( cdot frac{3}{16} )
B.
( c cdot frac{5}{16} )
D. ( frac{7}{16} )
12
250Let ( f(x)=frac{2025^{x}}{45+2025^{x}}, ) then the value of ( fleft(frac{1}{2025}right)+fleft(frac{2}{2025}right)+ )
( fleft(frac{3}{2025}right)+ldots+fleft(frac{2024}{2025}right) ) is equal to
A. 1011
в. 1012
( c .1013 )
D. 2024
11
25125.
Ifb> a, then the equation (x – a) (x – b)-1 = 0 has (2000)
(a) both roots in (a, b)
(b) both roots in (-00, a)
(c) both roots in (6, +00)
(d) one root in (-0, a) and the other in (b, too)
34
11
252Find the domain of the function ( boldsymbol{f}(boldsymbol{x})=frac{mathbf{1}}{sqrt{|boldsymbol{x}|-boldsymbol{x}}} )
A. ( [0, infty) )
(n)
в. ( (-infty, 0) )
( c cdot[1, infty) )
D. ( (-infty, 0 )
12
253The number of solutions of ( [sin x+ ) ( cos x]=3+[-sin x]+[-cos x] ) in the
interval ( [mathbf{0}, mathbf{2} boldsymbol{pi}] ) is (where [.] denotes the greatest integer function.)
( mathbf{A} cdot mathbf{0} )
B. 4
( c .3 )
D. Infinitely many
11
254For ( a, b in R-{0}, ) let ( f(x)=a x^{2}+ )
( b x+a ) satisfies ( fleft(x+frac{7}{4}right)= )
( fleft(frac{7}{4}-xright) forall x epsilon R )
Also the equation ( f(x)=7 x+a ) has
only one real distinct solution.
The value of ( (a+b) ) is equal to
A . 4
B. 5
c. 6
( D )
11
255Durks)
defined by
11.
A relation R on the set of complex numbers is define
is real. Show that Ris
+ R is an
2, Rz, if and only if 1 2
Z1 + Z2
equivalence relation.
(1982 – 2 Marks)
12
25610.
The function f(x) = px – q + r l xl, xe(-0,00) where
p>0,q>0,r>0 assumes its minimum value only on one
point if
(1995)
(a) P09
(b) ruq
(c) rzp
(d) p=q=r
1
12
257Consider the binary operation o defined
by the following tables on set ( boldsymbol{S}= )
[
{a, b, c, d}
]
( begin{array}{llll}circ & text { a } & text { b } & text { c } & text { d }end{array} ) ( begin{array}{lllll}mathrm{a} & mathrm{a} & mathrm{a} & mathrm{a} & mathrm{a} \ mathrm{b} & mathrm{a} & mathrm{b} & mathrm{c} & mathrm{d} \ mathrm{c} & mathrm{a} & mathrm{c} & mathrm{d} & mathrm{b} \ mathrm{d} & mathrm{a} & mathrm{d} & mathrm{b} & mathrm{c}end{array} )
Show that the binary operation is commutative. Write down the identitie
and list the inverse of elements.
12
258Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be such that ( boldsymbol{f}(boldsymbol{2} boldsymbol{x}- )
1) ( =f(x) ) for all ( x epsilon R ) fif ( f ) is continuous
( operatorname{at} x=1 ) and ( f(1)=1, ) then
This question has multiple correct options
A. ( f(2)=1 )
B. ( f(2)=2 )
c. ( f ) is continuous only at ( x=1 )
D. ( f ) is continuous at all points
11
259Give an example of a relation. which is
reflexive and symmetric and Transitive
12
260( x^{2}=x y ) is a relation which is:
A. Symmetric
B. Reflexive
c. Transitive
D. All of these
12
261A large mixing tank currently contains
200 gallons of water into which 10 pounds of sugar have been mixed. A tap
will open pouring 20 gallons per minute
of water into the tank at the same time
sugar is poured into the tank at a rate
of 2 pound per minute. Find the concentration (pounds per gallon) of sugar in the tank after 14 minutes.
Then
A. the concentration is greater than at the beginning?
B. the concentration lesser than at the beginning?
c. the concentration equal to the concentration at the beginning?
D. None of the above
11
262Find whether the following functions are
one-one, onto or not:
( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) defined by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{3} ) for
all ( boldsymbol{x} epsilon boldsymbol{R} )
12
263Write the relations as sets of ordered
pairs
( {(x, y): y=3 x, x in{1,2,3}, y in{3,6 )
12
264Determine the domain and range of the relation R defined by ( boldsymbol{R}={(boldsymbol{x}, boldsymbol{x}+mathbf{5}) )
( boldsymbol{x} in{mathbf{0}, mathbf{1}, mathbf{2}, mathbf{3}, mathbf{4}, mathbf{5}}} )
12
265An “Anti-symmetric” relation need not be reflexive relation: give an example.12
266In three element group ( {e, a, b} ) where ( e )
is the identity, ( a^{5} b^{4} ) is equal to
( mathbf{A} cdot boldsymbol{a} )
в.
( c cdot a b )
D. ( b )
12
267A function ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) is defined as
( f(x)=x^{3}+4 . ) Is it a bijection or not? In
case it is a bijection, then find ( boldsymbol{f}^{-1}(mathbf{3}) )
12
26813. Let f:R → R be any function. Define g:R+R by
g(x)=f(x) for all x. Then gis
(2000)
(a) onto if f is onto
(b) one-one if f is one-one
(C) continuous if f is continuous
(d) differentiable iffis differentiable.
12
269If ( A={a, b, c}, B={1,2}, ) then find
( boldsymbol{A} times boldsymbol{B}, boldsymbol{B} times boldsymbol{A} boldsymbol{a} boldsymbol{n} boldsymbol{d} boldsymbol{A} times boldsymbol{A} )
12
270Show that the relation ( R ) on the set ( Z ) of
integers, given by ( boldsymbol{R}= )
( {(a, b): 2 text { divides } a-b} ) is an
equivalence relation.
12
271ff ( f(x)=2 x-5, ) then what is the value
of ( boldsymbol{f}(mathbf{2})+boldsymbol{f}(mathbf{5}) ? )
11
272( mathrm{R} ) is a relation in ( mathrm{A} ) and ( (mathrm{a}, mathrm{b}) notin mathrm{r}, ) implies
(b, a) ( notin mathrm{R} ) then ( mathrm{R} ) is said to be relation
A. symmetric
B. contradictory
c. skew symmetric
D. none of these
12
273Let ( R ) be the relation on ( Z ) defined by
( boldsymbol{R}={(boldsymbol{a}, boldsymbol{b}): boldsymbol{a}, boldsymbol{b} in boldsymbol{Z}, boldsymbol{a}-boldsymbol{b} ) is an
integer ( } . ) Find the domain and range of
( mathrm{R} )
12
274If ( 0<x<1000 ) and ( left[frac{x}{2}right]+left[frac{x}{3}right]+ )
( left[frac{x}{5}right]=frac{31}{30} x ) where ( [x] ) is the greatest
integer less than or equal to ( x ), the number of possible values of ( x ) is
A . 34
B. 32
( c .33 )
D. 30
11
27522. If the roots of the equation x2 – 2ax + a2 + a-3=0 are real
and less than 3, then
(1999 – 2 Marks)
(a) a<2
(b) 2 Sas 3
(c) 34
11
276Find the domain and range of the following real functions:
( boldsymbol{f}(boldsymbol{x})=-|boldsymbol{x}| )
12
277Let ( boldsymbol{R} ) be a relation in ( boldsymbol{N} ) defined by ( boldsymbol{R}= )
( {(x, y): x+2 y=8, x, y epsilon N} . ) The
range of ( boldsymbol{R} ) is
A ( cdot{2,4,6} )
в. {1,2,3}
( mathbf{c} cdot{1,2,3,4,5,6} )
D. None of these
12
278Let ( boldsymbol{f}: boldsymbol{R}-{boldsymbol{n}} rightarrow boldsymbol{R} ) be a function
defined by ( f(x)=frac{x-m}{x-n} ) such that
( boldsymbol{m} neq boldsymbol{n}, ) then
A. ( f ) is one one into function
B. ( f ) is one one onto function
c. ( f ) is many one into function
D. ( f ) is many one onto function
12
279The function ( f(x) ) defined on the real numbers has the property that ( boldsymbol{f}(boldsymbol{f}(boldsymbol{x})) cdot(1+boldsymbol{f}(boldsymbol{x}))=-boldsymbol{f}(boldsymbol{x}) ) for all ( boldsymbol{x} ) in
the domain of ( f ). If the number 3 is in
the domain and range of ( f, ) compute the
value of ( boldsymbol{f}(mathbf{3}) )
A. ( -frac{3}{4} )
в. ( -frac{3}{2} )
( c cdot frac{2}{3} )
D. ( -frac{2}{3} )
12
280The set of points where the function
( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}]+|mathbf{1}-boldsymbol{x}|,-mathbf{1} leq boldsymbol{x} leq mathbf{3} )
where [.] denotes the greatest integer
function, is not differentiable, is
A ( cdot{-1,0,1,2,3} )
B ( cdot{-1,0,2} )
c. {0,1,2,3}
D ( cdot{-1,0,1,2} )
11
281Verify whether the function ( boldsymbol{f}: mathrm{AB} ) where ( boldsymbol{A}=boldsymbol{R}-{mathbf{3}} ) and ( boldsymbol{B}= )
( boldsymbol{R}-{1}, ) defined by ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}-boldsymbol{2}}{boldsymbol{x}-boldsymbol{3}} ) is
one-one and on-to or not. Give reason.
12
282( f ) is a function defined as ( sum_{k=1}^{n} f(a+ ) ( boldsymbol{k})=mathbf{1 6}left(mathbf{2}^{n}-mathbf{1}right) ) and ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})= )
( f(x) cdot f(y) ) and ( f(1)=2 ) then integral
value of ( boldsymbol{a} )
A . 3
B. 0
( c cdot 2 )
D.
11
283Which of the following are not
equivalence relations on ( I ? )
A. ( a R b ) if ( a+b ) is an even integer
B. ( a R b ) if ( a-b ) is an even integer
c. ( a R b ) if ( a<b )
D. ( a R b ) if ( a=b )
12
284( ” * ” ) is said to be commutative in ( A ) for
all ( a, b epsilon A )
A ( . a+b=b+a )
B. ( a * b=b * a )
( c cdot a-b=b-a )
( a )
D. ( a * b neq b * a )
12
285For real numbers ( x ) and ( y ) we write ( _{x} R_{y} ) iff ( boldsymbol{x}-boldsymbol{y}+sqrt{mathbf{2}} ) is an irrational numbers.
Then relation ( R ) is reflexive.
12
286( operatorname{Let} A={a, b, c, d} ) and ( B={x, y, z} )
What is the number of elements in ( boldsymbol{A} times )
( B ? )
( mathbf{A} cdot mathbf{6} )
B. 7
c. 12
D. 64
12
287A function
( f:[3,7) rightarrow R ) is defined as
follows
( f(x)=left{begin{array}{cc}4 x^{2}-1 ; & -3 leq x<2 \ 3 x-2 ; & 2 leq x leq 4 \ 2 x-3 ; & 4<x<7end{array}right. )
Find ( boldsymbol{f}(-mathbf{2})-boldsymbol{f}(mathbf{4}) )
12
288If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) and ( boldsymbol{g}: boldsymbol{R} rightarrow boldsymbol{R} ) defined by
( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}| ) and ( boldsymbol{g}(boldsymbol{x})=[boldsymbol{x}-boldsymbol{3}] ) for ( boldsymbol{x} in boldsymbol{R} )
[.] denotes greatest integer function, ( operatorname{then}left{boldsymbol{g}(boldsymbol{f}(boldsymbol{x})): frac{-boldsymbol{8}}{boldsymbol{5}}<boldsymbol{x}<frac{boldsymbol{8}}{boldsymbol{5}}right}= )
A ( cdot{0,1} )
в. {-1,-2}
( mathbf{c} cdot{-3,-2} )
D. {2,3}
11
289If ( f(x) ) is a polynomial function of the second degree such that ( boldsymbol{f}(-mathbf{3})= ) ( mathbf{6}, boldsymbol{f}(mathbf{0})=mathbf{6} ) and ( boldsymbol{f}(mathbf{2})=mathbf{1 1}, ) then the
graph of the function ( f(x) ) cuts the
ordinate ( x=1 ) at the point:
B. (1,4)
c. (1,-2)
D. None of these
12
290Let ( R ) be a relation from a set ( A ) to a set
( B, ) then:
( mathbf{A} cdot R=A cup B )
В . ( R=A cap B )
c. ( R subset A times B )
D. ( R subseteq B times A )
12
291Three years ago, a High School started charging an admission fee for volleyball
games to raise money for new sports. The initial price was ( $ 2 ) per person, now school raised this price of admission to ( $ 2.50 ) this year. Assuming this trend continues, which of the following equations can be used to described the ( operatorname{cost} ) of admission ( (c), y ) years after the school began charging for admission to games?
A. ( c=6 y+2 )
B ( cdot c=frac{y}{6}+2.5 )
( mathbf{c} cdot c=frac{y}{6}+2 )
D. ( c=frac{y}{2}+6 )
12
292If ( A ) is the set of even natural numbers
less than 8 and ( B ) is the set of prime
numbers less than ( 7, ) then the number
of relations from ( A ) to ( B ) is
A ( cdot 2^{9} )
B ( cdot 9^{2} )
( c cdot 3^{2} )
D. ( 2^{9}-1 )
12
293Solve the inequality with absolute value
( -3|-2 x+4|<=4 )
( ^{A} cdotleft(-infty, frac{7}{3}right) )
B ( cdotleft(frac{7}{3},+inftyright) )
( ^{c} cdotleft(frac{7}{3},+frac{7}{3}right) )
D ( cdotleft(0, frac{7}{3}right) )
E ( .(-infty,+infty) )
11
294( operatorname{does} f(x)=x ) is equivalence function?12
295If ( f(x)=frac{1-x}{1+x} ) then the value of
( (f o f)(x) )
A ( cdot frac{1-x}{1+x} )
B. ( x )
( c cdot frac{1}{x} )
D. none of these
11
296Let ( boldsymbol{A}={1,2,3,4,5} ) and ( B={1,4,5} )
Let ( R ) be a relation ‘is less than’ from ( A )
to ( B ). Find the domain, co domain and
range of ( boldsymbol{R} )
12
297Let ( boldsymbol{A}={boldsymbol{x} epsilon boldsymbol{Z}: mathbf{0} leq boldsymbol{x} leq mathbf{1 2}} . ) Show that
( mathrm{R}={(a, b): a, b epsilon A,|a-b| ) is divisible
by ( 4} ) is an equivalence relation. Find
the set of all elements related to ( 1 . ) Also
write the equivalence class [2]
12
298Number of one-one functions from A to
B where ( n(A)=4, n(B)=5 )
A .4
B. 5
( c cdot 120 )
D. 90
12
299If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{S} ) defined by ( boldsymbol{f}(boldsymbol{x})=sin boldsymbol{x}- )
( sqrt{3} cos x+1 ) is onto, then the interval of
( boldsymbol{S} ) is
A ( cdot[-1,3] )
в. [1,1]
c. [0,1]
D. [0,-1]
11
300Let ( boldsymbol{P}(boldsymbol{x}) ) be a polynomial, which when divided by ( (x-3) ) and ( (x-5) ) leaves
remainders 10 and ( 6, ) respectively. If the polynomial is divided by ( (x-3)(x-5) )
then the remainder is
A ( .-2 x+16 )
B . 16
( c cdot 2 x-16 )
D. 60
11
301(u)
*
(c) 4
20. If the function f:R- {1,-1) A defined by f(x)= 1- x2
surjective, then A is equal to: IJEEM 2019-9 April (M)]
(a) R-{-1}
(b) [0,00)
(c) R-(-1,0)
(d) R-(-1,0)
12
302If ( boldsymbol{R}=left{(boldsymbol{x}, boldsymbol{y}): boldsymbol{x}^{2}+boldsymbol{y}^{2} leq mathbf{4} ; boldsymbol{x}, boldsymbol{y} in mathbb{Z}right} )
a relation on ( mathbb{Z} ), write the domain of ( boldsymbol{R} )
12
303( f(x)=sin ^{-1}left[log _{2}left(frac{x^{2}}{2}right)right] ) where
denotes the greatest integer function.
12
304ff ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) is given by ( boldsymbol{f}(boldsymbol{x})=(boldsymbol{3}- )
( left.boldsymbol{x}^{3}right)^{1 / 3}, ) find ( boldsymbol{f} boldsymbol{o} boldsymbol{f}(boldsymbol{x}) )
12
305A mapping function ( boldsymbol{f}: boldsymbol{X} rightarrow boldsymbol{Y} ) is one-
one, if
A. ( fleft(x_{1}right) neq fleft(x_{2}right) ) for all ( x_{1}, x_{2} in X )
B. ( fleft(x_{1}right)=fleft(x_{2}right) Rightarrow x_{1}=x_{2} ) for all ( x_{1}, x_{2} in X )
C. ( x_{1}=x_{2} Rightarrow fleft(x_{1}right)=fleft(x_{2}right) ) for all ( x_{1}, x_{2} in X )
D. none of these
12
306If the relation ( boldsymbol{R}: boldsymbol{A} rightarrow boldsymbol{B} ) where ( boldsymbol{A}= )
( {1,2,3,4}, B={1,3,5} ) is defined by
( boldsymbol{R}={(boldsymbol{x}, boldsymbol{y}): boldsymbol{x}<boldsymbol{y}, boldsymbol{x} in boldsymbol{A}, boldsymbol{y} in boldsymbol{B}} ) then
find ( R ) and ( R^{-1} )
12
307what is Transitive relation?12
308Show that the function ( boldsymbol{f}: boldsymbol{R}-{boldsymbol{3}} rightarrow )
( R-{1} ) given by ( f(x)=frac{x-2}{x-3} ) is a
bijection.
12
309If ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{B} ) and ( boldsymbol{g}: boldsymbol{B} rightarrow boldsymbol{C} ) are one-one
functions, show that gof is a one-one function.
12
310X
34. The function f:R
defined as f(x)= –
1+x2 , is:
(JEE M 2017)
(a) neither injective nor surjective
(b) invertible
(c) injective but not surjective
(d) surjective but not injective
12
311Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) where ( boldsymbol{f}(boldsymbol{x})=boldsymbol{2}^{|boldsymbol{x}|}-boldsymbol{2}^{-boldsymbol{x}} )
then ( f(x) ) is
A. periodic function
B. one one function
c. many one function
D. None of these
12
312Range of values of the function ( y= ) ( log _{3}left[frac{3 sin x+4 cos x+10}{3 sin x+4 cos x}right] ) is equal to
( A cdot(-infty, 1] )
B . ( [1, infty) )
( c cdot[3, infty) )
D・ [0,1]
12
313Let ( boldsymbol{f}:(e, infty) rightarrow boldsymbol{R} ) be a function
defined by ( f(x)=log (log (log x)), ) the
base of the logarithm being ( e ). Then
A. ( f ) is one-one and onto
B. ( f ) is one-one but not onto
( c . f ) is onto but not one-one
D. the range of ( f ) is equal to its domain
12
314Number of solutions satisfying ( |boldsymbol{x}|+ ) ( |x-2| leq 2 ) for ( x in R ) is12
315If ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{B} ) is a bijective function then
( boldsymbol{f}^{-1} circ boldsymbol{f}= )
A ( cdot f circ f^{-1} )
B. ( f )
( c cdot f^{-1} )
D. ( I_{A} ) (Identity map of the set ( A ) )
12
316( operatorname{Let} boldsymbol{S}={(boldsymbol{a}, boldsymbol{b}, boldsymbol{c}) in boldsymbol{N} times boldsymbol{N} times boldsymbol{N}: boldsymbol{a}+ )
( boldsymbol{b}+boldsymbol{c}=mathbf{2 1}, boldsymbol{a} leq boldsymbol{b} leq boldsymbol{c}} ) and ( boldsymbol{T}= )
( {(a, b, c) in N times N times N: )
( a, b, c a r e i n A P}, ) where ( N ) is the set of
all natural numbers. Then, the number
of elements in the set ( S cap T ) is
( A cdot 6 )
B. 7
c. 13
D. 14
12
317If ( f(x+1)=3 x-9, ) then what will be
the value of ( boldsymbol{f}left(boldsymbol{x}^{2}-mathbf{1}right) ? )
A ( cdot 3 x^{2}-9 )
B. ( 3 x^{2}-15 )
c. ( x^{2}-10 )
D. ( 3 x^{2}-10 )
11
318Let ( f ) be a function such that ( f(3)=1 )
and ( f(3 x)=x+f(3 x-3) ) for all ( x )
Then find the value of ( boldsymbol{f}(boldsymbol{3 0 0}) )
A . 5500
B. 10100
( c .5050 )
D. 1,71,700
12
319Let the number of elements of the sets
( A ) and ( B ) be ( p ) and ( q ) respectively. Then
the number of relations from the set ( boldsymbol{A} )
to the set ( B ) is
( mathbf{A} cdot 2^{p+q} )
B. ( 2^{p q} )
c. ( p+q )
( mathbf{D} cdot p q )
12
320( operatorname{Let} A={1,2,3} ) and ( B={3,4} ) then
find ( A cap B ) and then find ( A times(A cap B) )
12
321If ( sin ^{4} theta+cos ^{4} theta=1-frac{1}{2} f(theta) ) then
( fleft(frac{pi}{4}right)= )
( A )
B.
( c cdot 1 / 2 )
D. ( 1 / 4 )
12
322If ( A={1,2,3}, B={3,4}, C={4,5,6} ). Find ( A times )
( (B cap C) )
12
32315. Let f(x)=(x + 1)2 -1, x 2-1
Statement-1: The set {x:f(x)=f-‘(x)= {0,-1}
Statement-2 :fis a bijection.
120091
(a) Statement-1 is true, Statement-2 is true.
Statement-2 is not a correct explanation for Statement-1.
(6) Statement-1 is true, Statement-2 is false.
(C) Statement-1 is false, Statement-2 is true.
(d) Statement-1 is true, Statement-2 is true.
Statement-2 is not a correct explanation for Statement-1.
1
12
324( N ) is the set of positive integers. The
relation ( R ) is defined on ( mathrm{N} times mathrm{N} ) as follows:
( (a, b) R(c, d) Longleftrightarrow a d=b c ) Prove that
This question has multiple correct options
A. ( R ) is an equivalence relation
B. ( R ) is symmetric relation
c. ( R ) is transitive relation
D. ( R ) is not an equivalence relation.
12
325Let ( * ) be the binary operation on ( N ) given by ( a * b=L . C . M ) of ( a ) and ( b . ) Find
(i) ( 5 * 7,20 * 16 ) (ii) Is ( * ) commutative?
iii) Is ( * ) associative?
12
326Determine the domain and range of the following relations. ( boldsymbol{S}={(boldsymbol{a}, boldsymbol{b}): boldsymbol{b}=|boldsymbol{a}-mathbf{1}|, boldsymbol{a} in boldsymbol{Z} ) and
( |a| leq 3} )
12
327Let ( boldsymbol{A}=left{boldsymbol{a}_{1}, boldsymbol{a}_{2}, boldsymbol{a}_{3}, boldsymbol{a}_{4} boldsymbol{a}_{5}, boldsymbol{a}_{6}right} ) and ( boldsymbol{B}= )
( left{b_{1}, b_{2}, b_{3}right} . ) The number of functions of
( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{B} ) such that it is onto and there
are exactly three elements in ( A ) such that ( boldsymbol{f}(boldsymbol{A})=boldsymbol{b}, ) is
A . 75
B. 90
c. 100
D. 120
12
328Binary operation ( ^{*} ) on ( mathrm{R}-{-1} ) defined by ( boldsymbol{a} * boldsymbol{b}=frac{boldsymbol{a}}{boldsymbol{b}+mathbf{1}} ) is
( A .^{*} ) is associative and cummutative
B. ( ^{*} ) is neither associative nor commutative
c. ( ^{*} ) is commutative but not associative
D. ( ^{*} ) is associative but not commutative
12
329If ( [x]=4, ) where [.] denotes greatest
integer function, then it lies in the interval
A. (2,3)
(年) (2,3,3)
в. [-1,2]
c. [4,5]
D. [4,5]
11
330If ( a+frac{1}{a}=6 ) and ( a neq 0 ; ) find ( a^{2}-frac{1}{a^{2}} )
( mathbf{A} cdot pm 24 sqrt{3} )
B. ( pm 4 sqrt{2} )
( c cdot pm 24 sqrt{2} )
D. ( pm 4 sqrt{3} )
11
331Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be defined as
( left{begin{array}{lll}2 x & text { if } & x>3 \ x^{2}, & text { if } & 1<x leq 3 text { . Then, find } \ 3 x, & text { if } & x leq 1end{array}right. )
( boldsymbol{f}(-1)+boldsymbol{f}(2)+boldsymbol{f}(4) )
( A cdot S )
B . 14
( c .5 )
D. none of these
12
332Solve:
( boldsymbol{f}(boldsymbol{x})=sqrt{mathbf{9}-boldsymbol{x}^{2}}=mathbf{0} )
12
333Find the domain of the function defined
( operatorname{as} f(x)=frac{x+1}{2 x+3} )
A ( cdot x_{epsilon} mathrm{R}-left{frac{3}{2}right} )
B. ( _{x in mathrm{R}-left{frac{-3}{2}right}} )
( c . x epsilon ) R
D. ( _{x in mathrm{R}-left{frac{2}{3}right}} )
12
334If ( boldsymbol{A}={mathbf{2}, mathbf{4}, mathbf{5}}, boldsymbol{B}={mathbf{7}, mathbf{8}, mathbf{9}} ) then
( boldsymbol{n}(boldsymbol{A} times boldsymbol{B}) ) is equal to
( A cdot 6 )
B. 9
( c .3 )
( D )
12
335Prove that every identity relation on a set is reflexive, but the converse is not
necessarily true.
12
336Let ( mathbf{f}: mathbf{R} rightarrowleft[mathbf{0}, frac{boldsymbol{pi}}{mathbf{2}}right) ) be defined by ( mathbf{f}(mathbf{x})= )
( tan ^{-1}left(x^{2}+x+aright) cdot ) Then the set of
values of ‘a’ for which ( f ) is onto is
B. [2,1]
c. ( left{frac{1}{4}right. )
D. ( left[frac{1}{4}, inftyright) )
12
337( boldsymbol{f}:(boldsymbol{0}, infty) rightarrow(boldsymbol{0}, infty) ) defined by
( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}2^{x} & boldsymbol{x} in(mathbf{0}, mathbf{1}) \ mathbf{5}^{boldsymbol{x}} & boldsymbol{x} in[mathbf{1}, infty)end{array}right. )
A. one-one but not onto
B. onto but not one-one
c. neither one – one nor onto
D. bijective
12
338Define a relation ( mathrm{R} ) on the set ( mathrm{N} ) of
natural numbers by ( boldsymbol{R}={(boldsymbol{x}, boldsymbol{y}): boldsymbol{y}= )
( x+5, x ) is a natural number less than
( mathbf{4}, boldsymbol{x}, boldsymbol{y} in boldsymbol{N}} )
Depict this relationship using
(i) roster
form (ii) an arrow diagram. Write down the domain and range or ( mathrm{R} )
12
339ff ( (x)=7 x^{2}-15, ) then find the value
of ( boldsymbol{f}(boldsymbol{3})-boldsymbol{f}(boldsymbol{5}) )
A . 102
B. 66
c. -112
D. -54
12
340If ( f(x)=left{begin{array}{ll}2 x+3 & x leq 1 \ a^{2} x+1 & x>1end{array}, ) then the right.
values of ( a ) for which ( f(x) ) is injective.
A . -3
B.
c. 0
D. none of these
12
341Let ( boldsymbol{A}=boldsymbol{R} times boldsymbol{R} ) and ( * ) be a binary
operation on ( A ) defined by ( (a, b) * ) ( (c, d)=(a+c, b+d) . ) Show that ( * ) is
commutative and associative. Find the
identity element for ( * ) on ( A, ) if any
12
342Let ( g(x)= ) ( int_{0}^{x} f(t) d t, quad f quad ) is such that ( quad frac{1}{2} leq )
( boldsymbol{f}(boldsymbol{t}) leq mathbf{1} )
for ( boldsymbol{t} boldsymbol{epsilon}[mathbf{0}, mathbf{1}] ) and ( mathbf{0} leq boldsymbol{f}(t) leq )
( frac{1}{2} ) for ( t epsilon[1,2] . ) Then
( boldsymbol{g}(boldsymbol{2}) ) satisfies the inequality
A ( cdot-frac{3}{2} leq g(2) leq frac{1}{2} )
в. ( 0 leq g(2) leq 2 )
c. ( frac{3}{2} leq g(2) leq frac{5}{2} )
D. ( 2 leq g(2) leq 4 )
11
343Find the domain and range of each of the following real value functions:
( boldsymbol{f}(boldsymbol{x})=-|boldsymbol{x}| )
11
344Find the value of ( f(x)=x^{3}-3 x+5 ) at
( boldsymbol{x}=mathbf{1 . 9 9} )
12
345The relation P defined from R to R as a P
( mathbf{b} Leftrightarrow 1+mathbf{a b}>mathbf{0}, ) for all ( mathbf{a}, mathbf{b} boldsymbol{epsilon} mathbf{R} ) is
A . reflexive only
B. reflexive and symmetric only
c. transitive only
D. equivalence
12
346Let ( *: boldsymbol{R} times boldsymbol{R} rightarrow boldsymbol{R} ) given by ( (boldsymbol{a}, boldsymbol{b}) rightarrow )
( a+4 b^{2} ) is a binary operation, compute
(-5)( *(2 * 0) )
12
347Show that the function ( f(x)=frac{x-2}{x+1} )
for ( boldsymbol{x} neq mathbf{0} ) is increasing.
12
348What is the first component of an ordered pair (1,-1)( ? )
( mathbf{A} cdot mathbf{1} )
B. –
( c cdot 2 )
( D )
12
349The point through which the curve ( y= )
( x^{2}-1 ) passes through is
A. (4,15)
В. (-4,-15)
c. (-4,15)
D. (4,-15)
12
350Given ( M=(0,1,2) ) and ( N=(1,2,3) )
Find ( (boldsymbol{N}-boldsymbol{M}) times(boldsymbol{N} cap boldsymbol{M}) )
( A cdot{(3,1),(3,2)) )
B. {(3,-1),(3,-2)
c. {(-3,-1),(-3,2)}
( mathbf{D} cdot{(3,-1),(-3,-2)} )
12
351Let ( f: N rightarrow N ) be defined by ( f(n)= ) ( left{begin{array}{l}frac{n+1}{2}, text { if } mathrm{n} text { is odd } \ frac{n}{2}, text { if } mathrm{n} text { is even }end{array}right} ) for all ( n in N )
State whether the function ( f ) is bijective.
Justify your answer
12
352If ( boldsymbol{A}={1,2}, boldsymbol{B}={mathbf{2}, boldsymbol{3}} ) and ( boldsymbol{C}= )
( {3,4}, ) then what is the cardinality of
( (boldsymbol{A} times boldsymbol{B}) cap(boldsymbol{A} times boldsymbol{C}) )
( mathbf{A} cdot mathbf{8} )
B. 6
( c cdot 2 )
D.
12
353The cartesian product ( boldsymbol{A} times boldsymbol{A} ) has ( mathbf{9} )
elements among which are found (-1,0) and ( (0,1) . ) Find the set ( A ) and
the remaining elements of ( boldsymbol{A} times boldsymbol{A} )
12
354f ( x, y in{1,2,3,4} ) then check whether
( boldsymbol{f}={(boldsymbol{x}, boldsymbol{y}): boldsymbol{x}+boldsymbol{y}=mathbf{5}} ) is a function?
12
355If two sets ( A ) and ( B ) are having 39
elements in common, then the number
of elements common to each of the sets
( A times B ) and ( B times A ) are
( mathbf{A} cdot 2^{3} )
в. ( 39^{2} )
c. 78
D. 35
12
356Express ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{N}, boldsymbol{f}(boldsymbol{x})=mathbf{2} boldsymbol{x} ) where
( boldsymbol{A}={boldsymbol{x}: boldsymbol{x} in boldsymbol{N}, boldsymbol{x} leq 10} . ) Determine
range of the function.
12
357If ( f(x)=frac{4 x+x^{4}}{1+4 x^{3}} ) and ( g(x)= )
( ln left(frac{1+x}{1-x}right), ) then what is the value of ( f cdot gleft(frac{e-1}{e+1}right) ) equal to?
A .2
B. 1
c. 0
D.
11
358( f(x)=-(x-1)^{2}+10 . ) Find ( f(1) )12
359( operatorname{Let} A={1,2,3} ) and ( R= )
{(1,2),(1,1),(2,3)} be a relation on ( A )
What minimum number of ordered
pairs may be added to ( R ), so that it may
become a transitive relation on ( A ) ?
12
360Which of the following is not an equivalence relation on ( Z ) ?
A ( . a R b Leftrightarrow a+b ) is an even integer
B. ( a R b Leftrightarrow a-b ) is an even integer
( c cdot a R b Leftrightarrow a<b )
D. ( a R b Leftrightarrow a=b )
12
361If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) satisfies ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})= )
( boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y}) ) for all ( boldsymbol{x}, boldsymbol{y} in boldsymbol{R} ) and
( f(1)=7, ) then ( sum_{r=1}^{n} f(r) )
( A cdot frac{7 n}{2} )
B. ( frac{7(n+1)}{2} )
( mathbf{c} cdot 7 n(n+1) )
D. ( frac{7 n(n+1)}{2} )
11
362Let ( f(x)=a x^{5}+b x^{3}+c sin x+d tan ^{3} x- )
1
f ( f(-alpha)=-5, ) then ( f(alpha) ) equals
12
363( boldsymbol{f}={(boldsymbol{3}, boldsymbol{4}),(boldsymbol{4}, boldsymbol{3})} boldsymbol{g}={(boldsymbol{3}, boldsymbol{7}),(boldsymbol{4}, boldsymbol{8})} )
Find ( boldsymbol{f}-boldsymbol{g} )
11
364Let ( A={x, y, z} ) and ( B={p, q, r, s} )
What is the number of distinct relations
from ( boldsymbol{B} ) to ( boldsymbol{A} ) ?
A . 4096
в. 4094
c. 128
D. 126
12
365Let ( boldsymbol{f}: boldsymbol{N} rightarrow boldsymbol{N} ) be defined by ( boldsymbol{f}(boldsymbol{x})= )
( boldsymbol{x}^{2}+boldsymbol{x}+mathbf{1}, boldsymbol{x} in boldsymbol{N} . ) Then ( boldsymbol{f} ) is
A. one-one onto
B. many-one onto
c. one-one but not onto
D. none of these
12
366Let ( boldsymbol{A}={boldsymbol{x}, boldsymbol{y}, boldsymbol{z}}, boldsymbol{B}={boldsymbol{u}, boldsymbol{v}, boldsymbol{w}} ) and
( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{B} ) be defined by ( boldsymbol{f}(boldsymbol{x})= )
( boldsymbol{u}, boldsymbol{f}(boldsymbol{y})=boldsymbol{v}, boldsymbol{f}(boldsymbol{z})=boldsymbol{w} . ) Then ( boldsymbol{f} ) is
A. surjective but not injective
B. injective but not surjective
c. bijective
D. none of these
12
367Find whether ( f(x)=frac{a^{x}+1}{a^{x}-1} ) is even
odd or neither odd nor even.
11
368Let the function ( g(x)=sqrt[3]{x}-2+c )
and ( g(27)=-1, ) then find the value of
( c )
A . -10
B. –
c. 0
( D )
12
369The domain for, ( boldsymbol{f}(boldsymbol{x})=cos ^{-1}[boldsymbol{x}] ) is
A. ( x in[-1,2) )
B . ( x in[-1,1] )
c. ( x in(-1,2) )
D. ( x in(-1,1) )
12
370Which one of the following sets is true set of real values of ( x ) for which the
function, ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} ell boldsymbol{n} boldsymbol{x}-boldsymbol{x}+1 ) positive
( ? )
в. (0,1)
c. ( (e, infty) )
D. (0,1) and ( (1, infty) )
11
371The curve ( a y^{2}=x^{2}(3 a-x) ) cuts the ( y )
axis at coordinates ( (0, k) . ) Then ( k= )
12
372Let ( A={4,6,8,10} ) and ( B= )
{3,4,5,6,7} and ( f: A rightarrow B ) be defined
by ( f(x)=frac{1}{2} x+1, ) then represent ( f ) by:
(i) An arrow diagram
(ii) A set of
ordered pairs (iii) ( A ) table
12
373The function ( boldsymbol{f}: boldsymbol{N} rightarrow boldsymbol{N}, ) where ( boldsymbol{N} ) is
the set of natural numbers, defined by ( f(x)=7 x+11 ) is
A. injective
B. surjective
c. bijective
D. none of these
12
374f ( P=(1,2,3) ) and ( Q=(4) ) find sets
( boldsymbol{P} times boldsymbol{Q} boldsymbol{a} boldsymbol{n} boldsymbol{d} boldsymbol{Q} times boldsymbol{P} )
12
375Illustration 2.16 Suppose that the function F is defined for
all real numbers r by the formula f(r) = 2(n-1) + 3. Evaluate
Fat the input values 0, 2, x + 2, and f(2).
1
.
1
.
11
376If ( k(x)=4 x^{3} a ) and ( k(3)=27, ) what is
( k(2) ? )
( A cdot 9 )
B. 8
c. 17
D. 12
12
377If ( |2-|[x]-||1 leq 2, ) then ( x ) belongs to
the solution set (where [.] represents greatest integer function)
A. [-3,6)
()
в. [-4,6)
c. [-3,7)
D. [-3,6]
11
378If ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{B} ) is a constant function
which is onto then ( B ) is
A. a singleton set
B. a null set
c. an infinite set
D. a finite set
12
379For real values of ( x, ) the range of ( frac{x^{2}+2 x+1}{X^{2}+2 x-1} )
is
( mathbf{A} cdot(-infty, 0) cup(1, infty) )
B ( cdotleft[frac{1}{2}, 2right] )
( mathbf{c} cdotleft[-infty, frac{-2}{9}right] cup(1, infty) )
D. ( (-infty,-6) cup(-2, infty) )
12
380If ( boldsymbol{f}(boldsymbol{x})=left[mathbf{9}^{x}-mathbf{3}^{x}+mathbf{1}right], boldsymbol{x} in(-infty, mathbf{1}) )
then number of integers in the range of
( f(x) ) is (where [.] denotes greatest integer function)
11
381Let ( boldsymbol{A}={1,2,3} . ) Which of the following
is not an equivalence relation on A?
A ( cdot{(1,1),(2,2),(3,3)} )
B – {(1,1),(2,2),(3,3),(1,2),(2,1)}
C ( cdot{(1,1),(2,2),(3,3),(2,3),(3,2)} )
D cdot {(1,1),(2,1)}
12
382Let ( boldsymbol{A}= )
( {x in W, ) the set of whole numb
( boldsymbol{B}= )
( {boldsymbol{x} in boldsymbol{N}, ) the set of natural num
and ( C={3,4}, ) then how many
elements will ( (A cup B) times C ) conatin?
( mathbf{A} cdot mathbf{6} )
B. 8
c. 10
D. 12
12
383The domain of ( boldsymbol{f}(boldsymbol{x})=frac{1}{sqrt{(boldsymbol{x}-mathbf{1})(boldsymbol{x}-mathbf{2})(boldsymbol{x}-mathbf{3})}} )
is
( mathbf{A} cdot(-infty, 1) cup(3, infty) )
B. (1,2)( cup(3, infty) )
( mathbf{c} cdot(-infty, 2) )
D. R
12
384The domain of the function ( mathbf{f}(boldsymbol{x})=frac{mathbf{1}}{boldsymbol{x}^{log boldsymbol{x}}} )
is
A.
B. ( (0, infty) )
c. ( R-{1} )
D. R – {9
12
385Consider the function ( boldsymbol{f}(boldsymbol{x}) ) which satisfying the functional equation:
( 2 f(x)+f(1-x)=x^{2}+1, forall x in R ) and
( boldsymbol{g}(boldsymbol{x})=boldsymbol{3} boldsymbol{f}(boldsymbol{x})+1 )
The range of ( phi(x)=g(x)+frac{1}{g(x)+1} ) is
( A cdot[2, infty) )
B. ( [1, infty) )
( c cdot[-1, infty) )
D. ( R^{+} )
12
386Find ( g o f(x), ) if ( f(x)=8 x^{3} ) and ( g(x)= )
( boldsymbol{x}^{mathbf{1} / mathbf{3}} )
12
387If ( f(x)=frac{1}{sqrt{x-5}}, ) then find the domain
and range of ( boldsymbol{f}(boldsymbol{x}) )
11
388Let ( R ) be relation on I defined as mRn iff
( boldsymbol{m} leq boldsymbol{n} . ) Check if ( boldsymbol{R} ) is an equivalence
relation.
12
389Let ( boldsymbol{R}={(1, mathbf{3}),(2,2),(3,2)} ) and ( boldsymbol{S}= )
{(2,1),(3,2),(2,3)} be two relations on
( operatorname{set} A={1,2,3} . ) Then ( R_{0} S ) is equal
A ( cdot{(2,3),(3,2),(2,2)} )
B . {(1,3),(2,2),(3,2),(2,1),(2,3)}
c. {(3,2),(1,3)}
D. {(2,3),(3,2)}
12
390Which term of the A.P. ( 3,10,17, ldots ldots . ) wil
be 84 more than its 13 th term?
11
391State True or False
Let ( boldsymbol{A}={1,2} ) and ( boldsymbol{B}={2,3,4}, ) then ( boldsymbol{A} )
( times mathrm{B}=mathrm{B} times mathrm{A} ? )
A. True
B. False
12
392The domain of ( boldsymbol{f}(boldsymbol{x})=sqrt{log left(frac{mathbf{5} boldsymbol{x}-boldsymbol{x}^{2}}{mathbf{6}}right)} )
is
A. ( 2 leq x leq 3 )
в. ( 3 leq x leq 4 )
c. ( -3 leq x leq-2 )
D. ( 1 leq x leq 3 )
12
393If ( f(x)=frac{2 x+3}{3 x-2} )
Prove that fof is an identity function
11
39412.
A real valued function f(x) satisfies the functional equation
f(x – y)=f(x) f(y)-f(a-x)f(a+y)
where a is a given constant and f(0) = 1,(2a-x) is equal
[2005]
(a) f(x)
(b) f(x)
(c) f(a)+f(a-x)
(d) f(-x)
12
395Let ( boldsymbol{f}: boldsymbol{N} rightarrow boldsymbol{R} ) be such that ( boldsymbol{f}(mathbf{1})=mathbf{1} )
and ( boldsymbol{f}(mathbf{1})+mathbf{2} boldsymbol{f}(mathbf{2})+mathbf{3} boldsymbol{f}(boldsymbol{3})+ldots+ )
( boldsymbol{n} boldsymbol{f}(boldsymbol{n})=boldsymbol{n}(boldsymbol{n}+1) boldsymbol{f}(boldsymbol{n}), ) for all ( boldsymbol{n} in )
( N, n geq 2, ) where ( N ) is the set of natural
numbers and ( R ) is the set of real
numbers. Then, the value of ( f(500) ) is
A. 1000
B. 500
c. ( frac{1}{500} )
D. ( frac{1}{1000} )
12
396The number of relations from ( boldsymbol{A}= )
{2,6} to ( B={1,3,5,6,7} ) that are not
functions from ( A ) to ( B ) is
12
397Let ( M ) be the set of all ( 2 times 2 ) matrices
with entries from the set ( boldsymbol{R} ) of real
numbers. Then the function ( boldsymbol{f}: boldsymbol{M} rightarrow boldsymbol{R} )
defined by ( boldsymbol{f}(boldsymbol{A})=|boldsymbol{A}| ) for every ( boldsymbol{A} in boldsymbol{M} )
is
A. one-one and onto
B. neither one-one nor onto
c. one-one but not onto
D. onto but not one-one
12
398Consider the equation ( 2+mid x^{2}+4 x+ )
( mathbf{2} mid=boldsymbol{m}, boldsymbol{m} epsilon boldsymbol{R} ) so that the given
equation has three solutions is:
( A cdot{4} )
B ( cdot{2} )
( c cdot{1} )
( D cdot{0} )
11
399The function ( f(x)=x^{2}+b x+c, ) where
( b ) and ( c ) real constants, describes
A. One-to-one mapping
B. Onto mapping
c. Not one-to-one but onto mapping
D. Neither one-to-one nor onto mapping
12
400If ( x epsilon N, ) then range of function ( f(x)= ) ( frac{16-x}{20-3 x} C_{x-1} ) is
A. [1,15]
В ( cdot_{14} C_{4}, frac{^{10} C_{5}}{2} )
c. ( ^{11} C_{4} )
( mathbf{D} cdot[5,15 )
12
401Let ( A ) and ( B ) be two finite sets having ( m )
and ( n ) elements respectively. Then the
total number of mapping from ( boldsymbol{A} ) to ( boldsymbol{B} ) is:
A . ( m n )
B. ( 2^{text {mn }} )
( mathrm{c} cdot m^{n} )
D. ( n^{text {m }} )
12
402The function ( * ) on ( boldsymbol{N} ) as
( boldsymbol{a} * boldsymbol{b}=(boldsymbol{a}-boldsymbol{b})^{2} ) is a binary operator
A. True
B. False
12
403The domain of ( frac{1}{log _{10}(1-x)}+ )
( sqrt{(x+2)} ) is
A ( cdot(-2 leq x<-1) cup(0<x<1) )
B . ( (-2 leq x<2) )
c. ( (-2 leq x<1) )
D cdot ( (-2 leq x<0) cup(0<x<1) )
12
404Determine the domain and range of the following relation ( boldsymbol{R}={(boldsymbol{a}, boldsymbol{b}): boldsymbol{a} in boldsymbol{N}, boldsymbol{a}<mathbf{5}, boldsymbol{b}=boldsymbol{4}} )12
40542
If f(x)= 3x – 5, then f ‘(x)
(1998 – 2 Marks)
(a) is given by –
3x – 5
x+5
(b) is given by
(c) does not exist because fis not one-one
(d) does not exist because f is not onto.
12
406Determine the function which has a
domain of ( boldsymbol{x} leq mathbf{3} )
( mathbf{A} cdot )
[
f(x)=(2-x)^{frac{1}{2}}+frac{x}{2}
]
B. ( quad f(x)=(3-x)^{frac{1}{2}}+frac{x}{2} )
C
[
f(x)=(x-3)^{frac{1}{3}}+frac{x}{4}
]
D. ( quad f(x)=(2-x)^{frac{1}{4}}+frac{x}{3} )
E
12
407( f f={(4,5),(5,6),(6,-4) text { and } g={(4,-4) )
(6,5),(8,5)} then find
( boldsymbol{A}) boldsymbol{f}+boldsymbol{g} quad boldsymbol{B}) boldsymbol{f}-boldsymbol{g} )
( boldsymbol{C}) mathbf{2} boldsymbol{f}+ )
( begin{array}{ll}mathbf{4} boldsymbol{g} & boldsymbol{D}) boldsymbol{f}+end{array} )
( begin{array}{lll}boldsymbol{4} & boldsymbol{E}) boldsymbol{f} boldsymbol{g} & boldsymbol{F}) boldsymbol{f} / boldsymbol{g}end{array} )
( boldsymbol{G})|boldsymbol{f}| quad boldsymbol{H}) sqrt{boldsymbol{f}} quad boldsymbol{I}) boldsymbol{f}^{2} )
12
408If the function ( f(x)=frac{t+3 x-x^{2}}{x-4} )
where ( t ) is a parameter, has a minimum
and a maximum, then the greatest
value of ( t ) is
12
409( operatorname{Let} boldsymbol{y}=sqrt{frac{(boldsymbol{x}+mathbf{1})(boldsymbol{x}-mathbf{3})}{boldsymbol{x}-mathbf{2}}} )
Find all the real values of ( x ) for which ( y )
takes real values.
12
410If the range of ( boldsymbol{f}(boldsymbol{x})=mathbf{2}+sqrt[3]{boldsymbol{x}},-boldsymbol{3} leq )
( boldsymbol{x}<-1 ) is ( [mathbf{0}, sqrt[3]{boldsymbol{n}}] ) where ( boldsymbol{n} in boldsymbol{N} ) then ( n= )
12
411Let ( boldsymbol{R}={(boldsymbol{a}, boldsymbol{b}): boldsymbol{a}, boldsymbol{b} in boldsymbol{N} text { and } boldsymbol{a}+boldsymbol{3} boldsymbol{b}= )
12 ( } . ) Find the domain and range of R.
12
412The cartesian product ( boldsymbol{A} times boldsymbol{A} ) has ( mathbf{9} )
elements among which are found (-1,0) and ( (0,1) . ) Find the set ( A ) and
the remaining elements of ( boldsymbol{A} times boldsymbol{A} )
12
413The domain of the function ( f(x)= ) ( left{begin{array}{l}left(x^{2}-9right) /(x-3), text { if } x neq 3 \ 6, quad text { if } x=3end{array}right. )
A ( .(0,3) )
в. ( (-infty, 3) )
( mathbf{C} cdot(-infty, infty) )
D cdot ( (3, infty) )
E ( .(-3,3) )
12
414If ( a-b=0.9 ) and ( a b=0.36 ; ) find ( a+b )
( A ldots pm 2 )
в. ±1.5
( c .pm 3.5 )
D. ±14
11
415Solve the following inequalities. ( log _{1 / 2}left(x^{2}+2 x+4right)>-2 )11
416If ( N ) denote the set of all natural
numbers and ( R ) be the relation on ( N times )
( N ) defined by ( (a, b) R(c, d) . ) if ( a d(b+ )
( c)=b c(a+d), ) then ( R ) is
A. Symmetric only
B. Reflexive only
c. Transitive only
D. An equivalence relation
12
417The number of ordered pairs ( (x, y) ) of
real numbers that satisfy the simultaneous equations.
( x+y^{2}=x^{2}+y=12 ) is.
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. 4
12
418If the function ( f ) is defined by ( f(x)= ) ( 3 x+4, ) then ( 2 f(x)+4= )
( mathbf{A} cdot 5 x+4 )
B. ( 5 x+8 )
c. ( 6 x+4 )
D. ( 6 x+8 )
E ( .6 x+12 )
11
419If functions ( boldsymbol{f}(boldsymbol{x}) ) and ( boldsymbol{g}(boldsymbol{x}) ) are defined
on ( R rightarrow R ) such that
[
begin{aligned}
boldsymbol{f}(boldsymbol{x}) &=boldsymbol{x}+boldsymbol{3}, boldsymbol{x} in text { rational } \
&=mathbf{4} boldsymbol{x}, boldsymbol{x} in text { irrational } \
boldsymbol{g}(boldsymbol{x}) &=boldsymbol{x}+sqrt{mathbf{5}}, mathbf{x} in text { irrational } \
&=-boldsymbol{x}, boldsymbol{x} in text { rational }
end{aligned}
]
then ( (boldsymbol{f}-boldsymbol{g})(boldsymbol{x}) ) is
A. one-one & onto
B. neither one-one nor onto
c. one-one but not onto
D. onto but not one-one
12
420Let ( f(x)=x+frac{1}{2} . ) Then, the number of
real values of ( x ) for which the three
unequal terms ( boldsymbol{f}(boldsymbol{x}), boldsymbol{f}(mathbf{2} boldsymbol{x}), boldsymbol{f}(boldsymbol{4} boldsymbol{x}) ) are
in HP is
A . 1
B.
( c .3 )
D.
11
421If ( A={1,2} ) and ( B={3,4} ) then find
( A times B )
( mathbf{A} cdot A times B={(1,3),(1,2),(2,3),(2,4)} )
B ( . A times B={(1,3),(1,4),(2,3),(2,4)} )
( mathbf{c} cdot A times B={(1,3),(1,4),(2,1),(2,4)} )
D . ( A times B={(1,3),(1,4),(2,3),(2,1)} )
12
422Let ( boldsymbol{A}={1,2}, boldsymbol{B}={1,2,3,4}, C= )
{5,6} and ( D={5,6,7,8} ) Verify that
(i) ( boldsymbol{A} times(boldsymbol{B} cap boldsymbol{C})=(boldsymbol{A} times boldsymbol{B}) cap(boldsymbol{A} times boldsymbol{C}) )
(ii) ( A times C ) is a subset of ( B times D )
12
423The function ( boldsymbol{f}:[mathbf{2}, infty) rightarrow boldsymbol{Y} ) defined by
( f(x)=x^{2}-4 x+5 ) is one-one if (more
than one option can be correct)
A. ( Y=R )
B. ( Y=[1, infty) )
c. ( Y=[0, infty) )
D. ( Y=[-1, infty) )
11
424Find ( boldsymbol{m}, ) if ( boldsymbol{v}=mathbf{3}, boldsymbol{g}=mathbf{1 0}, boldsymbol{h}=mathbf{5} ) and ( boldsymbol{E}= )
( mathbf{1 0 9} )
A .
B . 2
( c .3 )
D.
11
425Given ordered pairs :(5,4),(5,5),(5,6) ( (6,4),(6,5),(6,6),(6,7),(8,4),(8,5),(8, )
6), (8,8)
Use these ordered pairs to find the following relation:
( R_{3}= ) “is one less than”
12
426The function ( boldsymbol{f}:[mathbf{0}, infty) rightarrow boldsymbol{R} ) given by
( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}}{boldsymbol{x}+mathbf{1}} ) is
A. one-one and onto
B. one-one but not onto
c. onto but not one-one
D. neither one-one nor onto
12
427( R ) is a relation defined in ( R times T ) by ( (a, b) R(c, d) ) iff ( a-c ) is an integer and
( b=d . ) The relation ( R ) is
A. an identity relation
B. an universal relation
c. an equivalence relation
D. None of these
12
428ff ( f, g: R rightarrow R ) be two functions defined
as ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}|+boldsymbol{x} ) and ( boldsymbol{g}(boldsymbol{x})=|boldsymbol{x}|-boldsymbol{x} ) for
all ( x in R ). Then, find ( f circ g ) and ( g circ f ) Hence, find ( boldsymbol{f} circ boldsymbol{g}(-boldsymbol{3}), boldsymbol{f} circ boldsymbol{g}(boldsymbol{5}) ) and ( boldsymbol{g} )
( boldsymbol{f}(-2) )
12
429If ( boldsymbol{A}={1,2,3} ) and ( boldsymbol{B}={3,8}, ) then
( (boldsymbol{A} cup boldsymbol{B}) times(boldsymbol{A} cap boldsymbol{B}) ) is
A ( cdot{(3,1),(3,2),(3,3),(3,8)} )
в. {(1,3),(2,3),(3,3),(8,3)}
( c cdot{(1,2),(2,2),(3,3),(8,8)} )
D. {(8,3),(8,2),(8,1),(8,8)}
12
430If ( boldsymbol{A}={boldsymbol{x}, boldsymbol{y}} ) and ( boldsymbol{B}={mathbf{3}, mathbf{4}, mathbf{5}, mathbf{7}, mathbf{9}} ) and
( boldsymbol{C}={mathbf{4}, mathbf{5}, mathbf{6}, mathbf{7}}, ) find ( boldsymbol{A} times(boldsymbol{B} cap boldsymbol{C}) )
A ( cdot{(x, 4),(x, 5),(x, 7),(y, 4),(y, 5),(y, 7)} )
B – ( {(x, 3),(x, 5),(x, 7),(y, 4),(y, 5),(y, 7)} )
c ( cdot{(x, 4),(x, 5),(x, 7),(y, 4),(y, 5),(y, 9)} )
D. none of the above
12
431For the same signal emitted by a radio antenna, Observer A measures its
intensity to be 16 times the intensity measured by Observer B. The distance of Observer A from the radio antenna is
what fraction of the distance of
Observer B from the radio antenna?
A ( cdot frac{1}{4} )
B. ( frac{1}{16} )
c. ( frac{1}{64} )
D. ( frac{1}{256} )
12
432The minimum number of elements that
must be added to the relation ( boldsymbol{R}= )
( {(1,2)(2,3)} ) on the set of natural
numbers so that it is an equivalence is
A . 4
B. 7
( c .6 )
D.
12
433( f(x)=frac{cos x}{left[frac{2 x}{pi}right]+frac{1}{2}}, ) where ( x ) is not an
integral multiple of ( pi ) and [.] denotes the
greatest integer function, is
A. an odd function
B. an even function
c. neither odd nor even
D. none of these
12
434For ( boldsymbol{x} in boldsymbol{R},[boldsymbol{x}] ) denotes the greatest
integer less than or equal to ( x ). The
largest natural number ( n ) for which ( boldsymbol{E}=left[frac{boldsymbol{pi}}{2}right]+left[frac{mathbf{1}}{mathbf{1 0 0}}+frac{boldsymbol{pi}}{mathbf{2}}right]+ )
( left[frac{2}{100}+frac{pi}{2}right] ldots+left[frac{n}{100}+frac{pi}{2}right]<43 )
is
A . 41
B . 42
( c .43 )
D. 97
11
435( operatorname{Let} boldsymbol{f}(boldsymbol{x})=mathbf{1}+mathbf{2} boldsymbol{x}^{2}+mathbf{2}^{2} boldsymbol{x}^{boldsymbol{4}}+ldots ldots+ )
( 2^{10} x^{20} . ) Then ( f(x) ) has
A. more than one minimum
B. exactly one minimum
c. at least one maximum
D. none of these
11
436Domain of the function ( log left|x^{2}-9right| ) is
A. ( R )
B. ( R-[-3,3] )
c. ( R-{-3,3} )
D. None of these
12
437The range of ( mathbf{f}(mathbf{x})=frac{sin boldsymbol{pi}left[mathbf{x}^{2}-mathbf{1}right]}{mathbf{x}^{4}+mathbf{1}} ) is
A. ( R )
в. [-1,1]
c. {0,1}
D. {0
12
438( f:left(-frac{pi}{2}, frac{pi}{2}right) rightarrow(infty, infty), f(x)=tan x )
is.
A. one-one and onto
B. one-one but not onto
c. onto but not one-one
D. neither one-one nor onto
12
439If ( boldsymbol{f}: boldsymbol{Z} rightarrow boldsymbol{Z} ) is such that ( boldsymbol{f}(boldsymbol{x})=boldsymbol{6} boldsymbol{x}- )
11 then ( f ) is
A. injective but not surjective
B. surjective but not injective
c. bijective
D. neither injective nor surjective
12
440Let ( R ) be a relation from ( A={1,2,3,4} ) to ( mathrm{B}={1,3,5} ) such that
( R=[(a, b): a<b, text { where } a varepsilon A text { and } b varepsilon B] )
What is RoR ( ^{-1} ) equal to?
A ( cdot(1,3),(1,5),(2,3),(2,5),(3,5),(4,5) )
B. (3,1),(5,1),(3,2),(5,2),(5,3),(5,4)
c. (3,3),(3,5),(5,3),(5,5)
D. (3,3),(3,4),(4,5)
12
441Let ( A ) and ( B ) be two sets such that
( boldsymbol{A}= )
( boldsymbol{x}: boldsymbol{f}(boldsymbol{x})=left[boldsymbol{x}^{2}right], ) is discontinuous in ( [mathbf{0}, mathbf{2}] )
( boldsymbol{B}= )
( boldsymbol{x}: boldsymbol{f}(boldsymbol{x})=left[tan ^{2} boldsymbol{x}right]left[cot ^{2} boldsymbol{x}right], ) is non ( – ) diff
If ( m ) is the total number of onto
functions from ( A ) to ( B ) then find the
total number of divisors of ( boldsymbol{m} .[mathrm{Note}:[boldsymbol{k} )
denotes the greatest integer less than or equal to ( k .] )
12
442The function ( boldsymbol{f}:[mathbf{2}, infty) rightarrow boldsymbol{Y} ) defined by
( f(x)=x^{2}-4 x+5 ) is both one-one ( & )
onto if:
A ( . Y=R )
В. ( Y=[1, infty) )
c. ( Y=[4, infty) )
D. ( Y=[5, infty) )
12
443If x satisfies |x-1| + | x-2|+| x-326, then
(1983 – 1 Mark)
(a) 05×54
(b) XS-2 or x>4
(C) x4
(d) None of these
12
444If ( boldsymbol{A}={mathbf{1}, mathbf{2}, mathbf{3}}, boldsymbol{B}={mathbf{3}, mathbf{4}} )
find ( (A times B) cup(B times A) )
( mathbf{A} )
[
{(1,3),(2,3),(3,3),(1,4),(2,4),(3,4),(3,1),(4,1),(3,2)
]
B.
[
{(2,3),(3,3),(1,4),(2,4),(3,4),(3,1),(4,1),(3,2),(4,2)
]
( mathbf{c} )
{(1,3),(2,3),(3,3),(1,4),(2,4),(3,4),(3,1),(4,1),(4,2)
D. None of these
12
445Let ( boldsymbol{R}=(boldsymbol{a}, boldsymbol{a}),(boldsymbol{b}, boldsymbol{c}),(boldsymbol{a}, boldsymbol{b}) ) be a relation
on a set ( A=a, b, c . ) Then the minimum
number of ordered pairs which when added to make it an equivalence
relation are
12
446Let ( R ) and ( S ) be two equivalence relations
on set A. Prove that ( boldsymbol{R} cap boldsymbol{S} ) is an
equivalence relation.
12
447If ( h(x)>[2 x-f n(f(x))] f(x) ) and
( f(x)>0 ) satisfy for ( forall x>0, ) if ( h(x)= ) ( int_{0}^{x}(f(t)-t) d t, ) then
A. ( h(x) ) is increasing function
B. ( h(x) ) is decreasing function
( c cdot h(1)>1 )
D. ( h(1)>e-1 )
11
448The following relations are defined on the set of real numbers check them for
Reflexivity,Symmetry, Transitivity. ( 1+a b>0 )
12
449If ( left(frac{x}{3}+1, y-frac{2}{3}right)=left(frac{5}{3}, frac{1}{3}right) ) find the
values of ( x ) and ( y )
12
450L1,
Let
:-1, 1)
→ B, be a function defined by
f(x) = tan-1_2x
2, then fis both one-one and onto when
B is the interval
[2005]
12
45132. Let f(x)=x2 and g(x) = sin x for all x € R. Then the set of all
x satisfying (fo gogof)(x)=(gogof)(x), where (fog)(x)
=f(g(x)), is
(2011)
(a) Evnt,n e{0,1,2,….}
(b) IJnt,ne {1,2,…}
(c) + 2nt,n € {…-2,-1,0,1,2….}
(d)
2nt,n e{… -2,-1,0,1,2,…}
12
452The domain of Function ( sqrt{9-x^{2}} )12
453Let ( A={0,1} ) and ( N ) the set of all
natural numbers. Then show the
mapping ( boldsymbol{f}: boldsymbol{N} rightarrow boldsymbol{A} ) defined by
( boldsymbol{f}(boldsymbol{2 n}-mathbf{1})=mathbf{0}, boldsymbol{f}(boldsymbol{2 n})=mathbf{1} forall boldsymbol{n} in boldsymbol{N} ) is
many-one onto.
12
454Show that the relation ( R ) defined by
( boldsymbol{R}= )
( {(a, b): 3 text { divides } a-b text { for } a, b in Z} ) is
an equivalence relation
12
455The range of ( boldsymbol{f}(boldsymbol{x})= )
( sec left(frac{pi}{4} cos ^{2} xright) cdot-infty<x<infty ) is
A ( cdot[1, sqrt{2}] )
B . ( [1, infty] )
c. ( [-sqrt{2},-1] cup[1, sqrt{2}] )
D ( cdot[-infty,-1] cup[1, infty] )
12
456The domain of the functions ( f(x)= ) ( sqrt{log left(2 x-x^{2}right)} ) is
A ( .(0,2) )
B. [0,2]
( c cdot{1} )
D. none
12
457Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be defined by ( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}] )
and ( g(x)=frac{3-2 x}{4} . ) We can say:
This question has multiple correct options
A. ( f ) is neither one-one nor onto
B. ( g ) is one-one but ( f ) is not one-one
c. ( f ) is one-one and ( g ) is onto
D. neither ( f ) nor ( g ) is onto
12
458Let ( R_{0} ) denote the set of all non-zero real
numbers and let ( boldsymbol{A}=boldsymbol{R}_{0} times boldsymbol{R}_{0} . ) If ( ^{prime} *^{prime} ) is a
binary operation on A defined by ( (a, b) * ) ( (c, d)=(a c, b d) ) for all ( (a, b)(c, d) in A )
Show that ( ^{prime} *^{prime} ) is both commutative and
associative on ( mathbf{A} )
12
4597.
If f:
R S, defined by
f(x) = sin x-13 cos x +1, is onto, then the interval of Sis
[2004]
(a) [-1,3] (b) (-1,1] (c) [0,1] (d) [0,3]
matricol about the
12
460If ( f(x)=5 log _{5} x ) then ( f^{-1}(alpha-beta) ) where
( boldsymbol{alpha}, boldsymbol{beta} in boldsymbol{R} ) is equal to
A ( cdot f^{-1}(alpha)-f^{-1}(beta) )
B. ( frac{f^{-1}(alpha)}{f^{-1}(beta)} )
c. ( frac{1}{f(alpha-beta)} )
D. ( frac{1}{f(alpha)-f(beta)} )
12
461If ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}boldsymbol{x} & text { for }-boldsymbol{3}<boldsymbol{x} leq-1 \ |boldsymbol{x}| & text { for }-1<boldsymbol{x}<1 \ |boldsymbol{x}| & text { for } boldsymbol{x} geq mathbf{1}end{array}right. )
then find ( {boldsymbol{x}: boldsymbol{f}(boldsymbol{x}) geq mathbf{0}} )
A ( cdot(-1, infty) )
B. [-1,3]
c. (-1,3
D. ( I-1 )
12
462( boldsymbol{A}={1,2,3,5} ) and ( boldsymbol{B}={mathbf{4}, mathbf{6}, mathbf{9}} )
Define a relation ( mathrm{R} ) from ( A ) to ( mathrm{B} ) by ( boldsymbol{R}= )
( {(x, y): text { the difference between } x text { and } y )
is odd, ( x in A, y in B} . ) Write ( R ) in Roster
form
12
463If ( boldsymbol{f}(boldsymbol{theta})=2left(sec ^{2} boldsymbol{theta}+cos ^{2} boldsymbol{theta}right), ) then its
value always
A ( cdot 4>f(theta)>2 )
B ( cdot f(theta) geq 4 )
( mathbf{c} cdot f(theta)<2 )
D. ( f(theta)=2 )
12
464A function ( boldsymbol{f}:(-mathbf{3}, mathbf{7}) rightarrow boldsymbol{R} ) is defined as
follows:
( f(x)=left{begin{array}{cc}4 x^{2}-1 ; & -3 leq x<2 \ 3 x-2 ; & 2 leq x leq 4 \ 2 x-3 ; & 4<x leq 6end{array}right. )
Find:
( boldsymbol{f}(mathbf{1})-boldsymbol{f}(-boldsymbol{3}) )
11
465f ( ,(x-1, y+2)=(7,5) ) then values of
( x ) and ( y ) are
A . 5,8
в. 8,3
c. -1,5
D. 7,1
12
466If ( f(x)=sin x-cos x-a x+b )
decreases for a where ( x in R ) then?
This question has multiple correct options
( mathbf{A} cdot a1 )
c. ( asqrt{2} )
12
467The number of pairs ( (a, b) ) of positive
real numbers satisfying ( a^{4}+b^{4}1 ) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. More than 2
11
468For all numbers a and ,b let ( a odot b ) be
defined by ( a odot b=a b+a+b . ) For all
numbers ( x, y ) and ( z, ) which of the following must be true?
I. ( boldsymbol{x} odot boldsymbol{y}=boldsymbol{y} odot boldsymbol{x} )
Il. ( (x-1) odot(x+1)=(x odot x)-1 )
III. ( boldsymbol{x} odot(boldsymbol{x}+boldsymbol{z})=(boldsymbol{x} odot boldsymbol{y})+(boldsymbol{x} odot boldsymbol{z}) )
A. I only
B. II only
c. ॥ only
D. I and II only
E. I, II and III
12
469If ( boldsymbol{A}={mathbf{0}, mathbf{1}, mathbf{2}, mathbf{3}, mathbf{4}, mathbf{5}} ) and relation ( boldsymbol{R} )
defined by ( a R b ) such that ( 2 a+b=10 )
then ( R^{-1} ) equals
A ( cdot{(4,3),(2,4),(5,0)} )
B – {(3,4),(4,2),(5,0)}
c. {(4,3),(2,4),(0,5)}
D ( cdot{(4,3),(4,2),(5,0)} )
12
470The value of ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{3} ) when ( boldsymbol{x}=boldsymbol{3} )
is
12
471( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{B} ) defined by ( f(x)=2 x+3 ) and if
( A={-2,-1,0,1,2} B={-1,1,3,5,7} ) then which
type of function is ( f ? )
A. One-one
B. onto
c. Bijection
D. constant
12
472The domain of ( boldsymbol{f}(boldsymbol{x})= )
( frac{tan pi[x]}{1+sin pi[x]+left[x^{2}right]} ) is (where [.] denotes
greatest integer function)
A. ( (0, infty) )
в. ( R )
( c cdot(-infty, 0) )
D. ( z )
11
473Let ( A={1,2} ) and ( B={3,4} ). Find the number of relations from A to B.12
474f ( f: R rightarrow R ) be defined by ( f(x)=3 x+ )
( 2, ) find ( f(f(x)) )
12
475Let ( boldsymbol{R}={(boldsymbol{a}, boldsymbol{a}),(boldsymbol{b}, boldsymbol{c}),(boldsymbol{a}, boldsymbol{b})} ) be a
relation on a set ( A={a, b, c} . ) Then the
minimum number of ordered pairs
which when added to R make it an
equivalence relation are
12
476Given ( boldsymbol{A}={boldsymbol{b}, boldsymbol{c}, boldsymbol{d}} ) and ( boldsymbol{B}={boldsymbol{x}, boldsymbol{y}} )
find element of ( boldsymbol{A} times boldsymbol{B} )
A ( cdot{b, x} )
в. ( {b, y} )
c. ( {c, x} )
D. All of the above
12
477If ( alpha, beta ) be straight lines in a plane, then
check ( R_{1} ) and ( R_{2} ) for being reflexive,
symmetric and transitive ( boldsymbol{alpha} boldsymbol{R}_{1} boldsymbol{beta} boldsymbol{i} boldsymbol{f} boldsymbol{alpha} perp )
( boldsymbol{beta} boldsymbol{a} boldsymbol{n} boldsymbol{d} boldsymbol{alpha} boldsymbol{R}_{2} boldsymbol{beta} boldsymbol{i} boldsymbol{f} boldsymbol{alpha} )
12
478Sum of ( (267+345)+21 ) and ( 267+ )
( (345+21) ) will be same
A. True
B. False
12
479If ( G={7,8} ) and ( H={5,4,2}, ) find
( boldsymbol{H} times boldsymbol{G} )
12
48048.
Consider the following relations:
R= lr. lx, y are real numbers andxawy for some ration
number w};
S={
In 9
I m.n. p and q are integers such that n,
a
“, P
n
and qm=pn).
Then
[2010]
(a) Neither R nor S is an equivalence relation
(b) Sis an equivalence relation but Ris not an equivalence
relation
(c) Rand S both are equivalence relations
(d) Ris an equivalence relation but is not an equivalence
relation
12
481Write the smallest equivalence relation
on the ( operatorname{set} A={1,2,3} )
12
482(JEE Adv. 2014)
10. Let a eR and let f:
R R be given by
f(x)=x-5x + a. Then
a) f(x) has three real roots if a>4
(b) f(x) has only real root if a>4
© f(x) has three real roots if a<-4
f(x) has three real roots if-4<a<4
12
483The range of ( left|tan ^{-1} xright| ) is
( mathbf{A} cdotleft[0, frac{pi}{2}right] )
В. ( left[0, frac{pi}{2}right) )
c. ( left(0, frac{pi}{2}right) )
D. ( left(frac{-pi}{2}, frac{pi}{2}right) )
12
484If ( f(x)=2 x-5, ) then what is the value
of ( boldsymbol{f}(mathbf{2})+boldsymbol{f}(mathbf{5}) ? )
A .
B. 2
( c cdot 3 )
D. 4
E . 5
12
485Make ( r ) the subject of formula.
A ( cdot_{r}=sqrt{frac{pi R^{2} l+V}{pi l}} )
в. ( r=sqrt{frac{pi R^{2} l-V}{pi l}} )
c. ( _{r}=sqrt{frac{pi R^{2} l+V}{2 pi l}} )
D. none of the above
11
486Let ( quad A={x:-1 leq x leq 1} ) and ( f )
( boldsymbol{A} rightarrow boldsymbol{A} ) such that ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}|boldsymbol{x}| ), then ( boldsymbol{f} ) is
A. a bijection
B. injective but not surjective
c. surjective but not injective
D. neither injective nor surjective
12
487If ( f(x)=ln frac{1+x}{1-x} ) and ( g(x)=frac{3 x+x^{3}}{1+3 x^{2}} )
then ( boldsymbol{f}[boldsymbol{g}(boldsymbol{x})] ) equals.
A. ( f(x) )
B . ( [f(x)]^{3} )
( mathbf{c} cdot 3 f(x) )
D. ( f(x)^{2} )
12
488If ( boldsymbol{f}: boldsymbol{R} rightarrow(-1,1) ) is defined by ( boldsymbol{f}(boldsymbol{x})= )
( frac{-boldsymbol{x}|boldsymbol{x}|}{mathbf{1}+boldsymbol{x}^{2}} ) then ( boldsymbol{f}^{-1}(boldsymbol{x}) ) equals
( ^{A} cdot sqrt{frac{|x|}{1-|x|}} )
в. ( quad-operatorname{sgn}(x) sqrt{frac{|x|}{1-|x|}} )
( ^{mathrm{c}} cdot sqrt{frac{x}{1-x}} )
D. None of these
11
489If ( frac{4}{x}<frac{1}{3}, ) what is the possible range of
value for ( x ? )
11
490Let ( A={1,2,3} . ) Then number of
equivalence relations containing (1,2) is:
( A )
B. 2
( c cdot 3 )
D. 4
12
491The function ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) defined by
( f(x)=frac{x^{2}}{1+x^{2}} forall x in R ) is
A. one one but not onto
B. onto but not one one
c. a bijection
D. neither one one nor onto
12
492Day Number of coffee sold
3
21
4
35
5
[
53
]
6
[
75
]
7
101
The above data shows the number of
coffee sold by the famous coffee shop at
a particular venue. If the number is
shown by ( v ) and days as ( t, ) then they are
related by
( mathbf{A} cdot v(t)=2 t^{2}+3 )
( B )
[
v(t)=frac{t^{2}}{2}+3
]
C ( cdot v(t)=2 t^{2}+21 )
D.
[
v(t)=frac{t^{2}}{2}+21
]
12
493If ( f(x)=left(a-x^{n}right)^{1 / n} ) where ( a>0 ) and ( n )
is a positive integer then ( (text { fof })(x) ) is
A ( . f(x) )
B.
( c cdot 0 )
D.
12
494If ( f(x)=frac{1}{x}, g(x)=sqrt{x} ) and
( (g o sqrt{f})(16)= )
( A cdot 2 )
B.
( c cdot frac{1}{2} )
D.
12
495Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+boldsymbol{2} boldsymbol{x}^{2}+boldsymbol{3} boldsymbol{x}+boldsymbol{4}, ) then the
quation ( frac{1}{x-f(1)}+frac{2}{x-f(2)}+ )
( frac{3}{x-f(3)}=0, ) has
A. 1 real root
B. 2 real roots
c. all three roots are real
D. no real root exist
11
496Which one of the following functions is
not one-one?
( mathbf{A} cdot f:(-1, infty) rightarrow R ) given by ( f(x)=x^{2}+2 x )
B ( cdot g:(2, infty) rightarrow R ) given by ( g(x)=e^{x^{3}-3 x+2} )
C ( . h: R rightarrow R ) given by ( h(x)=2^{x(x-1)} )
D ( quad phi:(-infty, 0) rightarrow R ) given by ( phi(x)=frac{x^{2}}{x^{2}+1} )
12
497Define many-one function. Give an
example of function.
12
498Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+boldsymbol{c}, ) where ( boldsymbol{a}, boldsymbol{b}, boldsymbol{c} )
are rational, and ( boldsymbol{f}: boldsymbol{Z} rightarrow boldsymbol{Z}, ) where ( boldsymbol{Z} ) is
the set of integers. Then ( a+b ) is
A. a negative integer
B. an integer
c. nonintegral rational number
D. none of these
11
499If ( boldsymbol{A}={1,2,3}, ) then a relation ( boldsymbol{R}= )
{(2,3)} on ( A ) is
A. symmetric and transitive only
B. symmetric only
C. transitive only
D. none of these
12
500If a relation ( ^{prime} boldsymbol{R}^{prime} ) is defined by ( boldsymbol{R}= )
( left{(x, y) / 2 x^{2}+3 y^{2} leq 6right}, ) then the
domain of ( boldsymbol{R}(boldsymbol{x}, boldsymbol{y}) ) is
This question has multiple correct options
A ( cdot x in[-3,3] )
B . ( x in[-sqrt{3}, sqrt{3}] )
c. ( y in[-sqrt{2}, sqrt{2}] )
D ( cdot y in[-2,2] )
12
501The domain of ( frac{1}{sqrt{x-x^{2}}}+ ) ( sqrt{3 x-1-2 x^{2}} ) is
( ^{mathbf{A}} cdotleft(frac{1}{2}, 11right) )
в. ( left(frac{1}{2}, 31right) )
c. ( left(frac{1}{2}, 17right) )
D ( cdotleft(frac{1}{2}, 41right) )
12
502Prove that intersection of equivalence
relations on a set is also an equivalence relation.
12
503The number of ordered pair ( (x, y) ) such
that ( 0 leq x leq 2 pi ) and satisfying the inequality ( 2^{sec ^{2} x} sqrt{y^{2}-2 y+2} leq 2 ) is
A .2
B. 3
c. 1
D. None of these
12
504Number of roots of ( |sin | x||=x+|x| ) in
( [-2 pi, 2 pi] ) is
( A cdot 2 )
B. 3
( c cdot 4 )
D. 6
11
505Range of ( f(x)=frac{sec x+tan x-1}{tan x-sec x+1} )
( boldsymbol{x} epsilonleft(mathbf{0}, frac{pi}{2}right) ) is
A ( .(0,1) )
B. ( (0, infty) )
c. (-1,0)
D. ( (-infty,-1) )
12
506If ( f(x)=x^{2}+9 ) and ( g(x)=24+4 x )
what is the value of ( frac{f(4)}{g(-1)} ? )
A . 0.75
B. 0.8
c. 1.2
D. 1.25
E . 1.75
12
507120041
The range of the function f(x) ==* P,-zis
(a) {1,2,3,4,5)
(b) {1,2,3,4,5,6)
(c) {1,2,3,4}
(d) {1,2,3,
12
508If ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{c}{[boldsymbol{x}] forall boldsymbol{x} geq mathbf{0}} \ mathbf{1}+[boldsymbol{x}] forall boldsymbol{x}<mathbf{0}end{array}right. )
A. ( f(x) ) is a periodic
B. ( f(x) ) is a many one
c. ( f(x) ) is a one-one
D. None of these
12
509Prove that the function ( f(x)=[x] ) is not
continuous at ( boldsymbol{x}=mathbf{0} . ) Where ( [boldsymbol{x}] ) is the
greatest integer function.
11
510Let a relation ( boldsymbol{R} ) be defined by ( boldsymbol{R}= )
( {(4,5),(1,4),(4,6),(7,6),(3,7)} . ) The
relation ( R^{-1} circ R ) is given by
A ( cdot{(1,1),(4,4),(7,4),(4,7),(7,7)} )
B – {(1,1),(4,4),(4,7),(7,4),(7,7),(3,3)}
c. {(1,5),(1,6),(3,6)}
D. None of these
12
511If ( boldsymbol{A}={boldsymbol{a}, boldsymbol{b}}, boldsymbol{B}={1,2,3}, ) find ( mathbf{B} times mathbf{A} )
A. ( B times A={(1, a),(2, a),(3, a),(1, b)(2, b),(3, b)} )
B. ( B times A={(2, a),(3, a),(1, b)(2, b),(3, b)} )
C. ( B times A={(1, a),(2, a),(3, a),(1, b)(2, b)} )
D. None of these
12
512If ( : n(A)=m, ) then number of relations
in ( A ) are
( mathbf{A} cdot 2^{m} )
B. ( 2^{m}-2 )
( mathbf{c} cdot 2^{m^{2}} )
D. None of these
12
513Given, ( boldsymbol{f}(boldsymbol{x})=mathbf{2} boldsymbol{x}^{2}+boldsymbol{a} )
If ( a ) is a constant and ( f(2)+f(3)= )
( f(5), ) calculate the value of ( a )
( A cdot 6 )
B. 12
( c cdot 24 )
D. 48
12
514A mapping from ( N ) to ( N ) is defined as follows
( boldsymbol{f}: boldsymbol{N} longrightarrow boldsymbol{N} )
( boldsymbol{f}(boldsymbol{n})=(boldsymbol{n}+mathbf{5})^{2}, boldsymbol{n} in boldsymbol{N} )
( (N ) is the set of natural numbers). Then,
A. ( f ) is not one to one
B. ( f ) is onto
c. ( f ) is both one to one and onto
D. ( f ) is one to one but not onto
12
515If the ordered pairs ( (x,-1) ) and ( (5, y) )
belong to the set ( (a, b): b=2 a-3, ) find the values of ( x ) and ( y ? )
12
516If ( boldsymbol{A}={mathbf{1}, mathbf{2}, mathbf{3}}, boldsymbol{B}={mathbf{1}, mathbf{4}, mathbf{6}, mathbf{9}} ) and ( mathbf{R} ) is
a relation from ( A ) to ( B ) defined by ( ‘ x ) is
greater than ( y ) ‘. The range of R is
A ( cdot{1,4,6,9} )
в. {4,6,9}
c. {1}
D. {4,6}
12
517If ( a-b=4 ) and ( a+b=6, ) find ( a^{2}+b^{2} )
A . 26
B . 13
c. 53
D. 40
11
51812
)
2403
then k is
19. If the fractional part of the number 16 IS 15
equal to:
JEEM 2019-9 Jan (M)]
(a) 6
(b) 8
(c) 4
(d) 14
12
519The function ( f(x)=tan ^{-1}(sin x+ )
( cos x) ) is strictly increasing function in
A ( cdotleft(frac{pi}{4}, frac{pi}{2}right) )
B ( cdotleft(-frac{pi}{2}, frac{pi}{4}right) )
c. ( left(0, frac{pi}{4}right) )
D. ( left(-frac{pi}{2}, frac{pi}{2}right) )
12
520State whether the following statements are true of false. Justify.
(i) For an arbitrary binary operation ( * ) on as ( operatorname{set} N, a * a=a forall a epsilon N )
(ii) If ( * ) is a commutative binary
operation on ( N, ) then ( a *(b * c)=(c * )
( b) * a )
A. True
B. False
12
521Suppose ( boldsymbol{f}:[-mathbf{2}, mathbf{2}] rightarrow boldsymbol{R} ) is defined by
( boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}-1 text { for }-2 leq x leq 0 \ x-1 text { for } 0 leq x leq 2end{array}right. )
then the ( {x in(-2,2): x leq )
( operatorname{Oand} f(|x|)=x}= )
A . {-1}
в. {0}
c. ( left{frac{-1}{2}right} )
D.
12
522Solve the
( 2 frac{1}{5}-frac{-1}{3} )
12
523Show that the function ( boldsymbol{f}: mathbb{R} rightarrow )
( {x in mathbb{R}:-1<x<1} ) defined by
( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}}{mathbf{1}+|boldsymbol{x}|}, boldsymbol{x} in mathbb{R} )
is one to one and onto function.
12
524If ( (x)=x^{2}-2 x+4 ) then the set of
values of ( x ) satisfying ( f(x-1)= )
( boldsymbol{f}(boldsymbol{x}+mathbf{1}) ) is
( mathbf{A} cdot{-1} )
B . {-1,1}
( c cdot{1} )
D ( cdot{1,2} )
11
525If ( f(x+1)=3 x-9, ) then what will be
the value of ( fleft(x^{2}-1right) ? )
A ( cdot 3 x^{2}-9 )
B. ( 3 x^{2}-15 )
c. ( x^{2}-10 )
D. ( 3 x^{2}-10 )
11
526Let ( R ) be a reflexive relation on a finite
set ( A ) having ( n ) elements, and let there
be ( m ) ordered pairs in ( R, ) then:
A ( . m geq n )
в. ( m leq n )
c. ( m=n )
D. ( m<n )
12
527Let ( boldsymbol{A}={mathbf{1}, mathbf{2}} ) and ( boldsymbol{B}={mathbf{2}, mathbf{3}, mathbf{4}}, ) then
( boldsymbol{B} times boldsymbol{A}= )
{(2,1),(2,2),(3,1),(3,2),(4,1),(4,2)}
If true enter 1 , or else enter 0 .
12
528( boldsymbol{f}: boldsymbol{N} rightarrow boldsymbol{N}, ) then show that ( boldsymbol{f}(boldsymbol{n})= )
( 2 n+3 forall n in N )
This question has multiple correct options
A. many one, into.
B. one-one, into
c. one-one, not onto
D. many-one, onto
12
529Let ( A ) be a non-empty set such that ( A times ) ( A ) has 9 elements among which are
found (-1,0) and ( (0,1), ) then
( mathbf{A} cdot A={-1,0} )
В ( cdot A={0,1} )
C ( cdot A={-1,0,1} )
D ( cdot A={-1,1} )
12
530( operatorname{Let} f(x)=frac{x^{2}-4}{x^{2}+4} ) for ( |x|>2, ) then the
function ( boldsymbol{f}:(-infty,-2) cup[mathbf{2}, infty) rightarrow )
(-1,1) is
A. One-one into
B. One-one onto
c. Many one into
D. Many one onto
12
531If ( f(x)=x^{2}, g(x)=x^{2}-5 x+6, ) then
( boldsymbol{g}(mathbf{2})+boldsymbol{g}(mathbf{3})+boldsymbol{g}(mathbf{0})-boldsymbol{f}(mathbf{0})-boldsymbol{f}(mathbf{1})- )
( boldsymbol{f}(-2) )
A . 2
B.
( c cdot frac{5}{6} )
( D )
11
532If ( f(x)=frac{x}{sqrt{1-x^{2}}}, g(x)=frac{x}{sqrt{1+x^{2}}} )
( operatorname{then}(f o g)(x)= )
( A )
В. ( frac{x}{sqrt{1+x^{2}}} )
c. ( sqrt{1+x^{2}} )
D. 2x
12
533Show that if ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{B} ) and ( boldsymbol{g}: boldsymbol{B} rightarrow boldsymbol{C} )
are ( 1-1, ) then ( g o f: A rightarrow C ) is also 1
( mathbf{1} )
12
534Find ( boldsymbol{f} circ boldsymbol{g} ) and ( boldsymbol{g} circ boldsymbol{f}, ) if ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}+mathbf{1} )
( boldsymbol{g}(boldsymbol{x})=boldsymbol{e}^{boldsymbol{x}} )
12
535what could be the possible values of ( x )
when, ( [mathbf{2 x}]=[boldsymbol{x}] )
11
536Let ( boldsymbol{A}={-2,-1,0,1,2} ) and ( boldsymbol{f}: boldsymbol{A} rightarrow )
( Z ) be given by ( f(x)=x^{2}-2 x-3 . ) Find
the pre-image of 6,-3 and 5
12
537Let ( R ) be a relation from ( A= )
{1,2,3,4} to ( B={1,3,5} ) such that
( boldsymbol{R}=[(boldsymbol{a}, boldsymbol{b}): boldsymbol{a}< )
( b, text { where } a epsilon A text { and } b epsilon B] . ) What is ( R )
equal to?
A ( cdot(1,3),(1,5),(2,3),(2,5),(3,5),(4,5) )
B . (3,1),(5,1),(3,2),(5,2),(5,3),(5,4)
c. (3,3),(3,5),(5,3),(5,5)
D. (3,3),(3,4),(4,5)
12
538ff ( (x)=-x^{2}+1, g(x)=-sqrt[3]{x} ) then
(gofogofogogog) ( (x) ) is.
A. an odd function
B. an even function
c. a polynomial function
D. an identity function
12
539If the set ( A ) has 3 elements and the set
( boldsymbol{B}=(mathbf{3}, mathbf{4}, mathbf{5}) ) find the number of
elements in ( (boldsymbol{A} times boldsymbol{B}) ) ?
12
540Find the domain of ( cos ^{-1}[X] ) where ( [X] )
is the greatest integer functions.
12
541If ( R={(x, y) / 3 x+2 y=15 text { and } x, y epsilon N} )
the range of the relation ( mathrm{R} ) is
A ( cdot{1,2} )
B . ( {1,2, . ., 5} )
c. ( {1,2, ., 7} )
D. {3,6}
12
542The domain of ( frac{10^{x}+10^{-x}}{10^{x}-10^{-x}} ) is
A. ( R )
в. ( R-{0} )
c. ( R-{1} )
D. ( R^{+} )
12
543If ( boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{x}+boldsymbol{4})=boldsymbol{f}(boldsymbol{x}+boldsymbol{2})+boldsymbol{f}(boldsymbol{x}+ )
6) then find the period of ( f(x) )
11
544The domain of ( f(x)=log (sin x) ) is
A ( cdot(-pi, pi) )
В. ( (2 n pi,(2 n+1) pi) ) and ( n in Z )
с. ( [0,2 pi] )
]
D. ( left(-frac{pi}{2}, frac{pi}{2}right) )
12
54511.
If the equation anx” +an-1x”
tion anx” +an_xn-1 + …………. + d x =U
a 70, n > 2, has a positive root x = 0,
as a positive root x = a , then the equation
na,x”-+(n-1) an-jxn-2 +…….. + a1 = 0 has a positive
root, which is
[2005]
(a) greater than a
(b) smaller than a
(c) greater than or equal to a
(d) equal to a
12
546If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) defined by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+mathbf{1} )
then the values of ( boldsymbol{f}^{-}(mathbf{1 7}) ) and ( boldsymbol{f}^{-}(-mathbf{3}) )
respectively are
( mathbf{A} cdot phi,{4,-4} )
B ( cdot{3,-3}, phi )
( mathbf{c} cdot phi,{3,-3} )
( mathbf{D} cdot{4,-4}, phi )
12
547Find domain and range of the function
( f(x) ) given by ( f(x)=frac{x}{1+x^{2}} )
A. Domain ( =R ) and Range ( =N )
B. Domain ( =N ) and Range ( =R )
c. Domain ( =R ) and ( operatorname{Range}=R )
D. Domain ( =R-{0} ) and Range ( =N )
12
548A relation which satisfies reflexive
symmetric and transitive is relation
A. an identity
B. a constant
c. an equivalence
D. None of these
12
549If two sets ( A ) and ( B ) have 99 elements in
common, then the number of elements
common to each of the sets ( A times B ) and
( boldsymbol{B} times boldsymbol{A} ) are
A . ( 2^{99} )
B. ( 99^{2} )
( c cdot 100 )
D. 18
12
550If ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{n}, boldsymbol{n} in boldsymbol{N} ) and ( (boldsymbol{g} boldsymbol{o} boldsymbol{f})(boldsymbol{x})= )
( boldsymbol{n} boldsymbol{g}(boldsymbol{x}) operatorname{then} boldsymbol{g}(boldsymbol{x}) ) can be
A ( . n|x| )
B. ( 3 . sqrt[3]{x} )
( c cdot e^{x} )
( mathbf{D} cdot log |x| )
12
551If ( boldsymbol{A}={mathbf{3}, boldsymbol{4}} quad ) and ( quad boldsymbol{B}={1,2} ) then of
( boldsymbol{A} times boldsymbol{B} ) is
12
552The inequality. ( min f(x)>-7 / 18[-pi, pi] ) holds for
the function ( f(x)=(cos x)^{2}(sin x) )
A. True
B. False
12
553Let ( ^{*} ) be a binary operation on the set ( boldsymbol{Q} )
of rational numbers as follows:
( boldsymbol{a} * boldsymbol{b}=frac{boldsymbol{a} boldsymbol{b}}{boldsymbol{4}} )
Find which of the binary operations are commutative and which are
associative.
12
554Find the period of ( f(x)=[sin 3 x]+ )
( |cos 6 x| ) where ( [.] ) is a greatest integer
function and / is a modulus function.
11
555If ( frac{1}{2} leq log _{0.1} x leq 2, ) then:
A ( cdot ) maximum value of ( x ) is ( frac{1}{sqrt{10}} )
B. ( x ) lies between ( frac{1}{100} ) and ( frac{1}{sqrt{10}} )
C. minimum value of ( x ) is ( frac{1}{10} )
D. minimum value of ( x ) is ( frac{1}{100} )
E . maximum value of ( x ) is ( frac{1}{100} )
12
556( operatorname{Let} A={x, y, z} ) and ( B={1,2} . ) Find
the number of relations from ( boldsymbol{A} ) to ( boldsymbol{B} )
12
557The function ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) defined by
( boldsymbol{f}(boldsymbol{x})=boldsymbol{4}^{boldsymbol{x}}+boldsymbol{4}^{|boldsymbol{x}|} ) is
A. One-one and into
B. Many-one and into
c. one-one and onto
D. Many-one and onto
12
558If ( R ) be a relation defined from ( A= )
{1,2,3,4} to ( B={1,3,5}, ) i.e. ( (a, b) in )
( R ) iff ( a<b ) then ( R o R^{-1} ) is
A ( cdot{(1,3),(1,5),(2,3),(2,5),(3,5),(4,5)} )
B・ ( {(3,1),(5,1)(3,2),(5,2),(5,3),(5,4)} )
c. {(3,3),(3,5),(5,3),(5,5)}
D ( cdot{(3,3),(3,4),(4,5)} )
12
559If ( boldsymbol{f}left(frac{boldsymbol{x}+boldsymbol{y}}{mathbf{2}}right)=frac{boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y})}{2} ) for all
( boldsymbol{x}, boldsymbol{y} in boldsymbol{R} ) and ( boldsymbol{f}^{prime}(boldsymbol{o})=-mathbf{1}, boldsymbol{f}(boldsymbol{o})=mathbf{1} )
then ( boldsymbol{f}(2)= )
A ( cdot frac{1}{2} )
B.
( c cdot-1 )
D. ( frac{-1}{2} )
11
560If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{S}, ) defined by ( boldsymbol{f}(boldsymbol{x})=sin boldsymbol{x}- )
( sqrt{3} cos x+1, ) in onto, then the interval
of ( boldsymbol{S} ) is
A . [0.1]
B . [-1,1]
( c cdot[0,3] )
D. [-1,3]
12
561Which of the following functions are
identity functions?
This question has multiple correct options
( mathbf{A} cdot f: R rightarrow R, f(x)=x )
( mathbf{B} cdot g: N rightarrow Z, g(p)=3 )
C ( . h: z rightarrow z, h(y)=y )
( mathbf{D} cdot g: N rightarrow N, g(z)=z )
11
562Find the value of the following modulus:
( |-11|-|3| )
11
563f ( boldsymbol{alpha}, boldsymbol{beta} neq mathbf{0}, ) and ( boldsymbol{f}(boldsymbol{n})=boldsymbol{alpha}^{n}+boldsymbol{beta}^{n} ) and
( left|begin{array}{ccc}mathbf{3} & mathbf{1}+boldsymbol{f}(mathbf{1}) & mathbf{1}+boldsymbol{f}(mathbf{2}) \ mathbf{1}+boldsymbol{f}(mathbf{1}) & mathbf{1}+boldsymbol{f}(mathbf{2}) & mathbf{1}+boldsymbol{f}(mathbf{3}) \ mathbf{1}+boldsymbol{f}(mathbf{2}) & mathbf{1}+boldsymbol{f}(mathbf{3}) & mathbf{1}+boldsymbol{f}(mathbf{4})end{array}right|= )
( boldsymbol{K}(mathbf{1}-boldsymbol{alpha})^{2}(mathbf{1}-boldsymbol{beta})^{2}(boldsymbol{alpha}-boldsymbol{beta})^{2}, ) then ( mathrm{K} ) is
equal to?
( A )
B.
( c cdot alpha beta )
D. ( frac{1}{alpha} )
12
564If ( boldsymbol{A}={1,2,3} ) and ( B={3,8}, ) then
( (A cup B) times(A cap B) ) is equal to
A ( cdot{(8,3),(8,2),(8,1),(8,8)} )
B – {(1,2),(2,2),(3,3),(8,8)}
c. {(3,1),(3,2),(3,3),(3,8)}
D・ {(1,3),(2,3),(3,3),(8,3)}
12
565If ( M={x: x in N, 1<x leq 4} ) and
( N={y: y in W, y<3} )
12
566The domain of ( f(x)=tan 2 x ) is
( A cdot(-infty, infty) )
B . ( R-left{(2 n+1) frac{pi}{4}, n in Zright} )
c. ( R-left{(n+1) frac{pi}{4}, n in Zright} )
D. ( R-{n pi, n in Z} )
12
567If ( r, s, t ) are prime numbers and ( p, q ) are the positive integers such that LCM of ( boldsymbol{p} ) ( q ) is ( r^{2} t^{4} s^{2}, ) then the number of ordered
pair ( (boldsymbol{p}, boldsymbol{q}) ) is
A . 252
в. 254
c. 225
D . 224
12
568Write a rational function ( f ) that has
vertical asymptote at ( x=4, ) a
horizontal asymptote at ( y=5 ) and ( a )
zero at ( x=-7 )
A ( quad f(x)=frac{5(x-7)}{(x-4)} )
B. ( f(x)=frac{5(x+7)}{(x-4)} )
c. ( f(x)=frac{5(x-7)}{(x+4)} )
D. ( f(x)=frac{(x+7)}{(x+4)} )
11
569In the equation ( y=k x+3, k ) is a
constant. If ( boldsymbol{y}=mathbf{1 7} ) when ( boldsymbol{x}=mathbf{2}, ) what is
the value of ( boldsymbol{y} ) when ( boldsymbol{x}=mathbf{4} ? )
A . 34
B. 31
c. 14
D. 11
( E .7 )
12
570Let ( N ) denote the set of all natural
numbers and ( R ) be the relation on ( N times )
( N ) defined by ( (a, b) R(c, d), ) if ( a d(b+ )
( c)=b c(a+d), ) then show that ( R ) is an
equivalence relation.
12
571What is the Cartesian product of ( boldsymbol{A}= )
{1,2} and ( B={a, b} ? )
A ( cdot{(1, a),(1, b),(2, a),(b, b)} )
B – ( {(1,1),(2,2),(a, a),(b, b)} )
c. ( {(1, a),(2, a),(1, b),(2, b)} )
D cdot ( {(1,1),(a, a),(2, a),(1, b)} )
12
572The domain of the function ( boldsymbol{f}(boldsymbol{x})= ) ( sin ^{-1}left(log _{2}left(frac{x^{2}}{2}right)right) ) is
A. [-2,-1)( cup(1,2] )
B. (-2,-1]( cup[1,2] )
c. [-2,-1]( cup[1,2] )
D. (-2,-1)( cup(1,2) )
12
573Are the following set of ordered pairs functions? If so, examine whether the
mapping is injective or surjective:
(i)
( {(x, y): x text { is a person, } y text { is the mother } o )
(ii)
( {(a, b): a ) is a person, b is the ancestor o
12
574If ( f(x)=x^{3} ) and ( g(x)=sin 2 x, ) then
A ( cdot g[f(1)]=1 )
B. ( f(g(pi / 12)=1 / 8 )
c. ( g f(2)=sin 2 )
D. none of these
12
575Let ( boldsymbol{A}={boldsymbol{x}, boldsymbol{y}, boldsymbol{z}}, boldsymbol{B}={boldsymbol{a}, boldsymbol{b}, boldsymbol{c}} ) and ( boldsymbol{f} )
( A rightarrow B ) be defined by ( f(x)=a, f(y)= )
( boldsymbol{b}, boldsymbol{f}(boldsymbol{z})=boldsymbol{c} . ) This function is
A. surjective but not injective
B. injective but not surjective
c. bijective
D. none of these
12
576The relation ( R ) on the set ( Z ) of all integer
numbers defined by ( (x, y) epsilon R Leftrightarrow x-y )
is divisible by ( n ) is
A. Equivalence
B. Symmetric only
c. Reflexive only
D. Transitive only
12
577Identify the first component of an ordered pair (2,1)
( mathbf{A} cdot mathbf{1} )
B . 2
( c cdot-1 )
( D )
12
578Let ( boldsymbol{A}={mathbf{1}, mathbf{2}} ) and ( boldsymbol{B}={mathbf{2}, mathbf{3}, mathbf{4}}, ) then
( boldsymbol{A} times boldsymbol{A}={(mathbf{1}, mathbf{1}),(mathbf{1}, mathbf{2}),(mathbf{2}, mathbf{1}),(mathbf{2}, mathbf{2})} . ) If
true enter ( 1, ) or else enter 0
12
579ff ( f(x+y)=f(x y) ) for all ( x, y, in R, ) and
( boldsymbol{f}(mathbf{2 0 0 3})=mathbf{2 0 0 3}, ) then ( boldsymbol{f}(mathbf{2 0 0 0}) ) equals
A . 2003
B.
c. -2003
D. none of these
12
5808.
The ora
The graph of the function y=f(x) is symmetrica
rical about the
line x=2, then
[2004]
(a) f(x) = -f(-x)
(b) f(2+ x) = f(2-x)
(c) f(x) = f(-x) (d) f(x+2) = f(x-2)
12
581Find the domain of the function ( f )
defined by ( f(x)=s q r tleft(x^{2}+1right)+2 )
where sqrt is the square root
A ( . R-{2} )
в. ( R )
( c cdot 2 )
D. ( R-{1} )
12
582Numbers of real roots of ( boldsymbol{P}(boldsymbol{x})=mathbf{0}, ) is
( mathbf{A} cdot mathbf{1} )
B. 3
c. 5
D. 2
11
583Let ( f(x) ) be a real -valued function such that ( f(0)=frac{1}{2} ) and ( f(x+y)= )
( boldsymbol{f}(boldsymbol{x}) boldsymbol{f}(boldsymbol{a}-boldsymbol{y})+boldsymbol{f}(boldsymbol{y}) boldsymbol{f}(boldsymbol{a}-boldsymbol{x}) forall boldsymbol{x}, boldsymbol{y} in boldsymbol{R} )
then for some real ( a )
This question has multiple correct options
A. ( f(x) ) is a periodic function
B. ( f(x) ) is a constant function
c. ( f(x)=frac{1}{2} )
D. ( f(x)=frac{cos x}{2} )
12
584Let ( *^{prime} ) be the binary operation on the set
{1,2,3,4,5} defined by ( a *^{prime} b= )
H.C.F. of ( a ) and ( b ). Is the operation ( *^{prime} )
same as the operation ( * ) defined in above? Justify your answer
12
585Check whether ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a function defined by ( f(x)=frac{x^{2}+2 x+5}{x^{2}+x+1} ) is one
one or not.
12
586In the set of triangles in a plane the relation ‘is similar to’ is an equivalence relation. Prove12
587Let ( A={1,2,3,4,5}, B=N ) and ( f: A rightarrow )
( B ) be defined by ( f(x)=x^{2} )
Find the range of ( f ). Identify the type of function.
12
588In the problem below, ( f(x)=x^{2} ) and
( boldsymbol{g}(boldsymbol{x})=boldsymbol{4} boldsymbol{x}-boldsymbol{2} )

Find the following function: ( (boldsymbol{f}+boldsymbol{g})(boldsymbol{x}) )
A ( cdot x^{2}+4 x-2 )
B. ( x^{2}-4 x-2 )
c. ( x^{2}-4 x+2 )
D. ( x^{2}-2 )

11
58915.
nd f (x)=
0, x= 0 . Then for all
Let g(x)=1+x-[x] and f(x) = { 0, x=0
| 1, x>0 (20015)
x, f(g(x)) is equal to
(a) x (b) 1 (c) Ax) (d) g(x)
12
590If ( left|boldsymbol{x}^{2}-mathbf{2} boldsymbol{x}-mathbf{8}right|+left|boldsymbol{x}^{2}+boldsymbol{x}-mathbf{2}right|= )
( mathbf{3}|boldsymbol{x}+mathbf{2}|, ) then the set of all real values
of x is
в. [1,4]
( mathbf{c} cdot[-2,1] cup[4, infty] )
D. ( [-infty,-2] cup[1,4] )
11
591Let ( f ) be a function defined by ( f(x)= )
( mathbf{5} boldsymbol{x}^{2}+mathbf{2}, boldsymbol{x} in boldsymbol{R} . ) Find a such that ( boldsymbol{f}(boldsymbol{a})= )
22
12
5921.
(1984 – 3 Marks)
If y = f(x) = *** then
(a) x=f()
(b) (1)=3
(c) y increases with x for x<1
(d) is a rational function of x
12
593Find the values of ( x ) satisfying ( [x]-1+ )
( x^{2} geq 0 ; ) where [.] denotes the greatest
integer function.
A ( . x in(-infty,-sqrt{3}] cup[1, infty) )
B . ( x in(-infty,-sqrt{2}] cup[2, infty) )
c. ( x in(-infty,-3] cup[3, infty) )
D. ( x in(-infty,-sqrt{3}] cup[3, infty) )
11
594An equation that defines ( y ) as a function
of ( x ) is given. Solve for ( y ) in terms of ( x )
and replace ( y ) with the function notation
( boldsymbol{f}(boldsymbol{x}) )
( boldsymbol{x}-boldsymbol{2} boldsymbol{y}=mathbf{1 8} )
A ( cdot f(x)=frac{1}{2} x-18 )
B ( cdot f(x)=frac{1}{2} x-9 )
c. ( f(x)=-x+9 )
D. ( f(x)=-frac{1}{2} x+9 )
11
5958.
l. If the ranges of the
fog and gof are R, and R
(1994 – 2 Marks)
Let f(x) = sinr and g(x) = Inx ). If the ran
composition functions fog and gof are
respectively, then
(a) R1 = {u: -1Su<1}, R = {v: – < v<0}
11 {u:-00<u<0), R ={v: -15v 0}
(c) R1 = {u:-1<u<l}, R2 = {v: – <v<0}
Ri = {u:-1<u<1), R ={v: – < < 0}
12
596( operatorname{Let} boldsymbol{R}= )
{(1,3),(4,2),(2,4),(2,3),(3,1)} be a
relation on the set ( A={1,2,3,4} . ) The
relation ( boldsymbol{R} ) is
A. A function
B. Transitive
C. Not symmetric
D. Reflexive
12
597Let ( boldsymbol{A}={mathbf{9}, mathbf{1 0}, mathbf{1 1}, mathbf{1 2}, mathbf{1 3}} ) and ( boldsymbol{f}: boldsymbol{A} rightarrow )
( N ) be defined by ( f(n)= ) highest prime
factor of ( n, ) then its range is
A ( .{13} )
в. {3,5,11,13}
c. {11,13}
D. {2,3,5,11}
12
598Let ( boldsymbol{A}={boldsymbol{u}, boldsymbol{v}, boldsymbol{w}, boldsymbol{z}} ; boldsymbol{B}={mathbf{3}, mathbf{5}}, ) then
the number of relations from ( A ) to ( B ) is
A . 256
B. 1024
c. 512
D. 64
12
599For ( x in R, ) the range of the given expression ( frac{x^{2}+34 x-71}{x^{2}+2 x-7} ) is12
600A function ( R ) on the set ( N ) of natural
numbers is defined as ( R={(2 n, 2 n+ )
( mathbf{1}): boldsymbol{n} in boldsymbol{N}} )
The domain of ( R={2,4,6,8, ldots, ldots, ldots} )
A. True
B. False
12
601For the binary operation ( times_{10} ) on ( operatorname{set} S= )
( {1,3,7,9}, ) find the inverse of 3
12
602One root of the equation ( cos x-x+ )
( frac{1}{2}=0 ) lies in the interval:
A ( cdotleft(0, frac{pi}{2}right) )
B ( cdotleft(-frac{pi}{2}, 0right) )
c. ( left(frac{pi}{2}, piright) )
D. ( left(pi, frac{3 pi}{2}right) )
12
603Find the range of ( boldsymbol{f}(boldsymbol{x})= ) ( frac{sin left(pileft[x^{2}+1right]right)}{x^{4}+1} ) where [.] is greatest
integer function
A . [0,1]
в. [-1,1]
c. 0
D. None of these
11
604If ( boldsymbol{A}={boldsymbol{x}: boldsymbol{x} in boldsymbol{W}, boldsymbol{3} leq boldsymbol{x}<mathbf{6}}, boldsymbol{B}= )
{3,5,7} and ( C={2,4} ; ) find ( :(A- )
( boldsymbol{B}) times boldsymbol{C} )
12
605The number of possible surjection from ( boldsymbol{A}={1,2,3, dots n} ) to ( B={1,2} ) (where
( boldsymbol{n} geq mathbf{2}) ) is ( mathbf{6 2}, ) then ( boldsymbol{n}= )
( mathbf{A} cdot mathbf{5} )
B. 6
( c cdot 7 )
D.
12
606If a language of natural numbers has a binary regularly of 0 and ( 1, ) then which one of the following strings represents
the natural number ( 7 ? )
A . 1
в. 101
c. 110
D. 111
12
60719.
The number of points of intersection of two curves
y=2 sinx and y=5×2 + 2x + 3 is
(1994)
(a) 0 (0) 1 ( 2 (d) oo
2)
11
608If the binary operation ( * ) is on set of
integers ( Z ) is defined as
( a * b=a+2 b^{2}, ) then the value of ( (8 * )
( mathbf{3}) * mathbf{2} )
A . 26
B. 22
( c .32 )
D. 34
12
609The domain of the function ( boldsymbol{f}(boldsymbol{x})= )
( log _{10} log _{10}left(1+x^{3}right) ) is
A. ( (-1,+infty) )
(i) ( (-1,-infty) )
B. ( (0,+infty) )
( c cdot[0,+infty) )
D. (-1,0)
12
610If ( A ) and ( B ) have ( n ) elements in common,
then the number of elements common
to ( A times B ) and ( B times A ) is
A ( . n )
B. ( 2 n )
( c cdot n^{2} )
D.
12
611If ( f(x)=x+int_{0}^{1}left[x y^{2}+s^{2} yright] f(y) d y )
where ( x ) and ( y ) are independent varibles. Find ( f(x) )
A ( quad f(x)=x+frac{61}{119} x+frac{80}{119} x^{2} )
в. ( f(x)=frac{80 x^{2}+180 x}{119} )
c. ( f(x)=frac{80 x+180 x}{119} )
D. ( f(x)=frac{80 x^{2}+180 x}{19} )
12
612State the reason for the relation ( boldsymbol{R} ) on
the set {1,2,3} given by ( R= )
{(1,2),(2,1)} not to be transitive
12
613The domain of ( boldsymbol{f}(boldsymbol{x})=sqrt{log left(frac{boldsymbol{7} boldsymbol{x}-boldsymbol{x}^{2}}{mathbf{1 2}}right.}) )
is
в. ( (-infty, 4] )
c. ( [3, infty) )
()
D. [3,4]
12
614Let ( R ) be a reflexive relation in a finite
set having ( n ) elements and let there be
( m ) ordered pairs in ( R ). Then
A ( . m geq n )
в. ( m leq n )
c. ( m=n )
D. None of these
12
615Let ( boldsymbol{A}={1,2,3,4,5,6} . ) Let ( R ) be a relation on ( A ) defined by ( R={(a, b) )
( a, b in A, b text { is exactly divisible by a }} ) Find
the range of ( R )
12
616The set of real values of ( x ) satisfying the equality ( left[frac{mathbf{3}}{boldsymbol{x}}right]+left[frac{mathbf{4}}{boldsymbol{x}}right]=mathbf{5} ) (where [.]
denotes the greatest integer function) belongs to the interval ( left(a, frac{b}{c}right] ) where ( boldsymbol{b} )
( a, b, c in N ) and ( – ) is in its lowest form. ( boldsymbol{c} )
Find the value of ( a+b+c+a b c )
A . 25
B . 18
c. 20
D. 10
12
617Define a reflexive relation.12
618Let ( boldsymbol{f}(boldsymbol{x})=sin left(tan ^{-1} boldsymbol{x}right) . ) Then
( [boldsymbol{f}(-sqrt{mathbf{3}})], ) where [.] denotes the
greatest integer function, is
( ^{mathrm{A}}-frac{sqrt{3}}{2} )
B.
c. -1
D. none of these
11
619Let ( A={1,2} ) and ( B={3,4} . ) Find the
total number of relations from A into B.
12
620If ( mathbf{A}=left{x mid frac{pi}{6} leq x leq frac{pi}{3}right} ) and ( f(x)= )
( cos x-x(1+x), ) then ( f(A) ) is equal to
( mathbf{A} cdotleft[frac{pi}{6}, frac{pi}{3}right] )
( mathbf{B} cdotleft[-frac{pi}{3},-frac{pi}{6}right] )
( mathbf{C} cdotleft[frac{1}{2}-frac{pi}{3}left(1+frac{pi}{3}right), frac{sqrt{3}}{2}-frac{pi}{6}left(1+frac{pi}{6}right)right] )
( mathrm{D} cdotleft[frac{1}{2}+frac{pi}{3}left(1-frac{pi}{3}right), frac{sqrt{3}}{2}+frac{pi}{6}left(1-frac{pi}{6}right)right] )
12
621Then
9. Letf:( – be given by f(x)= (log(sec x + tan x)3.
(JEE Adv. 2014)
(a) f(x) is an odd function
(b) f(x) is one-one function
(c) f(x) is an onto function
(d) f(x) is an even function
12
622Let ( boldsymbol{f}(boldsymbol{x})= )
( max {1+sin x, 1,1-cos x}, x in )
( [0,2 pi] ) and ( g(x)= )
( max {1,|x-1|}, x in R, ) then
This question has multiple correct options
A. ( g(f(0))=1 )
B. ( g(f(1))=1 )
c. ( f(g(1))=1 )
D. ( f(g(0))=sin 1 )
12
623If ( P={a, b, c} ) and ( Q={b, c, d} ; ) find number of elements in ( (boldsymbol{P} cap boldsymbol{Q}) times boldsymbol{P} )12
624Ordered pairs ( (x, y) ) and (-1,-1) are
equal if ( boldsymbol{y}=-mathbf{1} ) and ( boldsymbol{x}= )
( mathbf{A} cdot mathbf{1} )
B. –
c. 0
D. 2
12
625Consider the equation ( boldsymbol{p}=mathbf{5}-|mathbf{2 q – 3}| )
f ( p=|r|+5, ) then number of possible
ordered pair ( (r, q) ) is?
12
626ff ( f(x) ) is a function such that ( f(x y)= ) ( boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y}): boldsymbol{x}, boldsymbol{y}[boldsymbol{R}, text { then } boldsymbol{f}(mathbf{1} / boldsymbol{e}) ) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. 3
11
627( = )
( + )
( +infty )
11
628Find the ranges of the following
functions:
( f(x)=1-x-x^{2} )
( f(x)=frac{3 x+7}{x-8} )
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{x}+boldsymbol{8} )
( f(x)=frac{4 x-7}{x-3} )
12
629Find domain and range of
( boldsymbol{R}={(boldsymbol{a}, boldsymbol{b}) / boldsymbol{a} in boldsymbol{N}, boldsymbol{a}<mathbf{5} text { and } boldsymbol{b}=mathbf{2}} )
12
630Find ( g circ f ) and ( f circ g ) when ( f: R rightarrow R ) and ( g: R rightarrow R ) are defined by ( f(x)= )
( 2 x+3 ) and ( g(x)=x^{2}+5 )
11
631State the reason why the relation ( boldsymbol{S}= )
( (a, b) in R times R: a leq b^{3} ) on the set ( R ) of
real numbers is not transitive.
12
632Let
( |X| ) denote the number of elements in a set ( X ).let ( S={1,2,3,4,5,6} ) be a
sample space,where each elements is equally likely to occur.If ( A ) and ( B ) are
independent events associated with ( S ) then the number of ordered pairs ( (A, B) ) such that ( 1 leq|B|<|A| ) equals.
12
633The domain of the function ( f(x)=^{24-x} )
( boldsymbol{C}_{3 x-1}+^{40-6 x} boldsymbol{C}_{8 x-10} ) is
A ( cdot{2,3} )
3 ( } )
в. {1,2,3}
D. None of these
12
634Write the domain of the real function
( boldsymbol{f}(boldsymbol{x})=sqrt{boldsymbol{x}-[boldsymbol{x}]} )
12
635Show that the relation ( geq ) on the set ( mathbb{R} ) of
all real numbers is reflexive and
transitive but not symmetric.
12
636f ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{A}, boldsymbol{g}: boldsymbol{A} rightarrow boldsymbol{A} ) are two
bijections, then prove that ( f circ g ) is an
injection.
12
63719. Let f(x)=x|x|and g(x)=sin x.
Statement-1: gof is differentiable at x = 0 and its derivative
is continuous at that point.
Statement-2: gof is twice differentiable at x = 0. [2009]
(a) Statement-1 is true, Statement-2 is true;
Statement-2 is not a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is false.
© Statement-1 is false, Statement-2 is true.
Statement-1 is true, Statement-2 is true;
Statement-2 is a correct explanation for Statement-1.
12
638( f(x)left(x+frac{1}{x}right)=x^{2}+frac{1}{x^{2}}, ) find ( fleft(x^{3}right) )12
639If ( n(A)=4, n(B)=3, n(A times B times C)= )
( 24, ) then ( n(C) ) is equal to
A . 288
в.
( c cdot 2 )
D. 12
12
640If
( int frac{d x}{sqrt[3]{cos ^{12} x+3 cos ^{10} x+3 cos ^{8} x+cos ^{6}}} )
then ( f(x) ) is
A. Bounded & periodic
B. Bounded & aperiodic
c. Unbounded & periodic
D. Unbounded & aperiodic
12
641If ( f(x)=frac{x^{2}-x}{x^{2}+2 x}, ) find the domain of ( f(x) ) Show that ( f ) is one-one.12
642Which of the following statements are incorrect?

If ( f(x) ) and ( g(x) ) are one to one then
( boldsymbol{f}(boldsymbol{x})+boldsymbol{g}(boldsymbol{x}) )
II If ( f(x) ) and ( g(x) ) are one to one then
( boldsymbol{f}(boldsymbol{x}) cdot boldsymbol{g}(boldsymbol{x}) )
III If ( f(x) ) is odd then it is necessarily
one to one
A. I and II only
B. I land III only
( mathrm{c} ). ।, ॥ and ॥|
D. None of the above

12
643A function ( f ) is defined for all positive
integers and satisfies ( boldsymbol{f}(mathbf{1})=mathbf{2 0 0 5} ) and
( boldsymbol{f}(1)+boldsymbol{f}(2)+ldots+boldsymbol{f}(boldsymbol{n})=boldsymbol{n}^{2} boldsymbol{f}(boldsymbol{n}) ) for
all ( n>1 . ) Find the value of ( f(2004) )
12
644Find ( g circ f ) and ( f circ g ) when ( f: R rightarrow R )
and ( g: R rightarrow R ) are defined by ( f(x)=x )
and ( g(x)=|x| )
11
645The relation “congruence modulo ( m ” ) is:
A. reflexive only
B. symmetric only
c. transitive only
D. an equivalence relation
12
646If ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}, boldsymbol{g}(boldsymbol{x})=boldsymbol{2} boldsymbol{x}^{2}+mathbf{1} ) and ( boldsymbol{h}(boldsymbol{x})= )
( x+1 ) then ( (text {hogof})(x) ) is equal to
A ( cdot x^{2}+2 )
B. ( 2 x^{2}+1 )
c. ( x^{2}+1 )
D. ( 2left(x^{2}+1right) )
12
647If ( f(x)=left(frac{x-1}{x+1}right), ) then which of the following statements is/are correct? This question has multiple correct options
( ^{mathrm{A}} cdot fleft(frac{1}{x}right)=f(x) )
B ( cdot fleft(frac{1}{x}right)=-f(x) )
c. ( fleft(-frac{1}{x}right)=frac{1}{f(x)} )
D ( fleft(-frac{1}{x}right)=-frac{1}{f(x)} )
11
648Let there be a function ( f(x) ) that is
greater than the cube of ( x ) when
subtracted to the fourth root of the sum
of 2 and square of 2 multiplied with ( x ) Then the function id depicted by
A ( cdot f(x)sqrt[4]{2+4 x^{2}}-x^{3} )
C ( cdot f(x) geq sqrt[4]{2+4 x^{2}}-x^{3} )
D. ( f(x) leq sqrt[4]{2+2 x^{2}}-x^{3} )
12
649The total number of injective mappings from a set with ( m ) elements to a set
with ( n ) elements, ( m leq n ), is
( mathbf{A} cdot m^{n} )
B . ( n^{text {m }} )
c. ( frac{n !}{(n-m) !} )
D. ( n ! )
12
650Let ( boldsymbol{E}={1,2,3,4} ) and ( boldsymbol{F}={1,2} ) then
the number of onto functions from E to
is
A . 14
B . 16
c. 12
D. 8
12
651( (x, y) ) and ( (p, q) ) are two ordered pairs. Find the values of ( p ) and ( y, ) if ( (4 y+ )
( mathbf{5}, mathbf{3} boldsymbol{p}-mathbf{1})=(mathbf{2 5}, boldsymbol{p}+mathbf{1}) )
A. ( p=0, y=5 )
В. ( p=1, y=5 )
c. ( p=0, y=1 )
D. ( p=1, y=1 )
12
652The domain of ( f(x)=cos (log x) ) is
( A cdot(-infty, infty) )
в. (-1,1)
c. ( (0, infty) )
D. ( (1, infty) )
12
653If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) and ( boldsymbol{g}: boldsymbol{R} rightarrow boldsymbol{R} ) are defined
by ( f(x)=2 x+3 ) and ( g(x)=x^{2}+7 )
then the values of ( x ) such that
( g(f(x))=8 ) are:
A . 1,2
в. -1,2
c. -1,-2
D. 1,-2
12
654( boldsymbol{x} * boldsymbol{y}=sqrt{frac{(boldsymbol{x}+boldsymbol{y})left(boldsymbol{y}^{2}-mathbf{1 2} boldsymbol{x}right)}{(boldsymbol{x}-mathbf{2})(boldsymbol{y}-mathbf{7})}} ) then what
will be the value of ( 5 * 9 ? )
( A cdot 7 )
B. 8
c. 10
D.
12
655The domain of the function ( boldsymbol{f}(boldsymbol{x})= ) ( (sqrt{1-sqrt{1-sqrt{1-x^{2}}}}) ) is
A . [0,1]
в. [-1,1]
( c cdot(-infty, infty) )
D. (-1,1)
12
656Range of the function ( boldsymbol{f}(boldsymbol{x})= )
( sin ^{2}left(x^{4}right)+cos ^{2}left(x^{4}right) ) is
( A cdot(-infty, infty) )
B . {1}
c. (-1,1)
D. (0,1)
12
657The domain of function satisfying ( f(x)+fleft(x^{-1}right)=frac{x^{3}+1}{x}, ) is
A. an empty set
B. a singleton
c. a finite set
D. an infinite set
12
658If ( * ) is defined by ( a * b=a-b^{2} ) and ( oplus ) is
defined by ( a oplus b=a^{2}+b, ) where ( a ) and
( b ) are integers, then ( (3 oplus 4) * 5 ) is equal
to
A . 164
B. 38
c. -12
D. -28
E .144
12
659Let ( R= ) the set of real numbers, ( Z= )
the set of integers, ( N= ) the set of
natural numbers. If ( S ) be the solution
set of the equation ( (x)^{2}+[x]^{2}=(x- )
( mathbf{1})^{2}+[boldsymbol{x}+mathbf{1}]^{2}, ) where ( (boldsymbol{x})= ) the least
integer greater than or equal to ( x ) and ( [x]= ) the greatest integer less than or
equal to ( x, ) then
A ( . S=R )
в. ( S=R-Z )
c. ( S=R-N )
D. ( S=phi )
11
660If ( f(x)=12 x^{5} ) and ( g(x)=-3 x^{2} )
determine ( (boldsymbol{f} cdot boldsymbol{g})(-1) )
( [text {Given }:(boldsymbol{f} cdot boldsymbol{g})(boldsymbol{x})=boldsymbol{f}(boldsymbol{x}) cdot boldsymbol{g}(boldsymbol{x})] )
A . 36
B . 42
( c cdot 32 )
D. -36
11
66113.
The largest interval lying in

TU
for which the function,
f(x)=4-+*+cos
-1+log (cos x), is defined, is
[2007]
·m (11)
(0.5)
(©) [0, 1]
12
662The function ( boldsymbol{f}:(-infty,-1] rightarrowleft(0, e^{5}right) )
defined by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{e}^{boldsymbol{x}^{3}-mathbf{3} boldsymbol{x}+mathbf{2}}, ) is
A. many-one and onto
B. many-one and into
c. one-one and onto
D. one-one and into
12
663If ( f(x)=x^{2}-3 x-frac{3}{x}+frac{1}{x^{2}}, ) then
( fleft(frac{1}{x}right) ) is
( A )
в. ( f(x) )
c. ( f(-x) )
D . ( -f(x) )
12
664Determine the domain and range of the relation R defined by ( boldsymbol{R}={(boldsymbol{x}, boldsymbol{x}+mathbf{5}) )
( boldsymbol{x} in{mathbf{0}, mathbf{1}, mathbf{2}, mathbf{3}, mathbf{4}, mathbf{5}}} )
12
665If ( f(x y-1)=f(-x) f(-y)+f(x)- )
( boldsymbol{y}+mathbf{1} ) and if ( boldsymbol{f}(mathbf{0})=mathbf{1}, ) then ( boldsymbol{f}(boldsymbol{x})=? )
( mathbf{A} cdot x-1 )
B. ( -x+1 )
( c cdot-x-1 )
D. ( x+1 )
E ( .1-2 x )
12
666f ( x, y ) are rational numbers such that
( (x+2 y)+(x-3 y) sqrt{6}= )
( (x-y-2) sqrt{5}+(2 x+y-2), ) then
A ( . x+y=6 )
В. ( x+y=5 )
c. ( x+y=4 )
D. ( x+y=3 )
11
667Let ( R ) be a relation in ( N ) defined as ( R= )
( {(x, y) in N times N: x+2 y=39} ) Find
the domain, co-domain and range of ( boldsymbol{R} )
Also write ( R^{-1} ) in roster form.
12
668Find ( boldsymbol{f} circ boldsymbol{g} ) and ( boldsymbol{g} circ boldsymbol{f}, ) if ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}| )
( g(x)=sin x )
12
669( operatorname{Let} A={1,2,3,4} ) and ( B={x, y, z} )
Consider the subset ( boldsymbol{R}= )
( {(1, x),(1, y),(2, z),(3, x)} ) of ( A times B . ) Is
R, a relation from A to B? If yes, find domain and range of R. Draw arrow
diagram of ( R ) and represent ( R ) in a tabular form.
12
670Determine all ordered pairs that satisfy ( (x-y)^{2}+x^{2}=25, ) where ( x ) and ( y ) are
integers and ( x geq 0 . ) Find the number of
different values of ( y ) that occur
A . 3
B. 4
( c .5 )
D. 6
12
671( A, B, C ) are three sets such that ( boldsymbol{n}(boldsymbol{A})=boldsymbol{2}, boldsymbol{n}(boldsymbol{B})=boldsymbol{3}, boldsymbol{n}(boldsymbol{C})=boldsymbol{4}, ) If ( boldsymbol{P}(boldsymbol{x}) )
denotes power set of ( boldsymbol{X}, boldsymbol{k}= ) ( frac{boldsymbol{n}(boldsymbol{P}(boldsymbol{P}(boldsymbol{C})))}{boldsymbol{n}(boldsymbol{P}(boldsymbol{P}(boldsymbol{A}))) times boldsymbol{n}(boldsymbol{P}(boldsymbol{P}(boldsymbol{B})))} ).then find
the Sum of digits of ( k )
12
672The domain of the function ( y= )
( -2 x^{2}+6 x+5 ) is
( mathbf{A} cdot x in R )
в. ( x in(-2,0] )
c. ( x in[1,3) )
D. ( x in(1,3] )
12
673( mathbf{n}[mathbf{0}, mathbf{1}] ) Lagranges Mean Value theorem is NOT applicable to
A. ( f(x)=x|x| )
B . ( f(x)=|x| )
c. ( f(x)=x )
D. none of these
11
674Let ( A ) and ( B ) be sets. Show that ( f: A times )
( B longrightarrow B times A ) such that ( f(a, b)=(b, a) )
is bijective function
12
675ff ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) and ( boldsymbol{g}: boldsymbol{R} rightarrow boldsymbol{R} ) are given by
( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}| ) and ( boldsymbol{g}(boldsymbol{x})=[boldsymbol{x}] ) for each ( boldsymbol{x} in )
( R, ) then
( {boldsymbol{x} in boldsymbol{R}: boldsymbol{g}(boldsymbol{f}(boldsymbol{x})) leq boldsymbol{f}(boldsymbol{g}(boldsymbol{x}))}= )
A ( cdot Z cup(-infty, 0) )
В ( cdot(-infty, 0) )
( c . z )
D. ( R )
12
676Show that the relation ( R ) defined in the
( operatorname{set} A ) of all polygons as ( R=left{left(P_{1}, P_{2}right):right. )
( P_{1} ) and ( P_{2} ) have same number of sides
}( , ) is an equivalence relation. What is the set of all elements in ( A ) related to
the right angle triangle ( T ) with sides 3,4
and ( 5 ? )
12
677Let ( R ) be a relation from a set ( A ) to a set ( B )
then
( mathbf{A} cdot R=A cup B )
в. ( R=A cap B )
c. ( R subseteq A times B )
D. ( R subseteq B times A )
12
678If ( x lambda y=2 x-y ) and ( x nu y=3 x times y )
then ( (5 nu 2) lambda 9 )
12
679If ( f(x)=a x^{2}+b x+c ) and ( f(0)=1 )
and ( boldsymbol{f}(-1)=mathbf{3},(boldsymbol{a}-boldsymbol{b})= )
( mathbf{A} cdot mathbf{3} )
B. 1
( c cdot 2 )
D.
E . -1
12
680If ( A={2,3} ) and ( B={1,2}, ) find ( A times )
( boldsymbol{B} )
A ( cdot{(2,1),(2,2),(3,1),(3,2)} )
в. {(2,1),(2,1),(3,1),(3,2)}
( mathbf{c} cdot{(2,1),(2,2),(2,1),(3,2)} )
D. ( {2,1),(2,2),(3,1),(2,2)} )
12
681The curve ( y^{2}=(x-1)(x-2)^{2} ) is not
defined for
( mathbf{A} cdot x geq 1 )
B. ( x geq 2 )
c. ( x<2 )
D. ( x<1 )
12
682Which one of the following relations on ( R( ) set of real numbers) is an
equivalence relation
( mathbf{A} cdot a R_{1} b Leftrightarrow|a|=|b| )
в. ( a R_{2} b Leftrightarrow a geq b )
( mathbf{c} cdot a R_{3} b Leftrightarrow a ) divides ( b )
D ( cdot a R_{4} b Leftrightarrow a<b )
12
683Calculate the numbers of times ( boldsymbol{f}(|boldsymbol{x}|) ) ( operatorname{crosses} boldsymbol{x}-boldsymbol{a x i s} ) where ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}-1 )
( A cdot 2 )
B.
( c .3 )
D. None of these
12
684If ( f(t)=frac{1+t}{1-t}, ) then ( f(-t) ) is
A ( cdot frac{1}{f(t)} )
в. ( frac{1}{f(-t)} )
c. ( f_{t} frac{1}{t} )
D. ( fleft(-frac{1}{t}right) )
E. ( 1+f(t) )
11
685The range of ( f(x)=^{16-x} C_{2 x-1}+^{20-3 x} )
( C_{4 x-5} ) is
B . {0,728}
C ( cdot{728,1617} )
D. None of these
12
686Classify the following function as injection, surjection or bijection:
( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R}, ) defined by ( boldsymbol{f}(boldsymbol{x})=mathbf{1}+boldsymbol{x}^{2} )
12
687On ( R-{1}, ) a binary operation ( * ) is
defined by ( a * b=a+b-a b . ) Prove
that ( * ) is commutative.
12
688f ( 2 f(x)+3 . fleft(frac{1}{2}right)=x^{2}-1 ) then find
( f(x) )
12
6897.
If 2a + 3b +6C = 0, then at least one root of the equation
ax2 + bx+c = 0 lies in the interval
[2004]
(a) (1,3) (b) (1,2) (c) (2,3) (d) (0,1)
12
690The domain of the function ( boldsymbol{f}(boldsymbol{x})= ) ( sqrt{10-sqrt{x^{4}-21 x^{2}}} ) is
B . ( [sqrt{-21}, sqrt{21}] )
c. [-5,5]
D. ( (-5, infty) )
12
691Consider ( boldsymbol{f}: boldsymbol{N} rightarrow boldsymbol{N}, boldsymbol{g}: boldsymbol{N} rightarrow boldsymbol{N} )
defined as ( boldsymbol{f}(boldsymbol{x})=boldsymbol{2} boldsymbol{x}, boldsymbol{g}(boldsymbol{x})=mathbf{3} boldsymbol{x}+boldsymbol{4} )
find ( boldsymbol{f} boldsymbol{o} boldsymbol{g}(boldsymbol{x}) )
12
692Check the commutativity and associativity of the following binary operation:
( *^{prime} ) on ( Q ) defined by ( a * b=(a-b)^{2} ) for
all ( boldsymbol{a}, boldsymbol{b} in boldsymbol{Q} )
12
693f ( boldsymbol{f}:[mathbf{0}, boldsymbol{pi}] rightarrow boldsymbol{R}, boldsymbol{f}(boldsymbol{x})=cos boldsymbol{x}, ) then ( boldsymbol{f} ) is
A. one-one onto
B. one-one into
c. many-one onto
D. many-one into
12
694If ( R ) is the relation ‘less than’ from ( A= )
{1,2,3,4,5} to ( B={1,4}, ) the set of
ordered pairs corresponding to ( R ), then the inverse of ( boldsymbol{R} ) is
A ( cdot{(3,1),(3,2),(3,3)} )
в. {(4,1),(4,2),(4,3)}
c. {(4,3),(4,4),(4,5)}
D. {(1,3),(2,4),(3,5)}
12
695Let ( A={1,2,3} . ) Then number of
relations containing (1,2) and (1,3) which are reflexive and symmetric but not transitive is
A .
B. 2
( c cdot 3 )
D. 4
12
696Give examples of two one-one functions
( f_{1} ) and ( f_{2} ) from ( R ) to ( R ) such that ( f_{1}+ )
( boldsymbol{f}_{2}: boldsymbol{R} rightarrow boldsymbol{R}, ) defined by ( left(boldsymbol{f}_{1}+boldsymbol{f}_{2}right)(boldsymbol{x})= )
( f_{1}(x)+f_{2}(x) ) is not one-one.
12
697Let ( R ) be the equivalence relation on the
( operatorname{set} Z ) of integers given by ( boldsymbol{R}= )
( {(a, b): 2 text { divides } a-b} . ) Write the
equivalence class {0}
12
698Let ( boldsymbol{f}:{mathbf{1}, mathbf{3}, mathbf{4}} rightarrow{mathbf{1}, mathbf{2}, mathbf{5}} ) and ( boldsymbol{g} )
{1,2,5}( rightarrow{1,3} )
given by ( boldsymbol{f}={(mathbf{1}, mathbf{2}),(mathbf{3}, mathbf{5}),(mathbf{4}, mathbf{1})} ) and
( boldsymbol{g}={(1, mathbf{3}),(mathbf{2}, mathbf{3}),(mathbf{5}, mathbf{1})} ) write down
gof
12
699Let ( A ) be the set of all points in a plane
and let 0 be the origin. Show that the relation ( boldsymbol{R}={(boldsymbol{P}, boldsymbol{Q}): boldsymbol{P}, boldsymbol{Q} in boldsymbol{A} ) and
( boldsymbol{O P}=boldsymbol{O Q}} ) is an equivalence relation.
12
700The relation ‘is a subset of in a set of
sets is not an equivalence relation.
Prove.
12
701The set of values of ( x ) for which the
inequality ( [boldsymbol{x}]^{2}-mathbf{5}[boldsymbol{x}]+boldsymbol{6} leq mathbf{0} ) (where [
denote the greatest integral function) holds good are
A ( .2 leq[x]<3 )
в. ( 2 leq x<4 )
c. ( 2 leq[x] leq 3 )
D. ( (b) ) and ( (c) ) both
11
702Which of the following functions (is) are injective map(s)?
A ( cdot f(x)=x^{2}+2, x in(-infty, infty) )
B. ( f(x)=|x+2|, x in[-2, infty) )
C. ( f(x)=(x-4)(x-5), x in(-infty, infty) )
D. ( f(x)=frac{4 x^{2}+3 x-5}{4+3 x-5 x^{2}}, x in(-infty, infty) )
12
703Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be defined as ( boldsymbol{f}(boldsymbol{x})= )
( frac{2 x-3}{4} . ) Find ( f oleft(f^{-1}right) )
12
704If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) and ( boldsymbol{g}: boldsymbol{R} rightarrow boldsymbol{R} ) are
functions defined by ( f(x)=3 x- )
( mathbf{1} ; boldsymbol{g}(boldsymbol{x})=sqrt{boldsymbol{x}+boldsymbol{6}}, ) then the value of ( (boldsymbol{g} circ )
( left.boldsymbol{f}^{-1}right)(mathbf{2 0 0 9}) ) is
A . 26
B . 29
c. 16
D. 15
12
705If the cardinality of a set ( A ) is 4 and that
of a set ( B ) is 3 , then what is the cardinality of the ( operatorname{set} A Delta B )
A .
B. 5
( c cdot 7 )
D. Cannot be determined
12
706Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{A}=left{boldsymbol{y}: boldsymbol{0} leq boldsymbol{y}<frac{boldsymbol{pi}}{2}right} ) be a
function such that ( f(x)=tan ^{-1}left(x^{2}+right. )
( boldsymbol{x}+boldsymbol{k}), ) where ( boldsymbol{k} ) is a constant. The value
of ( k ) for which ( f ) is an onto function is
A . 1
B.
c. ( frac{1}{4} )
D. none of these
12
707( operatorname{Given} f(x)=frac{a x+b}{x+1}, lim _{x rightarrow infty} f(x)=1 ) and
( lim _{x rightarrow 0} f(x)=2, ) then ( f(-2) ) is
( A cdot 0 )
B.
( c cdot 2 )
( D )
11
708Given two functions ( f(x) ) and ( g(x) ) such that ( f(x)=sin (a r c t a n x), g(x)= )
( tan (arcsin x), ) and ( 0 leq x<frac{pi}{2} . ) The
value of the composite function ( fleft(gleft(frac{pi}{10}right)right) ) is:
( mathbf{A} cdot 0.314 )
в. 0.354
c. 0.577
D. 0.707
E . 0.866
12
709The range of the function ( f(x)= )
( sin ^{-1}left(log _{2}left(-x^{2}+2 x+3right)right) ) is
A. ( left[-frac{pi}{2}, frac{pi}{2}right] )
B. ( left[-frac{pi}{2}, 0right] )
c. ( left[0, frac{pi}{2}right] )
D. [-1,1]
12
710Let ( * ) be an operation such that ( boldsymbol{a} * boldsymbol{b}= )
LCM of ( a ) and ( b ) defined on the ( operatorname{set} A= )
( {1,2,3,4,5} . ) Is ( * ) a binary operation?
Justify your answer.
12
711A relation ( R ) is defined on the set ( Z ) of
integers as follows: ( mathrm{R}=(x, y) in boldsymbol{R}: x^{2}+ )
( y^{2}=25 . ) Express ( R ) and ( R^{-1} ) as the sets
of ordered pairs and hence find their
respective domains
( mathbf{A} cdot mathbf{0} )
B. Domain of ( R={0,pm 3}= ) domain of ( R^{-1} ).
C. Domain of ( R={0,pm 3,pm 4}= ) domain of ( R^{-1} )
D. Domain of ( R={0,pm 3,pm 4,pm 5}= ) domain of ( R^{-1} )
12
712ff ( (x)=log (x-2)+log (x-3) ) and
( phi(x)=log (x-2)(x-3) )
Are both functions equal?
12
713If ( fleft(x_{1}right)-fleft(x_{2}right)=fleft(frac{x_{1}-x_{2}}{1-x_{1} x_{2}}right) ) for
( boldsymbol{x}_{1}, boldsymbol{x}_{2} in[-1,1] ) then ( boldsymbol{f}(boldsymbol{x}) ) is
This question has multiple correct options
( ^{mathbf{A}} cdot log left(frac{1-x}{1+x}right) )
B. ( tan ^{-1}left(frac{1-x}{1+x}right) )
( ^{mathrm{c}} cdot log left(frac{1+x}{1-x}right) )
D. ( tan ^{-1}left(frac{1+x}{1-x}right) )
11
714Given, ( f(x)= ) sum of all natural
numbers which can divide ( x ) completely Find ( : $ $ f(6) $ $ )
12
715If a binary operation is defined ( a star b= )
( a^{b} ) then ( 2 star 2 ) is equal to:
A . 4
B. 2
c. 9
D.
12
716If ( f(x) ) is continuous such that
( |boldsymbol{f}(boldsymbol{x})| leq mathbf{1}, forall boldsymbol{x} in boldsymbol{R} ) and ( boldsymbol{g}(boldsymbol{x})= )
( frac{e^{f(x)}-e^{|f(x)|}}{e^{f(x)}+e^{|f(x)|}} ) then range of ( g(x) ) is
A ( cdot[0,1] )
B. ( left[0, frac{e^{2}+1}{e^{2}-1}right] )
( ^{mathbf{C}} cdotleft[0, frac{e^{2}-1}{e^{2}+1}right] )
D. ( left[frac{1-e^{2}}{1+e^{2}}, 0right] )
12
717Let ( A={9,10,11,12,13} ) and let ( f ) ( A rightarrow N ) be defined by ( f(n)= ) the
highest prime factor of ( n ). Find the
range of ( boldsymbol{f} )
12
718Find number of all such functions ( boldsymbol{y}= )
( f(x) ) which are one-one?
A . 0
B . ( 3^{5} )
( mathrm{c} cdot^{5} P_{3} )
D. ( 5^{3} )
12
719Mark the incorrect statement by the
graph of the function ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}| )
A. ( f(0)=0 )
B. ( f(-4)=4 )
c. ( f(4)=-4 )
D. The domain of ( f(x) ) is all real numbers
12
720Let ( x in N ) Let ( x ) be a perfect square. Let
( f(x)= ) the quotient when ( x ) is divided
by 5 and ( g(x) ) the remainder when ( x ) is divided by 5. Then ( sqrt{boldsymbol{x}}=boldsymbol{f}(boldsymbol{x})+boldsymbol{g}(boldsymbol{x}) )
holds for ( x ) equal to
This question has multiple correct options
( mathbf{A} cdot mathbf{0} )
B . 16
c. 25
D. none of these
12
721The number of binary operation on ( {1,2, ) ( 3 ldots n} ) is..
( mathbf{A} cdot 2^{n} )
B ( cdot n^{2} )
c. ( n^{3} )
D. ( n^{2} )
12
722ff ( f(x)=x^{2}+2 x-1, ) find ( f(2) )
( A cdot 3 )
B. 5
( c cdot 7 )
D.
12
723If ( boldsymbol{f}:left[mathbf{0}, frac{boldsymbol{pi}}{mathbf{2}}right] rightarrow[mathbf{0}, infty] ) be a function
defined by ( y=sin left(frac{x}{2}right), ) then ( f ) is
A . Injective
B. Surjective
c. Bijective
D. None of these
12
724Check the commutativity and associativity of the following binary operation:
( *^{prime} ) on ( Q ) defined by ( a * b=a b+1 ) for all
( boldsymbol{a}, boldsymbol{b} in boldsymbol{Q} )
12
725Let ( S ) be the set of all real numbers
except -1 and let ( ^{prime} *^{prime} ) be an operation defined by ( a * b=a+b+a b ) for al
( a, b in S . ) Determine whether ( ^{prime} *^{prime} ) is a
binary operation on ( S . ) If yes, check its commutativity and associativity. Also, solve the equation ( (2 * x) * 3=7 )
12
726ff ( (x)=left{begin{array}{l}x^{2} x geq 0 \ x x<0end{array}right. )
then ( (f o f)(x) ) is given by
A ( cdot x^{2} ) for ( x geq 0 ) and ( x ) for ( x<0 )
B. ( x^{4} ) for ( x geq 0 ) and ( x^{2} ) for ( x<0 )
c. ( x^{4} ) for ( x geq 0 ) and ( -x^{2} ) for ( x<0 )
D. ( x^{4} ) for ( x geq 0 ) and ( x ) for ( x<0 )
12
727If ( f(x)=cos left[frac{1}{2} pi^{2}right] x+sin left[frac{1}{2} pi^{2}right] x )
( [x] ) denoting the greatest integer
function, then
A. ( f(0)=0 )
в. ( fleft(frac{pi}{3}right)=frac{1}{4} )
c. ( fleft(frac{pi}{2}right)=1 )
12
728( operatorname{Given}(boldsymbol{a}-mathbf{2}, boldsymbol{b}+mathbf{3})=(mathbf{6}, mathbf{8}), ) are equal
ordered pair. Find the value of ( a ) and ( b )
A ( . a=8 ) and ( b=5 )
B. ( a=8 ) and ( b=3 )
c. ( a=5 ) and ( b=5 )
12
729Consider the two quadratic
polynomials:
( C_{a}: y=frac{x^{2}}{4}-a x+a^{2}+a-2 ) and
( C: y=2-frac{x^{2}}{4} )
If the origin lies between the zeroes of
the polynomial ( C_{a}, ) then the number of
integral value(s) of ( ^{prime} boldsymbol{a}^{prime} ) is
A.
B. 2
( c .3 )
D. more than 3
11
730If ( f(n+1)=f(n) ) for all ( n in )
( N, f(7)=5 ) then ( f(35)= )
A . 25
B . 49
( c .35 )
D. 5
11
731If ( f(x)=frac{1}{1-x}, x neq 0,1 ) then the graph
of the function ( boldsymbol{y}=boldsymbol{f}{boldsymbol{f}(boldsymbol{f}(boldsymbol{x}))}, boldsymbol{x}>1 )
is
A . a circle
B. an ellipse
c. a straight line
D. a pair of straight lines
12
73218.
(C) R {-1, -2,-
Let E= {1,2,3,4} and F= {1, 2}. Then the number of
functions from E to Fis
(2) 14 (6) 16 (c) 12 (d) 8
(20015)
12
733( boldsymbol{y}=sin ^{-1}left[log _{3}left(frac{boldsymbol{x}}{mathbf{3}}right)right] Rightarrow-1 leq )
( log _{3}left(frac{boldsymbol{x}}{mathbf{3}}right) leq mathbf{1} )
12
734MISCELLULUU
22. Let R = {(3, 3), (6, 6), (9,9), (12, 12), (6, 12), (3,9).
(3, 12), (3,6)be a relation on the set
A = {3, 6, 9, 12). The relation is
[2005]
(a) reflexive and transitive only
(b) reflexive only
(c) an equivalence relation
(d) reflexive and symmetric only
12
735ff ( f(x)+f(1-x)=10, ) then the value
of ( boldsymbol{f}left(frac{1}{10}right)+boldsymbol{f}left(frac{2}{10}right)+ldots ldots ldots+boldsymbol{f}left(frac{9}{10}right) ) is
A . 45
B . 50
( c .90 )
D. Cannot be determined
11
736( f(x)=x^{2}-3 x-4 ) find ( f(2) )12
737Find the total number of binary
operations on ( {a, b} )
12
738The given diagram represents
A. An onto function
B. A constant function
c. An one-one function
D. Not a function
12
739Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}, boldsymbol{g}(boldsymbol{x})=tan boldsymbol{x} ) and ( boldsymbol{h}(boldsymbol{x})= )
( ln x )
For ( boldsymbol{x}=frac{sqrt{boldsymbol{pi}}}{mathbf{2}}, ) what is the value of
( [boldsymbol{h o}(boldsymbol{g} boldsymbol{o} boldsymbol{f})(boldsymbol{x})] ? )
( mathbf{A} cdot mathbf{0} )
B.
( c cdot frac{pi}{4} )
D.
11
740If ( f: R rightarrow R ) be a differentiable function and ( f(0)=0 ) and ( f^{prime}(0)=1 ) then
( lim _{x rightarrow 0} frac{1}{x}left[f(x)+fleft(frac{x}{2}right)+ldots+fleft(frac{x}{100}right)right] )
equals
A . 1
B cdot ( 1+frac{1}{2}+frac{1}{3}+ldots+frac{1}{100} )
D. Does not exist
12
741Let ( rho ) be a relation defined on ( N, ) the set
of natural numbers, as
( boldsymbol{rho}={(boldsymbol{x}, boldsymbol{y}) in boldsymbol{N} times boldsymbol{N}: mathbf{2} boldsymbol{x}+boldsymbol{y}=mathbf{4 1}} )
then
A. ( rho ) is an equivalence relation
B. ( rho ) is only reflexive relation
c. ( rho ) is only symmetric relation
D. ( rho ) is not transitive
12
742Find the range of the following function:
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{2}, boldsymbol{x} in boldsymbol{R} )
( A cdot(-2, infty) )
B. ( (2, infty) )
c. ( (3, infty) )
D. None of these
12
743The function ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{B} ) defined by
( boldsymbol{f}(boldsymbol{x})=-boldsymbol{x}^{2}+boldsymbol{6} boldsymbol{x}-boldsymbol{8} ) is a bijection, if
A ( . A=(-infty, 3] ) and ( B=(-infty, 1] )
B. ( A=[3, infty) ) and ( B=R )
C. ( A=(-infty, 3] ) and ( B=[1, infty) )
D. ( A=[3, infty) ) and ( B=[1, infty) )
12
744Prove that the Greatest Integer Function ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} operatorname{given} ) by ( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}], ) is
neither one-one nor onto, where ( [boldsymbol{x}] )
denotes the greatest integer less that or
equal to ( x )
12
745The value of ( 16-|-7|-|11-22| ) is
equal to
A . 56
B. – –
( c cdot 39 )
D. 26
11
746( boldsymbol{R}_{3}= )
{(1,1),(1,3),(3,5),(3,7),(5,7)}
Is it a mapping?
12
747For real values of ( x, ) the range of ( frac{x^{2}+2 x+1}{x^{2}+2 x-1} ) is
B ( cdotleft[frac{1}{2}, 2right] )
c. ( left(-infty, frac{-2}{9}right] cup(1, infty) )
D ( cdot(-infty,-6] cup(-2, infty) )
12
748ff ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) is given by ( boldsymbol{f}(boldsymbol{x})=(boldsymbol{3}- )
( left.boldsymbol{x}^{3}right)^{1 / 3}, ) find ( boldsymbol{f} boldsymbol{o} boldsymbol{f}(boldsymbol{x}) )
12
749Given the relation ( R={(1,2),(2,3)} ) on the set ( {1,2,3}, ) the minimum number of ordered pairs which when added to R make it an equivalence relation is
A. 5
B. 6
( c cdot 7 )
( D )
12
750If ( boldsymbol{A}={mathbf{5}, mathbf{7}}, boldsymbol{B}={mathbf{7}, mathbf{9}} ) and ( boldsymbol{C}= )
( {7,9,11}, ) find ( (A times B) cup(A times C) )
B . {(5,7),(5,9),(5,11),(7,7),(7,9),(7,11)}
c. {(5,5),(5,9),(5,11),(7,7),(7,9),(7,11)}
D. none of these
12
751Find the domain of:
( sec ^{-1} x-tan ^{-1} x )
12
752( f(x) ) is a function defined on the set of
Real Numbers ( boldsymbol{f}(mathbf{1})=mathbf{1} )

Also, ( f(x+5) geq f(x)+5 ) for all ( x epsilon R )
and ( f(x+1) leq f(x)+1 )
If ( , g(x)=(f(x))^{2}-f(x) )
This question has multiple correct options
A ( cdot g(2019)>(2019)^{2} )
B . ( g(2019)<(2019)^{2} )
C ( cdot g(2019)=(2019)^{2} )
D. ( g(2019)=2019 times 2018 )

11
753Relation ( boldsymbol{R} ) in the set ( boldsymbol{Z} ) of all integers
defined as ( boldsymbol{R}= )
[
{(x, y):(x-y) text { is an integer }}
]
enter 1 -reflexive and transitive but not
symmetric 2-reflexive only 3-Transitive only 4-Equivalence
5-None
12
754( boldsymbol{n} / boldsymbol{m} ) means that ( boldsymbol{n} ) is factor of ( boldsymbol{m}, ) then
the relation ‘ ( boldsymbol{f}^{prime} ) is.
B. Transitive and symmetricc
C. reflexive, transitive and symmetric
D. Reflexive,transitive and not symmetric
12
75526.
Iff(x)=sin x + cos x, g(x)= x2 – 1, then g (f(x)) is invertible
in the domain
(2004)
@) (0) (
AC
A
) (0,]
12
756Find the domain of ( f(x)=sqrt{log _{16} x^{2}} )12
757Let ( C ) be the set of all complex numbers
and ( C_{0} ) be the set of all non-zero
complex numbers. Let a relation ( boldsymbol{R} ) on
( C_{0} ) be defined as
( z_{1} R z_{2} Leftrightarrow frac{z_{1}-z_{2}}{z_{1}-z_{2}} ) is real for all ( z_{1}, z_{2} in )
( C_{0} )
Show that ( R ) is an equivalence relation.
12
758Function ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R}, boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}] ) is
A. one-one
B. onto
c. one-one onto
D. Many one into
12
759Let ( f(x)=sin ^{23} x-cos ^{22} x ) and
( boldsymbol{g}(boldsymbol{x})=mathbf{1}+frac{mathbf{1}}{mathbf{2}} tan ^{-mathbf{1}}|boldsymbol{x}| )
Then, the number of values of ( x ) in
interval ( [-10 pi, 20 pi] ) satisfying the equation ( f(x)=operatorname{sgn}(g(x)), ) is ( 5 a ) Then
( a ) is
11
760The function ( boldsymbol{f} ) defined by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}- )
( 3 x^{2}+5 x+7, ) is:
A. increasing in ( R )
B. decreasing in ( R )
C . decreasing in ( (0, infty) ) and increasing in ( (-infty, 0) )
D. increasing in ( (0, infty) ) and decreasing in ( (-infty, 0) )
12
761Show that the equation ( 10 x^{3}-17 x^{2}+ )
( x+6=0 ) has a root between 0 and -1
12
76224.
x²+x+2.
Range of the function f(x)=^
;X ER is (2003S)
x + x +1
(a) (1, 0) (b) (1,11/7] (c) (1,7/3] (d) (1, 113]
12
763The table below gives the distance, ( d )
feet, the cart was from a reference point
at 1 second intervals while it moves
from ( t=0 ) seconds to ( t=5 ) seconds.
[
begin{array}{cccccc}
t & 0 & 1 & 2 & 3 & 4 \
d & 14 & 20 & 26 & 32 & 38 \
88 & & & & \
& & & & \
& 20 & 26 & 32 & 38 & 38 & 3
end{array}
]
Find the correct relationship between ( d )
and ( t )
A. ( d=t+14 )
B. ( d=6 t+8 )
c. ( d=6 t+14 )
D. ( d=14 t+6 )
E ( . d=34 t )
12
764The range of the function ( boldsymbol{f}(boldsymbol{x})= )
( sqrt{4-x^{2}}+sqrt{x^{2}-1} ) is
A ( cdot[sqrt{3}, sqrt{7}] )
B ( cdot[sqrt{3}, sqrt{5}] )
c. ( [sqrt{2}, sqrt{3}] )
D. ( [sqrt{3}, sqrt{6}] )
12
765The range of the function ( f(x)=x^{2}+ )
( frac{1}{x^{2}+1} ) is
( A cdot[1,+infty) )
B. ( [2,+infty) )
( ^{mathbf{C}} cdotleft[frac{3}{2},+inftyright) )
D. None of these
12
766The domain of ( f(x)=sqrt{log _{x^{2}-1}(x) text { is }} )
( mathbf{A} cdot(sqrt{2},+infty) )
B. ( (0,+infty) )
( c cdot(1,+infty) )
D. none of these
12
767ff ( (x)=x+sin x ; g(x)=e^{-x} ; u= )
( sqrt{boldsymbol{c}+mathbf{1}}-sqrt{boldsymbol{c}} ; boldsymbol{v}=sqrt{boldsymbol{c}}-sqrt{boldsymbol{c}-mathbf{1}} ;(boldsymbol{c}>mathbf{1}) )
A ( . f o g(u) geq f o g(v) )
B. ( f o g(u) leq f o g(v) )
( mathbf{c} . f o g(u)>f o g(v) )
D. ( f o g(u)<f o g(v) )
12
768Let ( boldsymbol{f}:left(-frac{boldsymbol{pi}}{mathbf{2}}, frac{boldsymbol{pi}}{mathbf{2}}right) rightarrow boldsymbol{A} ) be defined by
( boldsymbol{f}(boldsymbol{x})=sin boldsymbol{x} . ) If ( boldsymbol{f} ) is a bijection, write set
( boldsymbol{A} )
12
769Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be defined as ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{5} )
show that it is a bijective function.
12
770A function ( f ) from the set of natural
numbers to integers defined by ( boldsymbol{f}(boldsymbol{n})= )
( left{begin{array}{l}frac{n-1}{2}, quad text { when } n text { is odd } \ -frac{n}{2}, quad text { when } n text { is even }end{array}right. )
A. neither one-one nor onto
B. one-one but not onto
c. onto but not one-one
D. one-one and onto both
12
771Suppose that ( f(n+1)=frac{2 f(n)+1}{2} ) for ( boldsymbol{n}=mathbf{1}, mathbf{2}, mathbf{3}, ldots ) and ( (mathbf{1})=mathbf{2} ) Then ( boldsymbol{f}(mathbf{1 0 1}) )
equals
A . 50
B. 52
c. 54
D. none of these
12
772An identity function is a?
A. Many to many function
B. One to One function
C. Many to one function
D. None
11
773If ( boldsymbol{f}={(boldsymbol{6}, boldsymbol{3}),(boldsymbol{8}, boldsymbol{9}),(boldsymbol{5}, boldsymbol{3}),(-mathbf{1}, boldsymbol{6})} )
then the pre-images of 3 are
A. 5 and -1
B. 6 and 8
c. 8 and -1
D. 6 and 5
12
774If ( boldsymbol{A}={1,2,3} ) and ( boldsymbol{B}={4,5,6} ) then
which of the following sets are relation from ( boldsymbol{A} ) to ( boldsymbol{B} )
(i) ( R_{1}={(4,2)(2,6)(5,1)(2,4)} )
(ii) ( boldsymbol{R}_{2}={(1,4)(1,5)(3,6)(2,6)(3,4)} )
(iii) ( R_{3}={(1,5)(2,4)(3,6)} )
(iv) ( boldsymbol{R}_{4}={(mathbf{1}, mathbf{4})(mathbf{1}, mathbf{5})(mathbf{1}, mathbf{6})} )
A. ( R_{1}, R_{2}, R_{3} )
в. ( R_{1}, R_{3}, R_{4} )
( mathbf{c} cdot R_{2}, R_{3}, R_{4} )
D. ( R_{1}, R_{2}, R_{3}, R_{4} )
12
775Write down the domain of the function
( csc ^{-1} x )
12
7762.
If2a + 3b +6c =0,(a, b,c e R) then the quadratic equation
ax2+bx+c=0 has
[2002]
(a) at least one root in [0, 1] (b) at least one root in [2,3]
(c) at least one root in [4, 5] (d) none of these
12
777Let the function ( f ) be defined by ( f(x)= ) ( 5 x-2 a, ) where ( a ) is a constant. If
( boldsymbol{f}(mathbf{1 0})+boldsymbol{f}(mathbf{5})=mathbf{5 5}, ) what is the value of
( boldsymbol{a} ? )
A . –
B. 0
( c cdot 5 )
D. 10
E. 20
12
778The domain of ( boldsymbol{f}(boldsymbol{x})= )
( sqrt{x-2-2 sqrt{x-3}}- )
( sqrt{x-2+2 sqrt{x-3}} )
11
779Let ( x=log _{4} 9+log _{9} 28 )
show that ( [x]=3, ) where ( [x] ) denotes the
greatest integer less than or equal to ( x )
11
780The domain of the relation ( R= )
( {(x, y): x, y epsilon N text { and } x+y leq 3} )
is
( A cdot{1,2,3} )
B. {1,2}
begin{tabular}{l}
c. ( (ldots-1,0,1,2,3] ) \
hline
end{tabular}
D. None of these
12
781If ( f^{prime prime}(x)=-f(x) ) and ( g(x)=f^{prime}(x) ) and
( boldsymbol{F}(boldsymbol{x})=left(boldsymbol{f}left(frac{x}{2}right)right)^{2}+left(boldsymbol{g}left(frac{x}{2}right)right)^{2} ) and given
that ( F(5)=5, ) then ( F(10) ) is equal to
A . 5
B. 10
c. 0
D. 15
12
782Let ( f: N rightarrow N ) defined by ( f(n)= )
( left{begin{array}{cc}frac{n+1}{2} & text { if } n text { is odd } \ frac{n}{2} & text { if } n text { is even }end{array}right. )
then ( boldsymbol{f} ) is.
A. Many-one and onto
B. One-one and not onto
c. onto but not one-one
D. Neither one-one nor onto
12
783( X, Y subset mathrm{R}, f: X rightarrow Y ) is a function and
( f(x)=x^{2} forall x in X ) then the condition
for it to be one-one but not onto is
A ( . X=Y=R^{+} )
B . ( X=R, Y=R^{+} )
c. ( X=R^{+}, Y=R )
D. ( X=Y=R )
( R )
12
784If ( boldsymbol{R}={(boldsymbol{a}, boldsymbol{b}): boldsymbol{b}=boldsymbol{a}+mathbf{1}, boldsymbol{a} in mathbb{Z}, boldsymbol{0}< )
( a<5} ) then find the range of ( R )
12
785f a non-zero function ( f ) satisfies the
relation ( f(x+y)+f(x-y)=2 f(x) cdot f(y) )
for all ( x, y ) in ( R ) and ( f(0) neq 0 ; ) then ( f(10)- )
( f(-10)= )
( A cdot 0 )
B.
( c cdot 2 )
D. 3
11
786If ( f:[0, infty) rightarrow[0, infty), ) and ( f(x)=frac{x}{1+x} )
then ( f ) is
A. one – one and onto
B. one – one but not onto
c. onto but not one-one
D. nether one-one nor onto
12
787Let ( boldsymbol{X} ) be the set of all persons living in a city. Persons ( x, y ) in ( X ) are said to be
related as ( x<y ) if ( y ) is at least 5 years
older than ( x ). Which one of the following
is correct?
A. The relation is an equivalence relation on ( X )
B. The relation is transitive but neither reflexive nor symmetric
c. The relation is reflexive but neither nor symmetric
D. The relation is symmetric but neither transitive nor reflexive
12
788Find ( boldsymbol{f}(-1) ) if ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-mathbf{1 0} boldsymbol{x}+mathbf{1 5} )
A . 15
B. -6
( c cdot 24 )
D. 4
12
789e
Show that 1+ x ln(x+Vx2 +1) V1+ x2 for all x > 0
(1083 2M
12
790Find the domain of definition and the
range of the following function:
( boldsymbol{f}(boldsymbol{x})=sqrt{boldsymbol{x}-mathbf{1}}+mathbf{2} sqrt{mathbf{3}-boldsymbol{x}} )
12
791Find ( boldsymbol{A}^{c} )12
792Let ( boldsymbol{R}=boldsymbol{g} boldsymbol{S}-mathbf{4} . ) When ( boldsymbol{S}=mathbf{8}, boldsymbol{R}=mathbf{1 6} )
When ( S=10, R ) is equal to
A . 11
B. 14
c. 20
D. 21
E. None of these
12
793( f(G={7,8} text { and } H={5,4,2}, ) find
( G times H ) and ( H times G )
12
794On differentiating an identity function,
we get?
A. Signum function
B. sinc function
c. constant function
D. None
11
795If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) defined by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{2} boldsymbol{x}^{2}- )
( 3 x+5, ) then find the value of
( frac{boldsymbol{f}(boldsymbol{x}+boldsymbol{h})-boldsymbol{f}(boldsymbol{x})}{boldsymbol{h}} ) at ( boldsymbol{h} neq mathbf{0} )
A. ( 4 x+2 h )
B. ( 4 x+2 h-3 )
c. ( 4 x-2 h+3 )
D. ( 4 x-2 h-3 )
11
796Given a function ( boldsymbol{f}(boldsymbol{x})=frac{1}{2} boldsymbol{x}-boldsymbol{4} ) and the
composite function ( boldsymbol{f}(boldsymbol{g}(boldsymbol{x}))=boldsymbol{g}(boldsymbol{f}(boldsymbol{x})) )
determine which among the following
( operatorname{can} operatorname{be} g(x): )
( 2 x-frac{1}{4} )
11. ( 2 x+8 )
III. ( frac{1}{2} x-4 )
A. I only
B. II only
c. III only
D. II and III only
E . ।, II, and III
12
797Assertion ( f: R rightarrowleft[0, frac{pi}{2}right) ) defined by ( f(x)= )
( tan ^{-1}left(x^{2}+x+aright) ) is onto for all ( a in )
( left(-infty, frac{1}{4}right) )
Reason
For onto function codomain of ( boldsymbol{f}= )
Range of ( boldsymbol{f} )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
798If ( boldsymbol{f}(1)=mathbf{1} ) and ( boldsymbol{f}(boldsymbol{x}+mathbf{1})=boldsymbol{2} boldsymbol{f}(boldsymbol{x})+mathbf{1} )
for ( boldsymbol{x} geq 1, ) then find the range of ( boldsymbol{f}(boldsymbol{x}) )
A. ( R )
B. A series like ( 1,3,5,7,9, ldots )
C. A series like ( 1,3,7,15, ldots )
D. ( R-{1,3,7,15, ldots} )
12
799If ( f(x)=x^{3}-frac{1}{x^{3}}, ) prove that ( f(x)+ )
( boldsymbol{f}left(frac{1}{x}right)=mathbf{0} )
11
800If a set contains ( m ) elements and ( B )
contains ( n ) elements, then find the
number of elements in ( boldsymbol{A} times boldsymbol{B} )
12
801Assertion
Let ( boldsymbol{f}: boldsymbol{X} rightarrow boldsymbol{Y} ) be a function defined by ( f(x)=2 sin left(x+frac{pi}{4}right)-sqrt{2} cos x+c )
Statement I: For set ( boldsymbol{X}, boldsymbol{x} inleft[mathbf{0}, frac{boldsymbol{pi}}{mathbf{2}}right] cup )
( left[pi, frac{3 pi}{2}right], f(x) ) is one-one function
Reason Statement II: ( boldsymbol{f}^{prime}(boldsymbol{x}) geq mathbf{0}, boldsymbol{x} inleft[mathbf{0}, frac{boldsymbol{pi}}{mathbf{2}}right] )
A. Statementl is true, statement II is also true:
Statement II is the correct explanation of Statement
B. Statement I is true, Statement II is also true:
Statement II is not the correct explanation of Statement
c. Statement I is true, Statement II is false
D. Statement l is false, Statement II is true
12
802Let ( R ) be a reflexive relation on a finite
set A having n elements, and let there be m ordered pairs in R. Then:
( mathbf{A} cdot m geq n )
в. ( m leq n )
c. ( m=n )
( mathbf{D} cdot m=-n )
12
803The range of the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}+frac{mathbf{1}}{boldsymbol{x}} )
is
12
80433. The function f : [0, 3] + [1, 29), defined by
f(x)=2×3 – 15×2 + 36x + 1, is
(2012)
(a) one-one and onto (b) onto but not one-one
(c) one-one but not onto(d) neither one-one nor onto
12
805Let the function ( boldsymbol{f}: boldsymbol{R}-{-boldsymbol{b}} rightarrow boldsymbol{R} )
{1} be defined by ( f(x)=frac{x+a}{x+b}, a neq b )
then
A. ( f ) is one-one but not onto
B. ( f ) is onto but not one-one
c. ( f ) is both one-one and onto
D. None of these
12
806The domain of ( mathbf{f}(mathbf{x})=frac{mathbf{3}}{mathbf{4}-mathbf{x}^{2}}+ )
( log _{10}left(x^{3}-xright) ) is
A ( .(1,2) )
B. (-1,0)( cup(1,2) )
c. (1,2)( cup(2, infty) )
D. (-1,0)( cup(1,2) cup(2, infty) )
12
807If the function ( f: R rightarrow R ) be such that
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}-[boldsymbol{x}], ) where ( [boldsymbol{x}] ) denotes the
greatest integer less than or equal to ( x )
then ( boldsymbol{f}^{-1}(boldsymbol{x}) ) is
( ^{mathrm{A}} cdot frac{1}{x-[x]} )
B cdot ( [x]-x )
c. not defined
D. none of these
11
808Let ( boldsymbol{A}=mathbf{1}, mathbf{2}, mathbf{3} . ) Then number of
equivalence relations containing ( (mathbf{1}, mathbf{2}) ) is
A .
B . 2
( c .3 )
D.
12
809For real ( boldsymbol{x} ), let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+mathbf{5} boldsymbol{x}+mathbf{1}, ) then
A. ( f ) is onto ( R ) but not one-one
B. ( f ) is one-one and onto ( R )
c. ( f ) is neither one-one nor onto ( R )
D. ( f ) is one-one but not onto ( R )
12
810If ( e^{x}=y+sqrt{1+y^{2}}, ) then the value of ( y )
is
A ( cdot frac{1}{2}left(e^{x}+e^{-x}right) )
в. ( frac{1}{2}left(e^{x}-e^{-x}right) )
c. ( quad frac{-x}{2} )
D. ( _{e^{x}}+e^{frac{-x}{2}} )
11
811Let ( n ) be a fixed positive integer. Define
a relation ( R ) on ( Z ) as follows:
( (a, b) in R Leftrightarrow n ) divides ( a-b )
Show that ( R ) is an equivalence relation
on ( Z )
12
812Let ( f(x)=left(a-x^{n}right)^{frac{1}{n}}, a>0 ) and ( n ) is positive integer, then prove that ( boldsymbol{f}[boldsymbol{f}(boldsymbol{x})]=boldsymbol{x} )12
813Given ( : a x+b y=d ) and ( y=m x+c )
Find ( x ) in terms of ( b, c, d ) and ( m )
A ( cdot x=frac{d-b c}{a-b m} )
B. ( x=frac{d+b c}{a+b m} )
c. ( _{x}=frac{d-b c}{a+c m} )
D. ( x=frac{d-b c}{a+b m} )
11
814( operatorname{Let} f(x)=frac{a x+b}{c x+d}, ) then ( f o f(x)=x )
provided
A ( . d=-a )
B. ( d=a )
c. ( a=b=c=d=1 )
D. ( a=b=1 )
12
815Define a binary operation ( * ) on the set
{0,1,2,3,4,5} as
( boldsymbol{a} * boldsymbol{b}=left{begin{array}{ll}boldsymbol{a}+boldsymbol{b}, & boldsymbol{i f} quad boldsymbol{a}+boldsymbol{b}<mathbf{6} \ boldsymbol{a}+boldsymbol{b}-boldsymbol{6} quad boldsymbol{i} boldsymbol{f} quad boldsymbol{a}+boldsymbol{b} geq boldsymbol{6} & boldsymbol{6}end{array}right. )
Show that zero is the identity for this
operation and each element ( a neq 0 ) of
the set is invertible with ( 6-a ) being the
inverse of ( a )
12
816A person on a high protein diet is interested in the random variable ( X ), the
gain in weight in a week. Then the random variable is
A. not continuous
B. Not discrete
c. continuous
D. discrete
12
817The number of elements of an identity
function defined on a set containing
four elements is
A ( cdot 2^{2} )
B. ( 2^{4} )
( c cdot 2^{8} )
D. ( 2^{16} )
11
818The number of reflexive relations of a
set with four elements is equal to
A ( cdot 2^{16} )
B. 2 ( ^{12} )
( c cdot 2^{8} )
( D cdot 2^{4} )
12
819Which of the following relations are false?
This question has multiple correct options
A ( cdot tan left|tan ^{-1} xright|=x )
B . ( cot left|cot ^{-1} xright|=|x| )
c. ( tan ^{-1}|tan x|=|x| )
D ( cdot sin left|sin ^{-1} xright|=|x| )
11
820Solve the equation
( |x+1|-|x|+3|x-1|-2|x-2|=x+2 )
11
821If ( f(x)=x^{3}+3 x^{2}+12 x-2 sin x )
such that ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) then
A. ( f(x) ) is one-one and onto
B. ( f(x) ) is many one onto
c. ( f(x) ) is one-one and into
D. ( f(x) ) is many one into
12
822Find the domain of ( boldsymbol{f}(boldsymbol{x})= )
( log _{10} log _{2} log _{pi / 2}left(tan ^{-1} xright)^{-1} )
12
823( mathbf{A}= ) the domain of ( mathbf{f} ) where ( mathbf{f}(mathbf{x})=log mathbf{x}^{2} )
and ( mathbf{B}= ) the domain of ( mathbf{g} ) where ( mathbf{g}(mathbf{x})= ) ( 2 log x, ) then ( mathbf{A}-mathbf{B}= )
( A cdot phi )
В ( cdot(-infty, 0) )
c. ( (0, infty) )
D. (0,1)
12
824If ( f(x)=3 x-2 ) and ( g(x)= )
( mathbf{7}, boldsymbol{f}[boldsymbol{g}(boldsymbol{x})]= )
A . ( 21 x-2 )
B. 7
c. 19
D. ( 7 x-2 )
E . 13
11
825If ( phi(x)=frac{1}{1+e^{-x}} ) then the value of
( boldsymbol{phi}(mathbf{5})+boldsymbol{phi}(mathbf{4})+ldots+boldsymbol{phi}(-mathbf{3})+boldsymbol{phi}(-mathbf{4})+ )
( phi(-5) )
( mathbf{A} cdot mathbf{5} )
B. ( frac{9}{2} )
c. ( frac{11}{2} )
D. none of these
12
826Find the domain and range of the
following function:
( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}2-boldsymbol{x} & boldsymbol{i f} quad mathbf{0}<boldsymbol{x}<mathbf{2} \ boldsymbol{x}-mathbf{1} & boldsymbol{i f} quad boldsymbol{3} leq boldsymbol{x}<mathbf{4}end{array}right. )
12
827Let ( A ) and ( B ) be two sets such that ( A times )
( B ) consists of 6 elements.lf three
elements of ( boldsymbol{A} times boldsymbol{B} ) are
( {(1,4),(2,6),(3,6)} . ) Find ( A times B ) and
( boldsymbol{B} times boldsymbol{A} )
12
828If ( f(x)=frac{1}{(1-x)} ) find ( (f o f o f)(x)=? )12
829Let ( A ) and ( B ) be two sets such that
( boldsymbol{n}(boldsymbol{A})=boldsymbol{3} ) and ( boldsymbol{n}(boldsymbol{B})=mathbf{2} . ) If
( (x, 1),(y, 2),(z, 1) ) are in ( A times B ) find ( A )
and ( B ) where ( x, y ) and ( z ) are distinct
elements
12
830If ( f(x)=frac{a x+b}{c x+d} ) and ( (f o f)(x)=x )
then
This question has multiple correct options
A ( cdot a^{2}+b c=1 )
B . ( d^{2}+b c=1 )
( mathbf{c} cdot a+d=0 )
D. none of these
11
831Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a function defined by
( f(x)=frac{e^{|x|}-e^{-x}}{e^{x}+e^{-x}}, ) then
A. ( f ) is a bijection
B. ( f ) is an injection only
c. ( f ) is surjection on only
D. ( f ) is neither injection nor a surjection
12
832If ( boldsymbol{A} * boldsymbol{B}=boldsymbol{A} cap boldsymbol{B} ) on ( boldsymbol{P}(boldsymbol{X}), ) then identify
for ( * ) is ( _{-}–_{-}—(X neq phi) )
( A cdot phi )
B. ( X )
( c cdot U )
D.
12
833Given ( : a x+b y=d ) and ( y=m x+c )
Find ( x ) in terms of ( b, c, d ) and ( m )
A ( cdot x=frac{d-b c}{a-b m} )
B. ( x=frac{d+b c}{a+b m} )
c. ( _{x}=frac{d-b c}{a+c m} )
D. ( x=frac{d-b c}{a+b m} )
11
834Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}, boldsymbol{g}(boldsymbol{x})=tan boldsymbol{x} ) and ( boldsymbol{h}(boldsymbol{x})= )
( ln x )
What is ( [f o(f o f)](2) ) equal to?
A .2
B. 8
c. 16
D. 256
11
835If ( f(x)=left(x^{2}-x+2right)left(x^{2}-4right) ) then the
( operatorname{set}{x mid f(x)>0, x in R} ) is equal to
( mathbf{A} cdot(-infty,-2) )
в. ( (2,+infty) )
c. (-2,2)
D ( cdot(-infty,-2) cup(2,+infty) )
12
836( l=lim _{x rightarrow alpha} frac{f(x)}{x(x-alpha)(x-2)} ) is
A . positive
B. negative
( c cdot 0 )
D. sign of ( l ) depends upon ( alpha pi )
12
837Period of ( boldsymbol{f}(boldsymbol{x})={boldsymbol{x}}+left{boldsymbol{x}+frac{1}{3}right}+ )
( left{x+frac{2}{3}right} ) is equal to (where ( {.} ) denotes fraction part function
( mathbf{A} cdot mathbf{1} )
B. ( frac{2}{3} )
( c cdot frac{1}{2} )
D.
11
83862. Let A and B be two sets containing four and two elements
respectively. Then the number of subsets of the set A B ,
each having at least three elements is : [JEE M 2015]
(a) 275 (6) 510 (c) 219 (d) 256
11
839Which of the following statements best
describes the data in the given table?
( begin{array}{cc}boldsymbol{x} & boldsymbol{y} \ frac{1}{2} & 4 \ 1 & 5 \ 3 & 9 \ 4 & 11end{array} )
A ( cdot ) The value of ( y ) is ( 3 frac{1}{2} ) more than the value of ( x )
B. The value ( y ) is 3 more than twice the value ( x ).
C. The value of ( y ) is 3 times the value of ( x ) increased by 1
D. The value of ( y ) is 3 less than twice the value of ( x ).
12
840Find composite of ( f ) and ( g ) and express
it by formula:
( boldsymbol{f}={(mathbf{1}, mathbf{3}),(mathbf{2}, mathbf{4}),(mathbf{3}, mathbf{5}),(mathbf{4}, mathbf{6})} )
( boldsymbol{g}={(mathbf{3}, mathbf{6}),(mathbf{4}, mathbf{8}),(mathbf{5}, mathbf{1 0}),(mathbf{6}, mathbf{1 2})} )
12
841If ( x ) is real, then ( frac{x^{2}+2 x+c}{x^{2}+4 x+3 c} ) can take all real values if?
A. ( 0<c<2 )
В. ( 0<c<1 )
c. ( -1<c<1 )
D. None of the above
12
842If ( f(x)=4 x^{2}-1 ) and ( g(x)=8 x+ )
( 7, g^{o} f(2)= )
A . 15
B. 23
( c cdot 127 )
D. 345
E . 2115
12
843toppr
Q Type your question
the ( x- ) coordinate.
( A )
B.
( c )
( D )
( E )
12
844If ( k(x)=4 x^{3} a ) and ( k(3)=27, ) what is
( k(2) ? )
( A cdot 9 )
B. 8
c. 17
D. 12
12
845The domain of ( boldsymbol{f}(boldsymbol{x})=cot frac{boldsymbol{x}}{mathbf{3}} ) is
( A cdot(-infty, infty) )
B . ( R-{n pi: n in Z} )
c. ( R-{3 n pi: n in Z} )
D. ( (0, infty) )
12
846If ( n(A)+n(B)=m, ) then the number
of possible bijections from ( boldsymbol{A} ) to ( boldsymbol{B} ) is
( mathbf{A} cdotleft(frac{m}{2}right) ! )
в. ( m^{2} )
( c . m ! )
D. ( 2 m )
12
847If ( {(7,11),(5, a)} ) represents a constant
function, then the value of ‘a’ is :
( A cdot 7 )
B. 11
( c .5 )
D.
11
848Let ( ^{*} ) be a binary operation on the set ( boldsymbol{Q} )
of rational numbers as follows:
( boldsymbol{a} * boldsymbol{b}=(boldsymbol{a}-boldsymbol{b})^{2} )
Find which of the binary operations are commutative and which are
associative.
12
849If the relation is defined on ( boldsymbol{R}-{mathbf{0}} ) by
( (x, y) in S Leftrightarrow x y>0, ) then ( S ) is
A. an equivalence relation
B. symmetric only
c. reflexive only
D. transitive only
12
850Ordered pairs ( (x, y) ) and (3,6) are equal
if ( boldsymbol{x}=mathbf{3} ) and ( boldsymbol{y}=? )
( A cdot 3 )
B. 6
c. -6
D. – 3
12
851Consider the following two binary
relations on the set ( A={a, b, c} )
( boldsymbol{R}_{1}= )
( {(c, a),(b, b),(a, c),(c, c),(b, c),(a, a)} )
and ( boldsymbol{R}_{2}= )
( {(a, b),(b, a),(c, c),(c, a),(a, a),(b, b) )
Then
A. ( R_{2} ) is symmetric but it is not transitive
B. Both ( R_{1} ) and ( R_{2} ) are transitive
c. Both ( R_{1} ) and ( R_{2} ) are not symmetric
D. ( R_{1} ) is not symmetric but it is transitive
12
852If ( g(x) ) is defined on [-1,1] and the area
of the equilateral triangle with two of its vertices (0,0) and ( (x, g(x)) ) is ( frac{sqrt{3}}{4}, ) then
A ( . g(x)=pm sqrt{left(1-x^{2}right)} )
в. ( g(x)=-sqrt{left(1-x^{2}right)} )
c. ( g(x)=sqrt{left(1-x^{2}right)} )
D. ( g(x)=sqrt{left(1+x^{2}right)} )
12
853If ( A={1,2}, B={3,4}, ) then ( A times B= )
B. {(1,1),(2,2),(3,3),(4,4)}
c. {(4,1),(3,1),(4,2),(3,2)}
D. All the above
12
854Let ( Z ) be the set of all integers and ( Z_{0} ) be
the set of all non-zero integers. Let a
relation ( boldsymbol{R} ) on ( boldsymbol{Z} times boldsymbol{Z}_{mathbf{0}} ) be defined as
follows:
( (a, b) R(c, d) Leftrightarrow a d=b c ) for all
( (a, b),(c, d) in Z times Z_{0} )
Prove that ( R ) is an equivalence relation
on ( Z times Z_{0} )
12
855Let ( N ) denote the set of all natural
numbers and R a relation on ( N times N )
Which of the following is an equivalence
relation?
( mathbf{A} cdot(a, b) R(c, d) ) if ( a d(b+c)=b c(a+d) )
B. ( (a, b) R(c, d) ) if ( a+d=b+c )
( mathbf{c} cdot(a, b) R(c, d) ) if ( a d=b c )
D. all the given
12
8561.
R is a function
(1979)
Let R be the set of real numbers. Iff: R
defined by / (x)=x, then fis:
(a) Injective but not surjective
(b) Surjective but not injective
(C) Bijective
(d) None of these
12
857( boldsymbol{A}={1,2,3,4} ) and ( boldsymbol{B}={boldsymbol{a}, boldsymbol{b}, boldsymbol{c}} . ) The
relations from ( boldsymbol{A} ) to ( boldsymbol{B} ) is
A ( cdot{(1,2),(1,3),(2,3),(2,4),(3,4),(3,1)} )
B ( cdot{(a, b),(a, c),(b, a),(b, c),(c, a)} )
c. ( {(1, a),(1, b)(1, c),(2, a),(2, b),(2, c),(3, a),(3, b),(3, c),(4 )
D cdot ( {(a, 1),(1,3),(b, 2),(c, 3),(b, 3),(b, 4)} )
12
858If ( P ) and ( Q ) are the sum and product
respectively of all integral values of ( x )
satisfying the equation ( |mathbf{3}[boldsymbol{x}]-mathbf{4} boldsymbol{x}|=mathbf{4} )
then
(where [.] denotes represents greatest integer function ( ) ) This question has multiple correct options
( mathbf{A} cdot P=0 )
B . ( P=8 )
c. ( Q=-16 )
D. ( Q=-9 )
11
859Assertion
Consider two functions ( boldsymbol{f}(boldsymbol{x})=mathbf{1}+ )
( e^{cot ^{2} x} ) and ( g(x)=sqrt{2|sin x|-1}+ )
( frac{1-cos 2 x}{1+sin ^{4} x} )
Statement I: The solution of the
equation ( f(x)=g(x) ) is given by ( x= ) ( (2 n+1) frac{pi}{2}, forall n in I )
Reason
Statement II: If ( f(x) leq k ) and ( g(x) leq k )
(where ( k in boldsymbol{R} ) ), then solutions of the
equation ( f(x)=g(x) ) is the solution
corresponding to the equation ( f(x)=k )
A. Statement I is true, Statement II is also true Statement II is the correct explanation of Stateme
B. Statement I is true, Statement II is also true Statement II is not the correct explanation of statement
c. Statement lis true, Statement II is false
D. Statement lis false, Statement II is true
11
860Suppose ( S={1,2} ) and ( T={a, b} )
( operatorname{then} boldsymbol{T} times boldsymbol{S} )
A ( cdot(a, 1),(a, 2),(b, 1),(b, 2) )
( mathrm{B} times mathrm{A}={(mathrm{a}, 1),(mathrm{a}, 2),(mathrm{b}, 1),(mathrm{b}, 2)} )
B. ( (1, a),(2, b),(b, 1),(b, 2) )
D. None of the above
12
861If ( boldsymbol{A}={1,2,3, ldots, 9} ) and ( boldsymbol{R} ) be the
relation in ( A times A ) defined by
( (a, b) R(c, d) . ) If ( a+d=b+c ) for
( (a, b),(c, d) ) in ( A times A . ) Prove that ( R ) is an
equivalence relation. Also, obtain the
equivalence class [(2,5)]
12
862Find the values of ( x ) which satisfy ( left|frac{x^{2}}{x-1}right| leq 1 )
( ^{mathbf{A}} cdot_{x} inleft[frac{-1-sqrt{3}}{2}, frac{-1+sqrt{3}}{2}right] )
в. ( _{x} inleft[frac{-1-sqrt{5}}{2}, frac{-1+sqrt{5}}{2}right] )
( ^{mathbf{c}} cdot_{x} inleft[frac{-1-sqrt{7}}{3}, frac{-1+sqrt{7}}{3}right] )
D. None of these
11
863If ( a+b+c=3 ) and ( a>0, b>0, c>0 )
the greatest value of ( a^{2} b^{3} c^{2} )
( ^{A} cdot frac{3^{10}, 2^{4}}{7^{7}} )
в. ( frac{3^{9}, 2^{4}}{7^{7}} )
c. ( frac{3^{9}, 2^{5}}{7^{7}} )
D. ( frac{3^{10}, 2^{5}}{7^{7}} )
11
864( (x, y) ) and ( (p, q) ) are two ordered pairs.
Find the values of ( x ) and ( p, ) if ( (3 x- )
( mathbf{1}, mathbf{9})=(mathbf{1 1}, boldsymbol{p}+mathbf{2}) )
A. ( x=4, p=9 )
В. ( x=6, p=7 )
c. ( x=4, p=5 )
D. ( x=4, p=7 )
12
865If ( f ) and ( g ) two function defined on ( N )
such that ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{l}2 n-1 i f n text { isven } \ 2 n+2 i f text { is odd }end{array} text { and } g(n)=f(n)+right. )
Then range of ( g ) is
A ( cdot{m epsilon N: m=text { multiple of } 4} )
B. setofevennaturalnumber
C ( cdot{m epsilon N: m=4 k+3, k i s text { a natural number }} )
D. ( m epsilon N: m= ) multiple of 3 or multiple of 4
12
866f ( boldsymbol{R}={(boldsymbol{x}, boldsymbol{y}): boldsymbol{x}+mathbf{2} boldsymbol{y}=mathbf{8}} ) is a
relation on ( N, ) write the range of ( R )
12
867If ( boldsymbol{n}(boldsymbol{A})=boldsymbol{4}, boldsymbol{n}(boldsymbol{B})=boldsymbol{3}, boldsymbol{n}(boldsymbol{A} times boldsymbol{B} times )
( boldsymbol{C})=boldsymbol{2} 4, ) then ( boldsymbol{n}(boldsymbol{C})= )
12
868The function ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) defined by
( f(x)=(x-1)(x-2)(x-3) ) is.
A. One-one but not onto
B. Onto but not one-one
c. Both one-one and onto
D. Neither one nor onto
12
869Prove that ( boldsymbol{f}^{prime}(mathbf{1})+boldsymbol{f}(-1)=-boldsymbol{f}(mathbf{0}), ) i
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{5}+boldsymbol{x}^{3}-boldsymbol{2} boldsymbol{x}-boldsymbol{3} )
11
87018. For xe R- {0,1), let f (x)=
12(x) = 1 – x and z(x)= be three given functions. Ifa
function, J(x) satisfies (foJof,) (x) = f(x) then J(x) is equal
[JEE M 2019-9 Jan (M)]
(a) f(x)
(b) f(x)
(c) 12(x)
(d) fix)
to:
12
871If ( f(x)=a cos (b x+c)+d ) then range
of ( boldsymbol{f}(boldsymbol{x}) ) is
A ( cdot[d+a, d+2 a] )
B . ( [a-d, a+d] )
c. ( [d+a, a-d] )
D cdot ( [d-a, d+a] )
12
872The arrangement of the following sets in the proper order is
A. Domain of ( frac{|boldsymbol{x}|}{boldsymbol{x}} )
B. Range of ( operatorname{cosec} x )
C. Range of cot ( x )
D. Domain of ( sqrt{x^{2}-4} )
A. ( C subset A subset B subset D )
в. ( D subset B subset C subset A )
c. ( C subset A subset D subset B )
D. ( D subset B subset A subset C )
12
87332. Let -1 sp<1. Show that the equation 4×3 – 3x – p=0
has a unique root in the interval[1/2, 1] and identify it.
(2001 – 5 Marks)
12
874Write the total number of one-one
functions from set ( boldsymbol{A}={mathbf{1}, mathbf{2}, mathbf{3}, mathbf{4}} ) to
( operatorname{set} B={a, b, c} )
12
875If ( B ) is the set of even positive integers
then ( f: N rightarrow B ) defined by ( f(x)=2 x ) is
A. one – one only
B. onto only
c. neither one-one nor onto
D. bijective
12
8767.
Which of the following functions is periodic?
(1983-1 Mark)
(a) f(x) = x – [x] where [x] denotes the largest integer less
than or equal to the real number x
(b) f(x) = sin – for x+ 0, f(0) = 0
(c) f(x)= x cosx
(d) none of these
12
877If ( boldsymbol{f}: boldsymbol{I} rightarrow boldsymbol{I}, boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+1, ) then ( boldsymbol{f} ) is
A. one-one but not onto
B. onto but not one-one
c. one-one onto
D. none of these
12
878State whether the following statement
is True or False.

The inverse of an identity function is the identity function itself.
A. True
B. False

11
879Let ( g:[1,3] rightarrow Y ) be a function defined
( operatorname{as} g(x)=ln left(x^{2}+3 x+1right) . ) Then
(a) Determine whether ( g(x) ) one-one or
many-one.
(b) Find the set ( Y ), so that ( g(x) ) is onto.
(c) Find ( g^{-1}(x) ), if it exists.
12
880Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R}: boldsymbol{f}(boldsymbol{x})=left(boldsymbol{x}^{2}-boldsymbol{3} boldsymbol{x}+boldsymbol{2}right) )
Find ( boldsymbol{f} boldsymbol{o} boldsymbol{f}(boldsymbol{x}) )
12
881A mapping is defined as ( boldsymbol{f}: boldsymbol{R} rightarrow )
( R, f(x)=cos x . ) Show that it is neither
one-one nor surjective.
12
882Check the commutativity and associativity of the following binary operation:
( *^{prime} ) on ( Q ) defined by ( a * b=a+a b ) for all
( boldsymbol{a}, boldsymbol{b} in boldsymbol{Q} )
12
883The domain of ( f(x)=sqrt{frac{log _{3}|x-2|}{|x|}} ) is
( A cdot(-infty,-1] cup[3, infty) )
в. [2,3]
c. [1,2)( cup(2,3] )
D. [0,3]
12
884If ( f(x)=frac{4^{x}}{4^{x}+2}, ) then ( f(x)+f(1-x) )
is equal to.
( mathbf{A} cdot mathbf{0} )
B. –
c. 1
D. 4
12
885f ( n(A)=4 ) and ( n(B)=5, ) then ( n(A x )
( B)= )
A . 20
B . 25
( c cdot 4 )
D. 15
12
886Prove that the function ( f(x)=x+|x| )
( x in R ) is not one-one.
12
887If ( f(x)=3 x+10, g(x)=x^{2}-1, ) then
( (f o g)^{-1} ) is equal to
( ^{mathbf{A}}left(frac{x-7}{3}right)^{frac{1}{2}} )
B. ( quadleft(frac{x+7}{3}right)^{frac{1}{2}} )
( left(frac{x-3}{7}right)^{frac{1}{2}} )
D. ( quadleft(frac{x+3}{7}right)^{frac{1}{2}} )
11
888What is general representation of
ordered pair for two variables ( a ) and ( b ) ?
A ( . a, b )
в. ( (a, b) )
c. ( (a), b )
is
D. ( a,(b) )
12
889The domain of ( cos ^{-1} frac{x-3}{2}-log _{10}(4- )
( x) ) is
в. [1,4]
c. (1,4]
D. [1,4]
12
890Number of integers in the domain of the
function ( boldsymbol{f}(boldsymbol{x})=log _{(6-x)}left(7 x-x^{2}right) ) is
A. 6
B. 5
( c cdot 4 )
D. infinite
12
891For each operation ( * ) defined below, determine whether ( * ) is binary, commutative or associative.
(i) On ( Z, ) define ( a * b=a-b )
(ii) On ( Q, ) defined ( a * b=a b+1 )
(iii) On ( Q, ) defined ( a * b=frac{a b}{2} )
(iv) ( operatorname{On} Z^{+}, ) defined ( a * b=2^{a b} )
(v) ( operatorname{On} Z^{+}, ) defined ( a * b=a^{b} )
( (v i) ) on ( R-{-1}, ) defined ( a * b=frac{a}{b+1} )
12
892Which one of the following is an elementary symmetric function of
( boldsymbol{x}_{1}, boldsymbol{x}_{2}, boldsymbol{x}_{3}, boldsymbol{x}_{4} )
A. ( x_{1} x_{2} x_{3}+x_{2} x_{3} x_{4} )
в. ( x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{1} )
C ( cdot x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2} )
D. ( x_{1} x_{2}+x_{1} x_{3}+x_{1} x_{4}+x_{2} x_{3}+x_{2} x_{4}+x_{3} x_{4} )
12
893Let ( A={1,2} ) and ( B={3,4} . ) Find the
number of relations from ( boldsymbol{A} ) to ( boldsymbol{B} )
12
894Find the number of solutions of the
equations;
( |cos x|=2[x](text { where }[] ) denotes
greatest integer function
11
895Given a relation ( boldsymbol{R}={(mathbf{1}, mathbf{2}),(mathbf{2}, mathbf{3})} ) on
the set of natural numbers,Find the
minimum number of ordered pairs should be added so that the enlarged
relation is symmetric,transitive and
reflexive.
12
896If ( boldsymbol{f}: boldsymbol{Z} rightarrow boldsymbol{Z} ) is defined by ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}frac{x}{2} & text { if } x text { is even } \ 0 & text { if } x text { is odd }end{array} text { then } f ) is right.
A. onto but not one to one
B. One to one but not onto
c. one to one and onto
D. Neither one to one nor onto
12
897On the set ( Q ) of all rational numbers, if a
binary operation ( * ) is defined by ( a * b= ) ( frac{a b}{5} . ) Prove that ( * ) is associative on ( Q )
12
898The domain of the function ( boldsymbol{f}(boldsymbol{x})= )
( log _{x} e ) is given by ( [0, infty]-k )
What is ( k )
This question has multiple correct options
A . 1
B. ( e )
c. 0
D. 10
12
899If ( boldsymbol{A}={1,2}, ) form the ( operatorname{set} boldsymbol{A} times boldsymbol{A} times boldsymbol{A} )12
900A relation ( R ) is defined on the set ( z ) of
integers as follows. ( (x, y) in R Leftrightarrow x^{2}+ )
( boldsymbol{y}^{2}=25 . ) Express ( boldsymbol{R} ) and ( boldsymbol{R}^{-1} ) as the set
of ordered pairs and hence find their respective domains.
12
901Let ( A ) and ( B ) be two finite sets having ( m )
and ( n ) elements respectively. Then the
total number of mapping from ( boldsymbol{A} ) to ( boldsymbol{B} ) is
( A cdot m n )
B. ( 2^{m} )
( c cdot m^{n} )
D. ( n^{text {m }} )
12
902Find the number of relations from
( {boldsymbol{m}, boldsymbol{o}, boldsymbol{t}, boldsymbol{h}, boldsymbol{e}, boldsymbol{r}} ) to ( {boldsymbol{c}, boldsymbol{h}, boldsymbol{i}, boldsymbol{l}, boldsymbol{d}} )
A . 30
B . ( 30^{2} )
( c cdot 2^{30} )
D. 60
12
903The graph of a constant function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{k} ) is?
A. A straight line parallel to ( X ) -axis
B. A straight line parallel to ( Y ) -axis
c. A straight line passing through orgin
D. None
11
904( f(x)=frac{3}{2} x+b )
In the function above, ( b ) is a constant. If
( boldsymbol{f}(boldsymbol{6})=mathbf{7}, ) what is the value of ( boldsymbol{f}(-mathbf{2}) ? )
A . –
B . – –
( c . )
( D )
12
905Let ( boldsymbol{f}(boldsymbol{x})=mathbf{2}^{100} boldsymbol{x}+mathbf{1} )
( boldsymbol{g}(boldsymbol{x})=boldsymbol{3}^{100} boldsymbol{x}+mathbf{1} )
Then the set of real numbers x such that
( boldsymbol{f}(boldsymbol{g}(boldsymbol{x}))=boldsymbol{x} ) is
A. Empty
B. A singleton
c. A finite se with more than one element
D. Infinite
12
9065.
If f(x) = cos(in x), then f(x)f(y). + f(xy) has
the value
(1983 – 1 Mark)
(a) -1
(b) 1/2
(c) – 2
(d) none of these
12
907Find the values of ( a ) and ( b ) if
( (3 a-2, b+3)=(2 a-1,3) )
12
90817. Let R be the real line. Consider the following subsets of the
plane R* R:
S={(x, y): y=x+1 and 0<x<2}
T={(x, y): x – y is an integer),
Which one of the following is true?
[2008]
(a) Neither Snor Tis an equivalence relation on R
(b) Both S and I are equivalence relation on R
(C) Sis an equivalence relation on R but Tis not
(d) T is an equivalence relation on R but Sis not
12
909If ( f(x)=log _{e} sec x ) and ( phi(x)= )
( log _{e} tan x ) then prove that ( e^{2 f(x)}- )
( e^{2 phi(x)}=1 )
11
910The range of function ( boldsymbol{f}(boldsymbol{x})= ) ( sqrt{e^{cos ^{-x}left(log _{4} x^{2}right)}} ) is12
911Let ( * ) be ( a ) binary operation on ( N ) given by ( a * b=H C F *(a, b) ) find ( 22 * 4 )12
912If ( A={2,3,5} ) and ( B={5,7}, ) find the set with highest number of elements:
( mathbf{A} cdot A times B )
в. ( B times A )
c. ( A times A )
D. ( B times B )
12
913Let ( boldsymbol{f}(boldsymbol{x}) ) be defined ( forall boldsymbol{x} in boldsymbol{R} ) and
continuous.
( operatorname{Let} boldsymbol{f}(boldsymbol{x}+boldsymbol{y})-boldsymbol{f}(boldsymbol{x}-boldsymbol{y})=boldsymbol{4} boldsymbol{x} boldsymbol{y} forall boldsymbol{x}, boldsymbol{y} in )
( boldsymbol{R} ) and ( boldsymbol{f}(mathbf{0})=mathbf{0}, ) then ( boldsymbol{f}(boldsymbol{x})<mathbf{2} boldsymbol{x}+mathbf{3} ) will
have solution:
A. (-1,3)
B . ( (1-sqrt{5}, 1+sqrt{5}) )
c. ( (1-sqrt{5},-1) cup(3,1+sqrt{5}) )
D. ( (3,1+sqrt{5}) )
12
914The set of all points of discontinuities of
fofofof, ( g ) where ( f ) and ( g ) are given by
11
915The domain of the function ( boldsymbol{f}(boldsymbol{x})= ) ( log _{4}left(log _{5}left(log _{3}left(18 x-x^{2}-77right)right) ) is right.
A. ( x in(4,5) )
в. ( x in(0,10) )
c. ( x in(8,10) )
D. None of these
12
916State whether the given statement is true or false. If true enter 1 or else enter
( mathbf{0} )
( (x, y) ) is an ordered pair, whose first component is ( x ) and the second
component is ( boldsymbol{y} )
12
917( mathbf{f} boldsymbol{A}={mathbf{1}, mathbf{2}}, ) from the ( operatorname{set} boldsymbol{A} times boldsymbol{A} times boldsymbol{A} )12
91813.
Let R= {(1,3), (4,2),(2,4),(2,3),(3,1)> be a relation on the
set A = {1,2,3,4).. The relation Ris
[2004]
(a) reflexive
(b) transitive
(c) not symmetric (d) a function
12
919Let ( mathbf{f}(mathbf{x})=(mathbf{x}+mathbf{1})^{2}-mathbf{1}, mathbf{x} geq-mathbf{1} )
Statement-1: The set ( left{mathbf{x}: mathbf{f}(mathbf{x})=mathbf{f}^{-1}(mathbf{x})right} )
( ={mathbf{0},-mathbf{1}} )
Statement-2: ( f ) is a bijection
A. Statement 1 is true, Statement 2 is true,Statement 2 is correct explanation for statement
B. Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for statement 1
c. statement 1 is true, statement 2 is false.
D. Statement 1 is false, Statement 2 is true
12
920A closed set with respect to some binary operation is called semi- group if
( A cdot * ) is associative
B. ( * ) is commutative
c. ( * ) is anti-commutative
D. identity element exists
12
921If ( R ) is relation from a finite ( operatorname{set} A )
having ( m ) elements to a finite set ( B )
having ( n ) elements, then the number of relations from ( boldsymbol{A} ) to ( boldsymbol{B} ) is
( A cdot 2^{m} )
B . ( 2^{m n}-1 )
( c cdot 2 m n )
D. ( m^{n} )
12
922On the set of all integers defined as ( boldsymbol{f} )
( Z rightarrow Z ) such that ( f(x)=[x], ) then it is:
( A cdot ) not a function
B. a many-to-one function
c. an into function
D. an identity function
12
923The function ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) given by
( f(x)=cos x ) for all ( x in R, ) is :
A. surjective bu nor injective
B. injective but nor surjective
c. neither injective nor surjective
D. injective
12
924Assertion
Let ( boldsymbol{f}(boldsymbol{x})=sin boldsymbol{x} )
Statement I ( boldsymbol{f} ) is not a polynomial
function
Reason
Statement II ( n ) th derivative of ( f(x), ) w.r.t.
( x ) is not a zero function for any positive
integer ( n )
A. Statement I is true,Statement II is also true; Statement II is the correct explanation of Statement
B. Statement I is true,Statement II is also true; Statement II is not the correct explanation of Statement
c. Statement l is true,Statement II is false
D. Statement I is false,Statement II is true
11
925Use elements of set ( P={2,3} ) to find the number of all possible ordered pairs.12
9269.
The domain of the function f(x) =
sin-‘(x-3) is
[2004]
(a) [1,2]
(b) (2,3)
(C) [1,2] (d) [2,3]
12
927Determine the domain and range of the
following relations. ( boldsymbol{S}={(boldsymbol{a}, boldsymbol{b}): boldsymbol{b}=|boldsymbol{a}-mathbf{1}|, boldsymbol{a} in boldsymbol{Z} ) and
( |a| leq 3} )
12
928Express ( boldsymbol{R}= )
( (x, y), y=3 x, x in(1,2,3) ) and ( y in(3,6 )
as a set of ordered pair.
12
929Let ( boldsymbol{f}: boldsymbol{N} rightarrow boldsymbol{R} ) be such that ( boldsymbol{f}(mathbf{1})=mathbf{1} )
and ( boldsymbol{f}(mathbf{1})=mathbf{2} boldsymbol{f}(mathbf{2})+mathbf{3} boldsymbol{f}(mathbf{3})+ldots+ )
( boldsymbol{n} boldsymbol{f}(boldsymbol{n})=boldsymbol{n}(boldsymbol{n}+1) boldsymbol{f}(boldsymbol{n}), ) for all
( boldsymbol{n} boldsymbol{epsilon} boldsymbol{N}, boldsymbol{n} geq 2, ) where ( boldsymbol{N} ) is the set of
natural numbers and ( R ) is the set of real
numbers. Then the value of ( boldsymbol{f}(mathbf{5 0 0}) ) is
A. 1000
в. 500
c. ( 1 / 500 )
D. ( 1 / 1000 )
11
930Show that the function ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} )
defined as ( f(x)=x^{2}, ) is not one-one.
12
931( operatorname{Let} a, b, c, epsilon R ) If ( f(x)=a x^{2}+b x+c ) is
such that ( a+b+c=3 ) and ( f(x+ )
( boldsymbol{y})=boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y})+boldsymbol{x} boldsymbol{y}, forall boldsymbol{x}, boldsymbol{y} boldsymbol{epsilon} boldsymbol{R}, ) the
( sum_{n=1}^{10} f(n) ) is equal to
A . 330
в. 165
c. 190
D. 255 55
11
932If ( A={a, b, c, d}, B={1,2,3}, ) find whether
or not the following sets of ordered pairs are relations from ( A ) to ( B ) or not
(a) ( R_{1}={(a, 1),(a, 3)} )
(b) ( R_{2}={(b, 1),(c, 2),(d, 1)} )
( (c) R_{3}={(a, 1),(b, 2),(3, c)} )
12
933If ( f(x)=3 x^{2}-5 x-4, ) then ( f(-2 x) ) is
equal to
B . ( -f(x) )
c. ( 4 f(x) )
D. ( -4 f(x) )
E. none of these
12
934Show that the function ( f(x)=frac{x-2}{x+1} )
for ( boldsymbol{x} neq mathbf{0} ) is increasing.
12
935Solve in
( R: frac{10 x}{x^{2}+9} leq 1 )
12
936Show that the relation ‘a ( R ) b if and only
if ( a-b ) is an even integer defined on the
( Z ) of integers is an equivalence relation.
12
937Prove that the Greatest integer function ( boldsymbol{f}: mathbb{R} rightarrow mathbb{R} ) given by ( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}], ) is neither
one-one nor onto, where ( [x] ) denotes the
greatest integer less than or equal to ( x )
12
938If ( f(x)=frac{1}{(1-x)} ) then ( (text { fofof })(x)=? )
A ( cdot frac{1}{(1-3 x)} )
в. ( frac{x}{(1+3 x)} )
c. ( x )
D. None of these
12
939Find the domain and range of :-
(i) ( sqrt{boldsymbol{x}}-mathbf{1} )
(ii) ( (boldsymbol{x}-mathbf{1}) )
12
940(a) If ( x ) be real, then the expression ( frac{2 a(x-1) sin ^{2} theta}{x^{2}-sin ^{2} theta} ) does not lie between
( 2 a sin ^{2}(theta / 2) ) and ( 2 a cos ^{2}(theta / 2) )
(b) A function ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R}, ) where ( mathrm{R} ) is the
set of real numbers, is defined by ( f(x)=frac{alpha x^{2}+6 x-8}{alpha+6 x-8 x^{2}} ) Find the interval of
values of ( alpha ) for which ( f ) is onto.

Is the function one-to-one for ( boldsymbol{alpha}=mathbf{3} ? )
Justify yout answer.

12
941Given a function ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{B} ; ) where ( boldsymbol{A}= )
{1,2,3,4,5} and ( B={6,7,8} )
The number of mappings of ( g(x): B rightarrow )
( A ) such that ( g(i) leq g(j) ) whenever ( i<j )
is
A . 55
в. 140
c. 10
D. 35 5
11
942topp
( Q ) тур
An ice-cream shop notices the variation between outside temperature and number of customers attending the
shop in the table given above. From the graphs, determine the best possible relationship between the temperature and number of customers attending the
shop
( A )
Number of Customers
3.
Number of Customers
( c )
Number of Customers
D.
Number of Customers
12
943Ordered pairs ( (a, 3) ) and ( (5, x) ) are equal the values of ( a ) and ( x ) are
A .2 and 4
B. 3 and 6
c. 5 and 3
D. 1 and -1
12
944Given ordered pairs :(5,4),(5,5),(5,6) ( (6,4),(6,5),(6,6),(6,7),(8,4),(8,5),(8, )
6), (8,8)
Use these ordered pairs to find the following relation:
( R_{2}= ) “is equal to”
12
945Let ( f(x)= ) ( left{begin{array}{l}1, quad text { if } x text { is a rational number } \ 0, quad text { if } x text { is an irrational number }end{array}right. )
be a function from ( R ) to ( R ) where ( R ) is
the set of real numbers. Find
(i) ( fleft(frac{1}{3}right) )
(ii) ( f(sqrt{7}) )
(iii) ( f(text { fof })(1.4327) )
(iv) ( f(text { fof })(sqrt{3}) )
A. 0,0,0,0
0
в. 1,1,1,1
c. 1,0,1,1
D. 0,1,0,1
12
946( f(x)=1, ) if ( x ) is rational and ( f(x)=0 )
if ( x ) is irrational
then ( (text { fof })(sqrt{5})= )
( A cdot 0 )
B.
( c cdot sqrt{5} )
D. ( frac{1}{sqrt{5}} )
12
947( ln operatorname{eq} x^{2}+y=6 x-14 )
f ( x=0,1,2 )
then the values of ( y ) are
11
948The solution set of ( |boldsymbol{3}-boldsymbol{x}|=boldsymbol{a} ) is?
A ( cdot{3-a, 3+a} ) for ( a geq 0 ) and ( phi ) for ( a<0 )
В. ( [3-a, 3+a] )
( c cdot phi )
D. None of these
11
949If ( f(x)=2^{x} ) then ( frac{f(x+3)}{f(x-1)}= )
A ( . f(x) )
в. ( frac{1}{f(x)} )
c. ( f(4) )
D. ( f(2) )
11
950On the set ( N ) of all natural numbers
define the relation ( R ) by ( a R b ) if and only
if the G.C.D. of ( a ) and ( b ) is 2 . Then ( R ) is:
A. Reflexive, but not symmetrica
B. Symmetric only
C. Reflexive and transitive
D. Reflexive, symmetric and transitive
12
951Determine whether the operation ( ^{prime} O^{prime} ) on
( Z ) defined by ( a O b=a^{b} ) for all ( a, b in Z )
is a binary operation or not:
12
952Find the domain of the following function. ( y=sqrt{-4 x^{2}+4 x+3} )12
953If ( boldsymbol{A}={mathbf{2}, boldsymbol{3}} ) and ( boldsymbol{B}={1,2}, ) then ( boldsymbol{A} times )
( B ) is equal to
A ( cdot{(2,1),(2,2),(3,1),(3,2)} )
B – {(1,2),(1,3),(2,2),(2,3)}
c. {(2,1),(3,2)}
D ( cdot{(1,2),(2,3)} )
12
954ff ( g(x) ) is a polynomial satisfying ( boldsymbol{g}(boldsymbol{x}) boldsymbol{g}(boldsymbol{y})=boldsymbol{g}(boldsymbol{x})+boldsymbol{g}(boldsymbol{y})+boldsymbol{g}(boldsymbol{x} boldsymbol{y})-2 ) for
all real ( x ) and ( y ) and ( g(2)=5 ) then ( g(3) ) is equal to –
A . 10
B . 24
( c cdot 21 )
D. none of these
11
955Consider the equation ( 2+mid x^{2}+4 x+ )
( mathbf{2} mid=boldsymbol{m}, boldsymbol{m} in boldsymbol{R} ) Set of all real values of
( m ) so that given equation has four distinct solutions, is
в. (1,2)
( c cdot(1,3) )
D. (2,4)
11
956If ( boldsymbol{x}=frac{mathbf{4} boldsymbol{a} boldsymbol{b}}{boldsymbol{a}+boldsymbol{b}}, ) where ( boldsymbol{a} neq boldsymbol{b} neq mathbf{0}, ) then the
value of ( frac{x+2 a}{x-2 a}+frac{x+2 b}{x-2 b} ) is
(Assume that everything is defined) This question has multiple correct options
A. Independent of ( a )
B. Independent of ( b )
c. Independent of ( x )
D. None of these
11
957If ( f(x)=x^{3}+3 x^{2}+12 x-2 sin x )
where ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R}, ) then prove that ( boldsymbol{f} ) is
bijective
12
958Let ‘o’ be a binary operation on the set
( Q_{0} ) of all non-zero rational numbers defined by ( a o b=frac{a b}{2}, ) for all ( a, b in Q_{0} )
Find the identity element in ( Q_{0} )
12
959Let ( boldsymbol{A}={boldsymbol{x}, boldsymbol{y}, boldsymbol{z}}, boldsymbol{B}={boldsymbol{u}, boldsymbol{v}, boldsymbol{w}} . ) Then
the function ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{B} ) defined by
( boldsymbol{f}(boldsymbol{x})=boldsymbol{u}, boldsymbol{f}(boldsymbol{y})=boldsymbol{v}, boldsymbol{f}(boldsymbol{z})=boldsymbol{w} ) is
A. Many-one into
B. One-one into
c. One-one onto
D. Many-one onto
12
960A real valued function ( f(x) ) satisfies the functional equation ( boldsymbol{f}(boldsymbol{x}-boldsymbol{y})= )
( boldsymbol{f}(boldsymbol{x}) boldsymbol{f}(boldsymbol{y})-boldsymbol{f}(boldsymbol{a}-boldsymbol{x}) boldsymbol{f}(boldsymbol{a}+boldsymbol{y}), ) where ( boldsymbol{a} )
is a given constant and ( boldsymbol{f}(mathbf{0})= )
( mathbf{1}, boldsymbol{f}(mathbf{2} boldsymbol{a}-boldsymbol{x}) ) is equal to
A ( .-f(x) )
в. ( f(x) )
c. ( f(a)+f(a-x) )
D. ( f(a)-f(a-x) )
12
961The domain of ( mathbf{f}(mathbf{x})=log _{mathbf{x}}left(mathbf{9}-mathbf{x}^{2}right) ) is
A. (-3,3)
B. ( (0, infty) )
c. (0,1)( cup(1, infty) )
D. (0,1)( cup(1,3) )
12
962If ( A={0,1} ) and ( B={1,2,3}, ) show that ( boldsymbol{A} times boldsymbol{B} neq boldsymbol{B} times boldsymbol{A} )12
963Given ( A={2,3,4}, B={2,5,6,7} )
Construct an example of each of the following
(i) an injective mapping from ( A ) to ( B )
(ii) a mapping from ( A ) to ( B ) which is not injective.
(iii) a mapping from ( mathrm{B} ) to ( mathrm{A} )
12
964A pool that is being drained contains ( 300-6 t ) gallons of water after ( t )
minutes of draining. Calculate the gallons of water the pool contains after 15 minutes of draining.
A . 210
B. 250
c. 260
D. 230
12
965Is ( * ) defined on the set {1,2,3,4,5} by
( a * b=L C M ) of ( a ) and ( b, a ) binary
operation ? Justify your answer
12
966If in ( N times N, R ) is a relation defined by
the formula ( (x, y) R(p, q) ) if and only if
( boldsymbol{x}+boldsymbol{q}=boldsymbol{y}+boldsymbol{p}, ) show that ( mathrm{R} ) is an
equivalence relation
12
967The Cartesian product ( boldsymbol{P} times boldsymbol{P} ) has 16
elements among which two elements ( operatorname{are}(a, b) ) and ( (c, d) . ) Find the set ( P ) and
the remaining elements of ( boldsymbol{P} times boldsymbol{P} )
12
968The set values of ( x ) for which function
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} ln boldsymbol{x}-boldsymbol{x}+mathbf{1} )
( A cdot(1, infty) )
B. ( left(frac{1}{e}, inftyright) )
c. ( [e, infty) )
( ( ) )
D ( cdot(0,1) cup(1, infty) )
11
969Let ( S ) be a relation on ( mathbb{R}^{+} ) defined by
( boldsymbol{x} boldsymbol{S} boldsymbol{y} Leftrightarrow boldsymbol{x}^{2}-boldsymbol{y}^{2}=boldsymbol{2}(boldsymbol{y}-boldsymbol{x}), ) then ( boldsymbol{S} ) is
A. Only reflecxive
B. Only symmetric
c. only Trasitive
D. Equivalence
12
970Is it true that ( y=e^{log x} ) for all real ( x )12
971Find the range of the following function:
( f(x)=frac{|x-4|}{x-4} )
( mathbf{A} cdot[-1,1] )
В. ( left[-frac{pi}{2}, frac{3 pi}{2}right] )
c. ( (-2 pi, 0] )
D. ( left[frac{-3 pi}{2}, frac{pi}{2}right. )
12
972Relation ( R ) in the set ( A ) of human beings
in a town at a particular time given by
[
boldsymbol{R}={(boldsymbol{x}, boldsymbol{y}): boldsymbol{x} text { is } boldsymbol{w} boldsymbol{i} f e boldsymbol{o} boldsymbol{f} boldsymbol{y}}
]
enter 1 -reflexive and transitive but not
symmetric 2-reflexive only 3-Transitive only 4-Equivalence 5-Neither reflexive, nor symmetric, nor transitive
12
973If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} boldsymbol{a} boldsymbol{n} boldsymbol{d} boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}}{1+|boldsymbol{x}|} cdot ) then
( f(x) ) is
A. injective but not surjective
B. surjective but not injective
c. injective as well as surjective
D. neither injective nor surjective
12
974Show that the Signum Function ( boldsymbol{f} )
( R rightarrow R, ) is neither one one nor onto?
[
f(x)=left{begin{array}{l}
1, text { if } x>0 \
0, text { if } x=0 \
-1, text { if } x<0
end{array}right.
]
12
975Find the inverse if ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R}_{1}, boldsymbol{f}(boldsymbol{x})= )
( frac{2 x+1}{3} )
12
976( operatorname{Let} f(x)=sin ^{2} frac{x}{2}+cos ^{2} frac{x}{2} ) and ( g(x)= )
( sec ^{2} x-tan ^{2} x . ) The two functions are
equal over the set
A. ( Phi )
в. ( R )
c. ( R-left{x mid x=(2 n+1) frac{pi}{2}, n in Zright} )
D. None of these
12
977Use the elements of ( operatorname{set} A={x, y, z} ) to
form all possible ordered pairs Find the number of elements in the
ordered pair.
12
978The complete solution set of ( |2 x-3|+ )
( |boldsymbol{x}+mathbf{5}| leq|boldsymbol{x}-mathbf{8}| ) is
( A cdotleft[-5, frac{3}{2}right] )
B ( cdot(-infty,-5] )
( mathbf{c} cdotleft[frac{3}{2}, inftyright] )
( mathbf{D} cdot(-infty,-5] cupleft[frac{3}{2}, inftyright] )
11
979The minimum value of ( |boldsymbol{x}|+left|boldsymbol{x}+frac{mathbf{1}}{mathbf{2}}right|+ )
( |boldsymbol{x}-mathbf{3}|+left|boldsymbol{x}-frac{mathbf{5}}{mathbf{2}}right| ) is
( A cdot 2 )
B. 4
( c cdot 6 )
D.
11
980Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be given by ( boldsymbol{f}(boldsymbol{x})= )
( x^{2}+3 . ) Find ( (text { a }){x: f(x)=28} )
the pre-images of 39 and 2 under
12
981Find domain of the function ( boldsymbol{f}(boldsymbol{x})= )
( frac{1}{log _{10}(1-x)}+sqrt{x+2} )
12
982A bank offers ( 5 % ) cash back on petrol
purchase up to ( $ 1500 ) and ( 2 % )
on amount spent on petrol thereafter, in
a calendar year. If functions ( g(x) ) and
( e(x) ) represents cash back earned on
petrol purchases up to ( $ 1,500 ) and in
excess of ( $ 1,500, ) respectively, then the sets of functions that could be used to
determine the amount of cash back
earned is
( mathbf{A} cdot g(x)=0.05 x, 0 leq x1,500 )
C ( cdot g(x)=0.05 x, 0 leq x leq 1,500 ; e(x)= )
[
0.02(x-1,500), x>1,500
]
D ( cdot g(x)=0.05 x, 0 leq x leq 1,500 ; e(x)= )
[
0.02(1,500-x), x>1,500
]
12
983If ( boldsymbol{R}=left{(boldsymbol{x}, boldsymbol{y}): boldsymbol{x}, boldsymbol{y} in boldsymbol{Z}, boldsymbol{x}^{2}+boldsymbol{y}^{2} leq mathbf{4}right} )
is a relation in ( Z ) then domain ( D ) is
( mathbf{A} cdot{-2,-1,0,1,2} )
B ( cdot{-2,-1,0} )
c. {0,1,2}
D. None of these
12
984There are four men and six women on
the city councils. If one council number
is selected for a committee at random,
how likely that it is a woman
12
985(
UPY
11.
Let f(x) be defined for all x > 0 and be continuous.
satisfy f1
=f(x)-f() for all x, y and f(e)=1. Then
(1995)
(a) f(x) is bounded
© *f(x) → 1 as x 70
(b) → O as x > 0
(d) f(x)= ln x
12
986In the group ( Q-{-1} ) under the binary
operation ( + ) defined by ( a * b=a+b+ )
( a b ) the inverse of 10 is
A ( cdot frac{1}{10} )
в. ( frac{11}{10} )
c. ( frac{-11}{10} )
D. ( frac{-10}{11} )
12
98721.
10
(2
-1), where the function f satisfies
21. Let f(a+k) = 16(210-1). where the funct
no k=1
f(x+y)=f(x) f(y) for all natural numbers x, y anda)
Then the natural number ‘a’ is:
tal number ‘a’ is:
JELI
JEEM 2019-9 April (M)
(a) 2 (6) 16 (c) 4 (d) 3
12
988Let ( boldsymbol{A}={mathbf{1}, mathbf{3}, mathbf{5}, mathbf{7}} ) and ( boldsymbol{B}={mathbf{2}, mathbf{4}, mathbf{6}, mathbf{8}} )
be two sets ( R ) be a relation from ( A ) to ( B )
defined by the phrase ( “(x, y) in R Rightarrow )
( boldsymbol{x}>boldsymbol{y}^{prime prime} . ) Find relation ( boldsymbol{R} ) and its domain
and range
12
989for all s’,*+ 2x + 10-3020, then the interval in which
30. For all ‘x’, x2 + 2ax + 10–3a>0, then the interval in which
‘a’ lies is
(2004S)
(a) a<-5
(b) -5<a5
(d) 2<a<5
11
990If ( f(x)=ln left(x^{2}-x+2right) ; R^{+} rightarrow R ) and
( boldsymbol{g}(boldsymbol{x})={boldsymbol{x}}+mathbf{1} ;[mathbf{1}, mathbf{2}] rightarrow[mathbf{1}, mathbf{2}], ) where
( {x} ) denotes fractional part of ( x ). Find
the domain and range of ( f(g(x)) ) when defined.
12
9914) V2 (U) -V2
Suppose f(x) = (x + 1)2 for x>-1. If g(x) is the function
whose graph is the reflection of the graph off (x) with respect
to the line y=x, then g(x) equals
(2002)
(b)
2,X > -1
(a) – VX – 1, x 20
(c) √x+1, xZ-1
+12.x,
(d) √x – 1, x 20
G… dhuf) – 2
+ sin r for
12
992Make n the subject of formula:
( N=90left(2-frac{4}{n}right) ) Find ( n, ) if ( N=108 )
A ( cdot n=frac{360}{180+N} ; n=4 )
в. ( n=frac{180}{180-N} ; n=3 )
c. ( n=frac{360}{180-N} ; n=5 )
D. none of the above
11
99327. If the functions f(x) and g(x) are defined on R → R such that
(0, xerational
O, X e irrational
i g(x) =
(x, X e irrational 81
1x, xe rational then
(f-g)(x) is
(20055)
(a) one-one & onto
(b) neither one-one nor onto
(C) one-one but not onto
(d) onto but not one-one
12
9942
( g(x) )
( 11 quad 19 quad 27 )
The table above defines several values
of the linear function ( g ). Calculate the
value of ( boldsymbol{a} )
( A .31 )
B. 33
c. 35
D. 37
E . 41
12
995Which of following statements is/are
INCORRECT?
I. If ( f(x) ) and ( g(x) ) are one-one, then ( f(x)+g(x) ) is also one-one.
II. If ( f(x) ) and ( g(x) ) are one-one, then ( f(x) cdot g(x) ) is also one-one.
III. If ( boldsymbol{f}(boldsymbol{x}) ) is odd, then it is necessarily
one to one.
A. I and II only
B. II and III only
c. III and I only
D. I, II and III
12
996Frame a formula for the following
statement:
The Fahrenheit temperature ( F ) decreased by 32 is equal to nine-fifths of the centigrade temperature ( C )
A ( cdot F+32=frac{9}{5} C )
в. ( F-32=frac{5}{9} C )
c. ( quad F-32=frac{9}{5} C )
D. none of the above
12
997If ( a neq R ) and the equation ( -3(x- )
( [boldsymbol{x}])^{2}+mathbf{2}(boldsymbol{x}-[boldsymbol{x}])+boldsymbol{a}^{2}=mathbf{0}(text { where }[boldsymbol{x}] )
denotes the greatest integer ( leq x ) ) has
no integral solution, then all possible values of a lie in the interval:
A ( cdot(-2,-1) )
B . ( (-infty,-2) cup(2, infty) )
c. (-1,0)( cup(0,1) )
D. (1,2)
12
998If ( f(x)=a x+b ) and ( f(f(f(x)))= )
( 27 x+13 ) where a and b are real
numbers, then-

This question has multiple correct options
( A cdot a+b=3 )
B. ( a+b=4 )
( c cdot f^{prime}(x)=3 )
D. f'(x)=-3

12
999( operatorname{Let} f(x)=cos ^{-1}left(frac{x^{2}}{1+x^{2}}right) . ) The range
of ( boldsymbol{f} ) is
( mathbf{A} cdotleft[0, frac{pi}{2}right] )
B. ( left[-frac{pi}{2}, frac{pi}{2}right] )
c. ( left[-frac{pi}{2}, 0right] )
D. none of these
12
1000If ( f(x)=x^{2}-7, ) then find the value
( boldsymbol{f}(boldsymbol{a}-boldsymbol{3}) )
A ( cdot a^{2}-6 a-16 )
B . ( a^{2}-10 )
( mathbf{c} cdot a^{2}+21 )
D. ( a^{2}-6 a+2 )
E . ( 2 a-13 )
12
1001Classify the following function ( boldsymbol{f}(boldsymbol{x}) )
defined in ( R rightarrow R ) as injective,
surjective, both or none:
( boldsymbol{f}(boldsymbol{x})=left(boldsymbol{x}^{2}+boldsymbol{x}+mathbf{5}right)left(boldsymbol{x}^{2}+boldsymbol{x}-boldsymbol{3}right) )
A. only injective
B. Only surjective
c. Neither injective not surjective
D. Both injective and surjective.
12
1002Find the integrals of the function:
( sin ^{4} x )
12
1003If ( boldsymbol{R} ) is the largest equivalence relation
on a set ( A ) and ( S ) is any relation on ( A )
then
A ( . R subset S )
в. ( S subset R )
c. ( R=S )
D. none of these
12
1004Range of ( boldsymbol{f}(boldsymbol{x})=left(sin ^{-1} boldsymbol{x}+tan ^{-1} boldsymbol{x}+right. )
( left.sec ^{-1} xright) ) is
A ( cdotleft(frac{pi}{4}, frac{3 pi}{4}right) )
В. ( left[frac{pi}{4}, frac{3 pi}{4}right] )
c. ( left{frac{pi}{4}, frac{3 pi}{4}right} )
D ( cdotleft{0, frac{pi}{4}right} )
12
1005If ( g(x)=x^{2}+x-2 ) and ( frac{1}{2} g o f(x)= )
( 2 x^{2}-5 x+2, ) then which is ( f(x) )
A ( .2 x-3 )
B. ( -2 x-3 )
c. ( x-3 )
D. ( x+3 )
12
1006If ( boldsymbol{A}={1,2,4,5} ) and ( B={a, b} ) Find
the total number of relations from ( A ) to
( boldsymbol{B} )
12
1007If ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-boldsymbol{a}|+|boldsymbol{x}+boldsymbol{b}|, boldsymbol{x} epsilon boldsymbol{R}, boldsymbol{b}> )
( boldsymbol{a}>mathbf{0} . ) Then
( mathbf{A} cdot f^{prime}left(a^{+}right)=1 )
B . ( f^{prime}left(a^{+}right)=0 )
c. ( f^{prime}left(-b^{+}right)=0 )
D. ( f^{prime}left(-b^{+}right)=1 )
11
1008If ( n geq 2 ) then the number of surjections
that can be defined from ( {1,2,3, dots n} ) onto {1,2} is
A ( cdot n^{2}-n )
B ( cdot n^{2} )
( c cdot 2^{n} )
D. ( 2^{n}-2 )
12
1009( boldsymbol{A}={boldsymbol{x} in boldsymbol{R}: boldsymbol{x} neq mathbf{0},-boldsymbol{4} leq boldsymbol{x} leq mathbf{4}} boldsymbol{f} )
( A rightarrow R ) is defined as ( f(x)=frac{|x|}{x} ) then
the range of ( boldsymbol{f} ) is
A ( cdot{1,-1} )
B . ( {x: 0 leq x leq 4} )
c. {1}
D cdot ( {x:-4 leq x leq 0} )
12
1010Classify the following function as injection, surjection or bijection:
( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} operatorname{given} operatorname{by} boldsymbol{f}(boldsymbol{x})=sin boldsymbol{x} )
12
1011If ( boldsymbol{R}={(1,-1),(2,-2),(3,-1)} ) is
relation, then find the range of ( R )
12
1012Let ( f ) and ( g ) be differentiable functions
on ( R ) such that fog is the identify
function. If for some ( a, b, epsilon R, g^{prime}(a)=5 )
and ( g(a)=b, ) then ( f^{prime}(b) ) is equal to
A ( cdot frac{1}{5} )
B. 5
( c cdot 1 )
D.
11
1013ff ( p(x) ) be a polynomial satisfying the
identity ( pleft(x^{2}right)+2 x^{2}+10 x=2 x p(x+ )
1) ( +3, ) then function ( p(x) ) is
A. One-one
B. Many-one
c. Periodic
D. Even
11
1014The range of ( boldsymbol{f}(boldsymbol{x})=mathbf{6}-|mathbf{2} boldsymbol{x}+mathbf{3}| ) is
( A cdot(-infty, 6] )
B. ( [6, infty) )
c. [2,6]
D. ( (0, infty) )
12
1015I total number of runs scored in n matches is
22
+
-n-2) where n>1, and the runs scored in
the kth match are given by k. 2n + 1-k, where 1 <k<n.
Find n.
(2005 – 2 Marks)
11
1016If ( f(x)=frac{2^{x}+2^{-x}}{2}, ) then ( f(x+ )
( y) cdot f(x-y) ) is equal to ( ? )
A ( left.cdot frac{1}{2}[f(x+y)]+f(x-y)right] )
B. ( frac{1}{2}[f(2 x)+f(2 y)] )
c. ( left.frac{1}{2}[. f(x+y)]+f(x-y)right] )
D. None of these
11
101756. Let A and B two sets containing 2 elements and 4 elements
respectively. The number of subsets of Ax B having 3 or
more elements is
[JEEM 2013]
(a) 256 (b) 220 (C) 219 (d) 211
11
1018Let ( f ) be a linear function for which
( boldsymbol{f}(boldsymbol{6})-boldsymbol{f}(2)=12 . ) The value of ( boldsymbol{f}(mathbf{1 2})- )
( f(2) ) is equal the
A . 12
B . 18
( c cdot 24 )
D. 30
11
1019If ( boldsymbol{A} times boldsymbol{B}={(boldsymbol{a}, boldsymbol{x}),(boldsymbol{a}, boldsymbol{y}),(boldsymbol{b}, boldsymbol{x}),(boldsymbol{b}, boldsymbol{y})} )
Find ( A ) and ( B )
12
1020If ( boldsymbol{p}=boldsymbol{q} ) then ( boldsymbol{p} boldsymbol{x}= )
( mathbf{A} cdot mathbf{q} )
B. ( q x )
( mathbf{c} cdot q+x )
D.
11
1021For any two real numbers ( theta phi, ) we define ( theta R phi ) if and only if ( sec ^{2} theta-tan ^{2} phi=1 )
The relation ( boldsymbol{R} ) is
A. Reflexive but not transitive
B. Symmetric but not reflexive
c. Both reflexive and symmetric but not transitive
D. An equivalence relation
12
1022Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) and ( boldsymbol{f}(boldsymbol{x})=log _{e} boldsymbol{x}, boldsymbol{R} )
being the set of real numbers then which of the following is not correct?
A. ( f ) is onto
B. ( f ) is one-one
c. ( f ) is invertible
D. ( f ) is into
12
1023If ( A={2,4} ) and ( B={3,4,5}, ) then
( (boldsymbol{A} cap boldsymbol{B}) times(boldsymbol{A} cup boldsymbol{B}) ) is
A ( cdot{(2,2),(3,4),(4,2),(5,4)} )
B – {(2,3),(4,3),(4,5)}
c. {(2,4),(3,4),(4,4),(4,5)}
D ( cdot{(4,2),(4,3),(4,4),(4,5)} )
12
1024f ( boldsymbol{A}=[mathbf{2}, mathbf{4}, mathbf{5}], boldsymbol{B}=[mathbf{7}, mathbf{8}, mathbf{9}] ) then ( boldsymbol{n}(boldsymbol{A} times )
( B ) ) is equal to?
12
1025Given a relation ( boldsymbol{R}=(mathbf{1}, mathbf{2}),(mathbf{2}, mathbf{3}) ) on the
set of natural numbers, add a minimum
number of ordered pairs so that the enlarged relation is symmetric, transitive and reflexive.
12
1026Let ( R ) be the relation over the set of all
straight lines in a plane such that ( l_{1} R )
( l_{2} Leftrightarrow l_{1} perp l_{2} . ) Then, ( R ) is
A. symmetric
B. reflexive
c. transitive
D. an equivalence relation
12
1027The domain of ( boldsymbol{f}(boldsymbol{x})= )
( sqrt{x-2-2 sqrt{x-3}}- )
( sqrt{x-2+2 sqrt{x-3}}, ) is
A ( .[3,5] )
в. (3,5)
( c cdot[5, infty) )
( D cdot[3, infty) )
12
1028Find ( g ) o ( f ) and ( f circ g ) if ( f: R rightarrow R ) and ( g: )
( R rightarrow R ) are given by ( f(x)=cos x ) and
( g(x)=3 x^{2} . ) Show that ( g circ f neq f circ g )
12
1029Find the second component of an ordered pair (2,-3)
A . 2
B. 3
c. 0
D. – 3
12
1030Total number of equivalence relations defined in the ( operatorname{set} S={a, b, c} ) is
A . 5
в. 3
( c cdot 2^{3} )
D. ( 3^{3} )
12
1031In order that a relation ( R ) defined on a
non-empty set ( A ) is an equivalence
relation
It is sufficient, if ( boldsymbol{R} )
A . is reflexive
B. is symmetric
( mathbf{c} ). is transitive
D. possesses all the above three properties.
12
1032Give a non-empty set ( X . ) Consider ( P(X) ) which is the set of all subsets of ( boldsymbol{X} )
Define the relation ( R ) in ( P(X) ) as follows:
For subsets ( boldsymbol{A} ) and ( boldsymbol{B} ) in ( boldsymbol{P}(boldsymbol{X}), boldsymbol{A} boldsymbol{R} boldsymbol{B} ) if
( boldsymbol{A} subset boldsymbol{B}, ) is ( boldsymbol{R} ) an equivalence relation on
( boldsymbol{P}(boldsymbol{X}) ? )
Justify your answer.
12
1033Let ( boldsymbol{A}={1,2} ) and ( boldsymbol{B}={3,4} . ) Write
( A times B ) and find how many subsets will
( A times B ) have? List them.
12
1034If ( f(x)=a x^{2}+b x+c ) satisfies the
identity ( f(x+1)-f(x)=8 x+3 ) for
all ( x in R . ) Then ( (a, b)= )
A. (2,1)
(i)
в. (4,-1)
c. (-1,4)
D. (-1,1)
11
1035Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be defined by ( boldsymbol{f}(boldsymbol{x})= )
( frac{boldsymbol{x}+mathbf{1}}{boldsymbol{x}^{2}+mathbf{2}}, boldsymbol{x} in boldsymbol{R} ).Find ( boldsymbol{f}(boldsymbol{4}) )
11
1036( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{c}x \ frac{a e^{|x|}+3 . e^{frac{-1}{x}}}{1} \ (a+2) e^{frac{1}{|x|}-e^{frac{-1}{x}}}end{array}right) begin{array}{l}x neq 0 \ 0end{array} )
differentiable at ( boldsymbol{x}=mathbf{0} ) then ( [boldsymbol{a}]=_{–}(mathbb{D} )
denotes greatest integers function
4
B. -1
( c cdot 2 )
( 0 .-3 )
12
1037Prove that the function ( f: R rightarrow R ) be
defined by ( f(x)=left(x^{2}+1right)^{35} ) is not one
one.
12
1038The complete graph of the function ( boldsymbol{f} ) is
shown in the ( x y ) -plane above. For what
value of ( x ) is the value of ( f(x) ) at its
minimum?
A. -5
B. – 3
( c .-2 )
( D )
12
1039If ( A ) and ( B ) are independent event such ( operatorname{that} boldsymbol{P}left(boldsymbol{A} cap boldsymbol{B}^{prime}right)=frac{boldsymbol{3}}{25} ) and ( boldsymbol{P}left(boldsymbol{A}^{prime} cap boldsymbol{B}right)= )
( frac{8}{25}, ) then ( P(A)= )
A . ( 1 / 5 )
в. ( 3 / 8 )
c. ( 2 / 5 )
D. ( 4 / 5 )
12
1040If ( boldsymbol{P}={1,2}, ) Find the ( operatorname{set} boldsymbol{P} times boldsymbol{P} times boldsymbol{P} )12
1041State whether the following statement
is True or False.
( f(x, y)=(3,5) ; ) then ( x=3 ) and ( y=5 )
A. True
B. False
12
1042If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) is defined by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}- )
( 3 x+2, ) then ( fleft(x^{2}-3 x-2right)= )
A ( cdot x^{4}+1 )
B. ( x^{4}-3 x+2 )
c. ( x^{4}-6 x^{3}+2 x^{2}+21 x+12 )
D. ( x^{4}+2 x+2 )
11
1043If ( a-b=4 ) and ( a+b=6, ) find ( a^{2}+b^{2} )
A . 26
B . 13
c. 53
D. 40
11
1044Classify the following function as injection, surjection or bijection:
( boldsymbol{f}: boldsymbol{N} rightarrow boldsymbol{N} ) given by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2} )
12
1045If ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{x}+boldsymbol{3}, ) then find
( frac{boldsymbol{f}(boldsymbol{x}+boldsymbol{h})-boldsymbol{f}(boldsymbol{x}-boldsymbol{h})}{boldsymbol{h}}, boldsymbol{h} neq mathbf{0} )
11
1046If is a binary operation such that ( a * )
( b=a^{2}+b^{2} ) then ( 3 * 5 ) is
A . 34
B. 9
( c cdot 8 )
D. 25
12
1047Determine the domain and range of the
relation R defined by ( boldsymbol{R}=left{left(boldsymbol{x}, boldsymbol{x}^{3}right): boldsymbol{x} ) is right.
a prime number less than ( 10} )
12
1048If ( f(x)=frac{x^{2}+2}{x-1} ) when ( x3, ) for what values of
is the function not defined
( A cdot 3 )
B. 4
( c cdot 5 )
D. 6
12
1049( f(x)=x^{3}-2 x^{2}+3 ) Find ( f(3) )12
1050Find the domain of the following real
functions :
(i) ( f(x)=sqrt{16-x^{2}} )
12
1051Find the maximum value of ( f(x)=1 )
( x^{2} ) if ( -2 leq x leq 2 )
A . 2
B.
c. 0
D. –
E . -2
11
1052Let ( boldsymbol{f}:(boldsymbol{0}, infty) rightarrow boldsymbol{R} ) and ( boldsymbol{g}: boldsymbol{R} rightarrow boldsymbol{R} ) be
defined by ( f(x)=sqrt{x} ) and ( g(x)=x ) Find ( boldsymbol{f}+boldsymbol{g}, boldsymbol{f}-boldsymbol{g}, boldsymbol{f} boldsymbol{g} ) and ( frac{boldsymbol{f}}{boldsymbol{g}} )
11
1053Let ( * ) be a binary operation on ( N ) given by ( a * b=operatorname{LCM}(a, b) ) for all ( a, b in N )
Find ( 5 * 7 )
12
1054Simplify ( frac{1}{1+a^{m-n}}+frac{1}{1+a^{n-m}} )12
1055Let ( A={1,2,3} ) and ( R= )
{(1,1),(1,3),(3,1),(2,2),(2,1),(3,3)
then the relation ( R ) and ( A ) is
A. reflexive
B. symmetric
c. transitive
D. equivalence
12
1056The number of reflexive relation in ( operatorname{set} A )
( ={a, b, c} ) is equal to
A ( cdot 2^{9} )
B ( cdot 2^{4} )
( c cdot 2^{7} )
D. ( 2^{text {6 }} )
12
1057Total number of equivalence relations defined in the ( operatorname{set} S={a, b, c} ) is ?12
1058if ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) is given by ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{4}^{boldsymbol{x}}}{boldsymbol{4}^{boldsymbol{x}}+mathbf{2}} )
for all ( x epsilon R ) then ( fleft(frac{1}{1997}right)+ ) ( boldsymbol{f}left(frac{mathbf{2}}{mathbf{1 9 9 7}}right)+—+boldsymbol{f}left(frac{mathbf{1 9 9 6}}{mathbf{1 9 9 7}}right)= )
( mathbf{A} cdot 997.5 )
в. 998
c . 998.5
D. 999
12
1059If ( f(x) ) is a polynomial in ( x(>0) )
satisfying the equation ( boldsymbol{f}(boldsymbol{x})+ ) ( fleft(frac{1}{x}right)=f(x) cdot fleft(frac{1}{x}right) ) and ( f(2)=-7 )
then ( f(3) ) is equal to
A . -26
в. -27
c. -28
D. -29
11
1060Show that, if ( boldsymbol{f}: boldsymbol{R}-left{frac{mathbf{7}}{mathbf{5}}right} rightarrow boldsymbol{R}-left{frac{mathbf{3}}{mathbf{5}}right} )
is defined by ( f(x)=frac{3 x+4}{5 x-7} ) and ( g: R- ) ( left{frac{3}{5}right} rightarrow R-left{frac{7}{5}right} ) is defined by ( g(x)= )
( frac{7 x+4}{5 x-3} )
then ( log =l_{A} ) and gof ( =l_{B}, ) where ( l_{A} ) and
( l_{B} ) are called identity functions on sets
( A ) and ( B )
11
106128. X and Y are two sets and f: X → Y. If {f(c) = y; c CX,
y CY} and {f}(d) = x; d c Y, X CX), then the true
statement is
(2005S)
(a) f(f(b)) = b
(b) f'(fa))=a
(c) f(f(b)) = b, bcy (d) f(fa)) = a, a cx
12
1062Find ( boldsymbol{f} circ boldsymbol{g} ) and ( boldsymbol{g} ) o ( boldsymbol{f}, ) if ( boldsymbol{f}(boldsymbol{x})= )
( e^{x}, g(x)=log _{e} x )
12
1063The relation ( R ) defined on the ( operatorname{set} A= )
{1,2,3,4,5} by ( R= )
( left{(a, b):left|a^{2}-b^{2}right|<16right}, ) is not given by
A ( cdot{(1,1),(2,1),(3,1),(4,1),(2,3)} )
B・ {(2,2),(3,2),(4,2),(2,4)}
c. {(3,3),(4,3),(5,4),(3,4)}
D. none of these
12
1064List the relation ( R ) defined by ( R= )
( left{(a, b): a leq b^{3}right} ) in Roaster form. ( a, b in )
( N )
12
1065Check the commutativity and associativity of the following binary
operation:
( ‘ *^{prime} ) on ( Z a * b=a-b ) for all ( a, b in Z )
12
1066The relation ( R ) in ( N times N ) such that
( (a, b) R(c, d) Leftrightarrow a+d=b+c ) is
A. reflexive but not symmetric
B. reflexive and transitive but not symmetric
c. an equivalence relation
D. none of these
12
1067Is ( a leq a^{2}, a in R ) true? Why or why not?12
1068Let ( Z ) be the set of integers and ( a R b ) where ( a, b in Z ) if an only if ( (a-b) ) is
divisible by 5 Consider the following statements:
1. The relation ( boldsymbol{R} ) partitions ( boldsymbol{Z} ) into five
equivalent classes.
2. Any two equivalent classes are either equal or disjoint. Which of the above statements is/are
correct?
A. 1 only
B. 2 only
c. Both 1 and 2
D. Neither 1 nor 2
12
1069( boldsymbol{f}: boldsymbol{R} longrightarrow boldsymbol{R} operatorname{are} boldsymbol{g}: boldsymbol{R} longrightarrow boldsymbol{R} ) are defined
by ( f(x)=2 x+3 ) and ( g(x)=x^{2}+7 )
then the values of ( x ) for which ( f[g(x)]= )
25 are
( A cdot pm 1 )
в. ±2
( c .pm 3 )
( mathrm{D} cdot pm 4 )
11
1070A relation ( rho ) on the set of real number ( boldsymbol{R} )
is defined as follows:
( x rho y ) if any only if ( x y>0 . ) Then which of
the following is/are true? This question has multiple correct options
A ( . rho ) is reflexive and symmetric
B. ( rho ) is symmetric but not reflexive
c. ( rho ) is symmetric and transitive
D. ( rho ) is an equivalence relation
12
1071ff ( boldsymbol{f}:[mathbf{0}, boldsymbol{Pi}] rightarrow[-mathbf{1}, mathbf{1}], mathbf{f}(mathbf{x})=cos mathbf{x}, ) then
is.
( A cdot ) one-one
B. onto
c. one-one onto
D. none of these
12
1072If a function such that ( boldsymbol{f}(mathbf{0})= )
( mathbf{2} ; boldsymbol{f}(mathbf{1})=mathbf{3} ) and ( boldsymbol{f}(boldsymbol{x}+mathbf{2})=mathbf{2} boldsymbol{f}(boldsymbol{x})- )
( boldsymbol{f}(boldsymbol{x}+mathbf{1}) forall boldsymbol{x} in boldsymbol{R} ) then ( mathrm{f}(5)= )
( A cdot 7 )
B. 13
( c )
D.
12
1073If ( f ) and ( g ) are two functions such that
( (f g)(x)=(g f)(x) ) for all ( x . ) Then ( f ) and
( g ) may be defined as
A ( cdot f(x)=sqrt{x}, g(x)=cos x )
B . ( f(x)=x^{3}, g(x)=x+1 )
( mathbf{c} cdot f(x)=x-1, g(x)=x^{2}+1 )
D. ( f(x)=x^{m}, g(x)=x^{n} ) where ( m, n ) are unequal integers
12
1074f ( boldsymbol{A}={mathbf{5}, boldsymbol{6}, boldsymbol{7}, boldsymbol{8}} ) and ( boldsymbol{B}={boldsymbol{6}, boldsymbol{8}, boldsymbol{1} boldsymbol{0}} )
find total elements in ( (boldsymbol{A} cup boldsymbol{B}) times(boldsymbol{A} cap )
( B) )
12
1075If ( A=(a, b, c, d), B=(p, q, r, s) . ) then which
of the following are relations from ( A ) to
B? Give reasons for your answer:
This question has multiple correct options
A. ( R_{1}={(a, p),(b, r),(c, s)} )
B. ( R_{2}={(q, b),(c, s),(d, r)} )
C. ( R_{3}={(a, p),(a, q),(d, p),(c, r),(b, r)} )
D. ( R_{4}={(a, p),(a, q),(b, s),(s, b)} )
12
1076Let ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-mathbf{1}|, ) then
( mathbf{A} cdot fleft(x^{2}right)=(f(x))^{2} )
B. ( f(x+y)=f(x)+f(y) )
( c . f(|x|)=mid f(x) )
D. None of these
11
1077The domain of the real valued sine
function ( f(x)=2 sin (x-1) ) is
( (-infty, infty) ). If true enter 1 else 0
12
1078If ( n(A)=5 ) and ( n(B)=7, ) then the
number of relations on ( boldsymbol{A} times boldsymbol{B} ) is :
( mathbf{A} cdot 2^{3} )
B. ( 2^{4} )
( c cdot 2^{25} )
D. ( 2^{70} )
E ( .2^{35 times 35} )
12
1079If ( A={0,1} ) and ( B={1,0}, ) then what is ( A x )
B equal to?
B. {(0,0),(1,1)}
( D cdot A times A )
12
1080A ball is tossed in the air in such a way
that the path of the ball is modeled by
the equation ( y=-x^{2}+6 x ) where ( y )
represents the height of the ball in feet
and ( x ) is the time in seconds. At what
time ( x ) will the ball reach the ground
again?
( mathbf{A} cdot mathbf{6} )
B. 2
( c cdot 3 )
D. 4
E .
12
1081If ( f(x)+fleft(frac{1}{1-x}right)=x ) for all ( x neq 0,1 )
then
A ( quad f(x)=frac{1}{2}left(x-1+frac{1}{x}-frac{1}{1-x}right) )
B. ( f(x)=frac{1}{2}left(1+x-frac{1}{x}+frac{1}{1-x}right) )
c. ( f(x)=frac{1}{2}left(1+x-frac{1}{x}-frac{1}{1-x}right) )
D ( f(x)=frac{1}{2}left(1-x+frac{1}{x}-frac{1}{1-x}right) )
12
1082What is the domain of the function
( f(x)=frac{1}{sqrt{|x|-x}} ? )
A ( cdot(-infty, 0) )
B. ( (0, infty) )
c. ( 0<x1 )
12
108329. If F(x)=(
where f”(x) = f(x) and
g(x)=f'(x) and given that F(5)=5, then F(10) is equal to
(2006 – 3M, -1)
(2) 5 (6) 10 (c) 0 (d) 15
12
1084Which of the following functions from ( Z )
to itself are bijections?
( mathbf{A} cdot f(x)=x^{3} )
B. ( f(x)=x+2 )
c. ( f(x)=2 x+1 )
( mathbf{D} cdot f(x)=X^{2}+x )
12
1085If ( f: R rightarrow R ) such that ( f(x+y)-K x y= )
( f(x)+2 y^{2} ) for all ( x, y in R ) and ( f(1)= )
( 2, f(2)=8 ) then ( f(20)-f(10) )
A. 600
в. 300
c. 60
D. 200
11
1086Given that ( boldsymbol{R}={(boldsymbol{a}, boldsymbol{b}) mid boldsymbol{3} text { divides } boldsymbol{b}} )
is an equivalence relation in steel in the
set of integers ( Z ). What is the number of
partitions of ( Z ? )
12
1087( operatorname{Let} A={1,2,3, dots dots 50} ) and ( B= )
( {2,4,6 dots dots .100} ). The number of
elements ( (x, y) in A times B ) such that ( x+ )
( boldsymbol{y}=mathbf{5 0} )
A .24
B . 25
c. 50
D. 75
12
1088Is inclusion of a subset in another, in
the context of a universal set, an
equivalence relation in the family of subsets of the sets? Justify your
answer
12
1089( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) such that ( boldsymbol{f}(boldsymbol{x})=boldsymbol{e}^{boldsymbol{x}} ) then ( boldsymbol{f} )
is
A. only one-one
B. only into
c. one-one into
D. not a function
12
1090A printer can print 54,000 pages in 24 hours. then the function ( f(x)=2,250 x )
represents
A. The number of pages the printer can print in ( x ) days
B. The number of pages the printer can print in ( x ) hours
c. The number of days it takes the printer to print ( x ) pages
D. The number of hours it takes the printer to print ( x ) pages
12
10913.
4 – x²
Domain of definition of the function f(x) = 32
+ log10 (r? – x), is
[2003]
(a) (-1,0) U (1,2) U (2,00) (b) (a, 2)
(c) (-1,0) (1,2)
(d) (1,2) (2,00).
12
1092If ( a-b=4 ) and ( a+b=6, ) find ( a b )
A .
B. 5
( c cdot 2 )
D. 6
11
109322.
If f :[0,00)— >[0,00), and f(x) = -_ then fis
1 + x
(a) one-one and onto
(b) one-one but not onto
onto but not one-one
(d) neither one-one nor onto
12
1094The range of the function ( boldsymbol{f}(boldsymbol{x})= ) ( frac{sin (pi[x])}{x^{2}+1} ) (where [.] denotes greatest
integer function) is
( A cdot{0} )
в. ( R )
c. (0,1)
() 5
( D cdot(0, infty) )
12
1095In the set of triangles in a plane the relation ‘is similar to’ is an equivalence relation. Prove12
1096Find which of the function are one-one
,onto, many-one,one-many. Justify your
answer
( boldsymbol{f}={(1, mathbf{3}),(mathbf{2}, mathbf{6}),(mathbf{3}, mathbf{9}),(mathbf{4}, mathbf{1 2})} ) define
from ( A ) and ( B ) where
( boldsymbol{A}={mathbf{1}, mathbf{2}, mathbf{3}, mathbf{4}} boldsymbol{B}={mathbf{3}, mathbf{6}, mathbf{9}, mathbf{1 2}, mathbf{1 5}} )
12
1097Let ( f(x)=x^{3}-12 x ) be function
such that the equation ( |boldsymbol{f}(|boldsymbol{x}|)|= ) ( boldsymbol{n}(boldsymbol{n} in boldsymbol{N}) ) has exactly ( boldsymbol{6} ) distinct real
roots then number of possible values of
( boldsymbol{n} ) are :
A . 15
B . 16
c. 17
D. 14
12
1098Let ( A={0,1,2,3} ) and define a relation ( R )
as follows
( R={(0,0),(0,1),(0,3),(1,0),(1,1),(2,2) )
(3,0),(3,3)}
Is ( mathrm{R} ) reflexive, symmetric and transitive
( ? )
12
1099( * ) is a binary operation on ( Z ) such that:
( boldsymbol{a} * boldsymbol{b}=boldsymbol{a}+boldsymbol{b}+boldsymbol{a} boldsymbol{b} )
The solution of ( (3 * 4) * x=-1 ) is
A . 1
B. –
( c cdot 4 )
D. 3
12
1100ff ( f(x)=a x+b ) and ( g(x)=c x+d )
( operatorname{then} f(g(x))=g(f(x)) ) implies
( mathbf{A} cdot f(a)=g(c) )
B. ( f(b)=g(b) )
c. ( f(d)=g(b) )
D. ( f(c)=g(a) )
12
1101( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{c}x \ frac{a e^{|x|}+3 . e^{frac{-1}{x}}}{1} \ (a+2) e^{frac{1}{|x|}-e^{frac{-1}{x}}}end{array}right) begin{array}{l}x neq 0 \ 0end{array} )
differentiable at ( boldsymbol{x}=mathbf{0} ) then ( [boldsymbol{a}]=_{–}(mathbb{D} )
denotes greatest integers function
4
B. -1
( c cdot 2 )
( 0 .-3 )
12
1102The three solutions of the equation ( boldsymbol{f}(boldsymbol{x})=mathbf{0} ) are ( -mathbf{2}, mathbf{0}, ) and ( mathbf{3} . ) Therefore, the
three solutions of the equation ( f(x- )
2) ( =mathbf{0} ) are
12
1103Let ( boldsymbol{f}: mathbb{R} rightarrow mathbb{R} ) defined by ( boldsymbol{f}(boldsymbol{x})= )
( frac{a x^{2}+a x+b}{a x+b} ) then
A. ( f ) is many one
B. ( f ) is one-one
c. ( f ) is onto
D. range of ( f ) is not a single-tone set
12
1104Equation 1:
and 2 ( boldsymbol{x} ) ; -2 ( mathbf{0} ) 4
( 4 quad 2 )
Equation 2:
( boldsymbol{x} ) ( -8 quad-4 ) ( mathbf{0} ) What is and 4
-8 -7 7 -6 -5
The linear equations represented by the data shown in the above tables form a
system, what is the ( x- ) coordinate of
the solution to that system?
( mathbf{A} cdot 4 )
B. 8
( mathbf{c} .0 )
D. -6
12
1105Let ( x ) be a real number ( [x] ) denotes the
greatest integer function, and ( {x} )
denotes the fractional part and ( (x) )
denotes the least integer function,then
solve the following. ( [x]^{2}+2[x]= )
( 3 x, 0 leq x leq 2 )
( mathbf{A} cdot{0,1} )
B . {-1,1}
c. {0,-1}
D ( cdot{2,1} )
11
1106Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be defined by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{4} )
then
A. ( f ) may be one-one and onto
B. ( f ) is one-one and onto
c. fis one-one but not onto
D. fis neither one-one nor onto
12
1107If ( f(x)=frac{x+1}{x-1} ) the show that ( f(x)+ ) ( fleft(frac{1}{x}right)=0 )11
1108( operatorname{Let} boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{9}^{boldsymbol{x}}}{mathbf{9}^{boldsymbol{x}}+mathbf{3}} cdot operatorname{Show} boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(mathbf{1}- )
( x)=1, ) and hence evaluate ( fleft(frac{1}{1996}right)+fleft(frac{2}{1996}right)+ )
( boldsymbol{f}left(frac{mathbf{3}}{mathbf{1 9 9 6}}right)+ldots+boldsymbol{f}left(frac{mathbf{1 9 9 5}}{mathbf{1 9 9 6}}right) )
11
110917.
log2 (x + 3).
The domain of definition of f(x)=
x² + 3x+2
(a) R{-1,-2}
(b) (-2, 0)
(20015)
(c) R{-1, -2,-3}
d) (-3,0){-1,-2)
12
1110Is it true that the every relation which is
symmetric and transitive is also reflexive ? Give reasons.
12
1111Find the domain of the function defined
by ( f(x)=frac{2}{3 x^{2}+1} )
A. R-II
B.
c. R-(-1)
D. R-IO
12
1112If ( boldsymbol{A}={1,2,3}, B={1,4,6,9} ) and ( R )
is a relation from ( A ) to ( B ) defined by ( x ) is
greater than ( y ). The range of ( R ) is
( mathbf{A} cdot{1,4,6,9} )
в. {4,6,9}
( c cdot{1} )
D. none of these
12
1113ff ( f(x)+f(1-x)=10, ) then the value
of ( boldsymbol{f}left(frac{1}{10}right)+boldsymbol{f}left(frac{2}{10}right)+ldots ldots ldots+boldsymbol{f}left(frac{9}{10}right) ) is
A . 45
B . 50
( c .90 )
D. Cannot be determined
11
1114Which of the following functions from ( boldsymbol{A} )
to ( B ) are one-one and onto?
(i) ( boldsymbol{f}_{1}={(mathbf{1}, mathbf{3}),(mathbf{2}, mathbf{5}),(mathbf{3}, mathbf{7})} ; boldsymbol{A}= )
( {1,2,3}, B={3,5,7} )
(ii) ( f_{2}={(2, a),(3, b),(4, c)} ; A= )
( {2,3,4}, B={a, b, c} )
(iii) ( boldsymbol{f}_{3}={(boldsymbol{a}, boldsymbol{x}),(boldsymbol{b}, boldsymbol{x}),(boldsymbol{c}, boldsymbol{z})} ; boldsymbol{A}= )
( {a, b, c, d} ; B={x, y, z} )
12
1115Given the function ( f(x) ) such that ( 2 f(x)+x fleft(frac{1}{x}right) )
( 2 fleft(left|sqrt{2} sin pileft(x+frac{1}{4}right)right|right)= )
( 4 cos ^{2} frac{pi x}{2}+x cos left(frac{pi}{x}right), ) then which one
of the following is correct? This question has multiple correct options
A ( cdot f(2)+fleft(frac{1}{2}right)=1 )
B. ( f(1)=-1, ) but the values of ( f(2), fleft(frac{1}{2}right) ) cannot be determined
c. ( f(2)+f(1)=fleft(frac{1}{2}right) )
D. ( f(2)+f(1)=0 )
12
1116If ( f(x+y)=f(x y) ) and ( f(1)=5, ) then
the value of ( sum_{k=0}^{6} f(k) )
A . 25
B. 35
( c .36 )
D. 24
11
1117On the set ( S ) of all real numbers, define a
relation ( boldsymbol{R}={(boldsymbol{a}, boldsymbol{b}): boldsymbol{a} leq boldsymbol{b}} . ) Show that
( mathrm{R} ) is reflexive.
12
1118If ( x ) and ( y ) coordinate of a point is (3,10)
then the ( x ) co-ordinate is
A . 0
B. 3
c. 10
D. None of the above
12
1119Find the domain of the following
function:
( f(x)=sqrt{1-sqrt{1-sqrt{1-x^{2}}}} )
the domain is ( [-a, a], ) then a is?
12
1120Let ( boldsymbol{A}={mathbf{1}, mathbf{2}, mathbf{3}, mathbf{4}, mathbf{5}}, boldsymbol{B}={mathbf{2}, mathbf{3}, mathbf{6}, mathbf{7}} )
Then the number of elements in
( (boldsymbol{A} times boldsymbol{B}) cap(boldsymbol{B} times boldsymbol{A}) ) is
A . 18
B. 6
( c cdot 4 )
( D )
12
1121The range of ( f(x)=frac{1-tan x}{1+tan x} ) is
( A cdot(-infty, infty) )
в. ( (-infty, 0) )
c. ( (0, infty) )
D. ( (-infty,-1) cup(-1, infty) )
12
1122Let ( boldsymbol{F}(boldsymbol{x})=boldsymbol{x}^{3}-boldsymbol{3} boldsymbol{x}+mathbf{1} )
find no. of real soln, of ( boldsymbol{F}(boldsymbol{F}(boldsymbol{x}))=mathbf{0} )
12
1123The length, I metres, of a garden is 78.5
metres, correct to the nearest half
metre. Complete this statement about
the value of I.
( leq 1< )
11
1124Let ( A ) and ( B ) be two sets such that ( A times )
( B ) Consists of 6 elements. If three
elements of ( A times B ) are
(1,4),(2,6),(3,6) then ( B times A . ) is ( = )
( {(a, 1),(6,1),(4, d),(c, 2),(4,3)(b, 3)} )
Find ( boldsymbol{a}+boldsymbol{b}+boldsymbol{c}+boldsymbol{d} )
12
1125The minimum value of ( f(x)=x^{2}+ )
( mathbf{2} boldsymbol{x}+mathbf{3}, boldsymbol{x} in boldsymbol{R} ) is equal to
( A cdot 2 )
B. 3
( c cdot 4 )
( D )
11
1126If ( a^{2}+b^{2}+c^{2}=1, ) find the range of
( boldsymbol{a} boldsymbol{b}+boldsymbol{b} boldsymbol{c}+boldsymbol{c} boldsymbol{a} )
12
1127Show that if ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{B} ) and ( boldsymbol{g}: boldsymbol{B} rightarrow boldsymbol{C} )
are onto, then ( g o f: A rightarrow C ) is also onto.
12
1128If ( boldsymbol{A}={1,2,3}, ) show that a one-one
function ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{A} ) must be onto.
12
1129The solution set of the equation ( mid x- ) ( mathbf{3} mid=boldsymbol{x}-mathbf{3} ) is?
B. [3,0]
( c cdot phi )
D. The set R of all real numbers
11
1130The domain of the functions ( f(x)= ) ( sqrt{log left(2 x-x^{2}right)} ) is
A ( .(0,2) )
B. [0,2]
( c cdot{1} )
D. none
12
1131Determine whether or not the definition
of ( * ) On ( Z^{+} ) define ( * ) by ( a * b=|a-b| )
gives a binary operation. If the event that ( * ) is not a binary operation give justification of this.
12
1132The function ( boldsymbol{f}:(-infty,-1) rightarrowleft(mathbf{0}, e^{5}right] )
defined by ( f(x)=e^{x^{3}-3 x+2} ) is
A. many-one and onto
B. many-one and into
c. one-one and onto
D. one-one and into
12
1133( boldsymbol{f}: boldsymbol{B} rightarrow boldsymbol{B}, ) if ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}, ) then the
function ( boldsymbol{f} ) is
11
1134Let ( S ) be a relation on the set ( R ) of all
real numbers defined by ( boldsymbol{S}= )
( left{(a, b) in R times R: a^{2}+b^{2}=1right} . ) Prove
that ( S ) is not an equivalence relation on
( boldsymbol{R} )
12
1135Let ( * ) be binary operation on the set of all non-zero real numbers, given by ( a * ) ( b=frac{a b}{5} ) for all ( a, b, in R-{0} . ) Find the
value of ( x, ) given that ( 2 *(x * 5)=10 )
12
1136Relation ( R ) in the set ( A ) of human beings
in a town at a particular time given by
[
boldsymbol{R}=
]
( {(x, y): x text { and } y ) work at the same plac
enter 1 -reflexive and transitive but not
symmetric 2-reflexive only 3-Transitive only
4-Equivalence 5-None
12
1137If ( f_{1}(x) ) and ( f_{2}(x) ) are defined on
domains ( D_{1} ) and ( D_{2} ) respectively, then
( f_{1}(x)+f_{2}(x) ) is defined on ( D_{1} cap D_{2} )
A. True
B. False
11
1138The relation ‘has the same father as’
over the set of children is:
A. Only reflective
B. Only symmetric
c. only transitive
D. An equivalence relation
12
1139( f(x)=left(a-x^{n}right)^{frac{1}{n}} ) then ( f o f(x) ) is?12
1140Find the range of the following function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{2}, boldsymbol{x} in boldsymbol{R} )12
1141f ( boldsymbol{f}: boldsymbol{x} rightarrow boldsymbol{p}+boldsymbol{q} boldsymbol{x}, boldsymbol{f}^{-1}(boldsymbol{9})=(-1) ) and
( boldsymbol{f}(mathbf{3})=(-11) . ) find the value of ( boldsymbol{p} ) and ( boldsymbol{q} )
12
1142Which of the following is an equivalence relation?
1) ( xy )
3) ( x-y ) is divisible by 5
4) ( x ) divides ( y )
12
1143Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be defined by ( boldsymbol{f}(boldsymbol{x})= )
( frac{boldsymbol{x}+mathbf{1}}{boldsymbol{x}^{2}+mathbf{2}}, boldsymbol{x} in boldsymbol{R} ).Find ( boldsymbol{f}(mathbf{0}) )
11
1144( boldsymbol{f}: boldsymbol{N} rightarrow boldsymbol{N}-{1,2,3} ) defined by
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}+mathbf{3} ) is
A. one-one only
B. onto only
c. bijective
D. neither one-one nor onto
12
114550. Let R be the set of real numbers.
Statement-1: A= {(x,y) ERR :y-x is an integer is an
equivalence relation on R.
Statement-2: B = {(x, y) E RⓇR:x=ay for some rational
number a} is an equivalence relation on R.
(2011
(a) Statement-1 is true, Statement-2 is true; Statement-2 is
not a correct explanation for Statement-1.
(b) Statement-l is true, Statement-2 is false.
Statement-1 is false, Statement-2 is true.
(d) Statement-1 is true, Statement-2 is true; Statement-2 is
a correct explanation for Statement-1.
12
1146If ( f(g(a))=0 ) where ( g(x)=frac{x}{4}+2 ) and
( boldsymbol{f}(boldsymbol{x})=left|boldsymbol{x}^{2}-boldsymbol{3}right|, ) find the possible value
of ( boldsymbol{a} )
This question has multiple correct options
A. ( -8+4 sqrt{3} )
(
B . ( -(8+4 sqrt{3}) )
( c cdot 6 )
D. 18
12
1147In the set ( Q^{+} ) of all positive rational
numbers, the operation ( * ) is defined by the formula ( a * b=frac{a b}{6} . ) Then, the
inverse of 9 with respect to ( * ) is
A . 4
B. 3
( c cdot frac{1}{9} )
D.
12
1148If ( f(x)=x^{2}-3 x+4 ), then find the
value of ( x ) satisfying ( f(x)=f(2 x+1) )
11
1149The relation ‘is a sister of in the set of
human beings is
A. only transitive
B. only symmetric
c. equivalent
D. None of these
12
1150A function ( f(x) ) is defined as ( f(x)= )
( boldsymbol{x}^{2}+mathbf{3} . ) Find ( boldsymbol{f}(mathbf{0}), boldsymbol{f}(mathbf{1}), boldsymbol{f}left(boldsymbol{x}^{2}right), boldsymbol{f}(boldsymbol{x}+mathbf{1}) )
and ( boldsymbol{f}(boldsymbol{f}(mathbf{1})) )
11
1151If ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} ) and ( boldsymbol{g}(boldsymbol{x})=|boldsymbol{x}|, ) then ( boldsymbol{f}(boldsymbol{x})+ )
( g(x) ) is equal to
( mathbf{A} cdot mathbf{0} )
B . ( 2 x )
( mathbf{c} cdot 2 x ) if ( x geq 0 )
D. ( 2 x ) if ( x leq 0 )
12
1152( boldsymbol{f}(boldsymbol{x})=mathbf{1} ) and ( phi(boldsymbol{x})=sin ^{2} boldsymbol{x}+cos ^{2} boldsymbol{x} )
Show that both functions are equal?
12
1153( boldsymbol{R}_{4}= )
{(1,3),(2,5),(4,7),(5,9),(3,1)}
Is it a mapping? IF True enter 1 else enter 0
12
1154Let ( * ) a binary operation on ( boldsymbol{Q}-{1} )
defined by ( a * b=a+b-a b ) for all
( boldsymbol{a}, boldsymbol{b} in boldsymbol{Q}-{1} )
Then, find the identity element in ( Q- )
{1}
12
1155Let ( boldsymbol{A}=boldsymbol{R}_{0} times boldsymbol{R}, ) where ( boldsymbol{R}_{0} ) denote the
set of all non-zero real numbers. A
binary operation ‘ ( O^{prime} ) is defined on ( A ) as follows: ( (a, b) O(c, d)=(a c, b c+d) ) for
all ( (a, b)(c, d) in R_{0} times R )
Find the identity element in ( boldsymbol{A} )
12
1156Let N denote the set of all natural
numbers. Define two binary relations on ( mathrm{N} ) as ( boldsymbol{R}_{1}={(boldsymbol{x}, boldsymbol{y}) boldsymbol{epsilon} boldsymbol{N} times boldsymbol{N}: boldsymbol{2} boldsymbol{x}+boldsymbol{y}= )
( mathbf{1 0}} ) and ( boldsymbol{R}_{2}={(boldsymbol{x}, boldsymbol{y}) boldsymbol{epsilon} boldsymbol{N} times boldsymbol{N}: boldsymbol{x}+ )
( mathbf{2} boldsymbol{y}=mathbf{1 0}} . ) Then
( A cdot ) Both ( R_{1} ) and ( R_{2} ) are transitive relations
B. Both ( R_{1} ) and ( R_{2} ) are symmetric relations
C. Range of ( R_{2} ) is {1,2,3,4}
D. Range of ( R_{1} ) is {2,4,8}
12
1157( operatorname{Let} f(x)=sin ^{-1} sin x+ )
( cos ^{-1} cos x, quad g(x)= )
( m x quad ) and ( quad h(x)=x ) are three
functions. Now ( g(x) ) divided area
between ( boldsymbol{f}(boldsymbol{x}), boldsymbol{x}=boldsymbol{pi} ) and ( boldsymbol{y}=boldsymbol{0} ) into two
equal parts. The value of m is:
A ( cdot frac{1}{4} )
B. ( frac{3}{4} )
( c cdot frac{5}{4} )
D. ( frac{7}{4} )
12
1158( boldsymbol{R}_{1}= )
( {(boldsymbol{a}, boldsymbol{a}),(boldsymbol{a}, boldsymbol{b}),(boldsymbol{a}, boldsymbol{c}),(boldsymbol{b}, boldsymbol{c}),(boldsymbol{c}, boldsymbol{a}),(boldsymbol{b}, boldsymbol{b}) )
12
1159The range of ( y=sin 3 x+sin x ) is-
A ( cdotleft[frac{-8}{3 sqrt{3}}, frac{8}{3 sqrt{3}},right] )
в. ( left[frac{-4}{3 sqrt{3}}, frac{4}{sqrt[3]{3}}right] )
c. ( left[frac{-3}{sqrt{2}}, frac{3}{sqrt{2}},right] )
D. ( left[frac{-3}{3 sqrt{2}}, frac{3}{2 sqrt{2}},right] )
12
1160If ( x ) and ( y ) co-ordinate of a point is ( (3,10), ) then ( y ) co-ordinate is
A . 0
B. 3
c. 10
D. None of the above
12
1161The domain of the function ( f(x)= ) ( frac{sin ^{-1}(x-3)}{sqrt{9-x^{2}}} ) is
A . [2,3]
в. [1,2]
c. [1,2]
D. [2,3)
12
1162Check the injectivity and surjectivity of the function ( boldsymbol{f}: boldsymbol{N} rightarrow boldsymbol{N} ) given by
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2} )
12
1163If ( f(x)=2 x-1, ) if ( x>1 ;=x^{2}+1, ) if
( -1 leq x leq 1, ) then
( frac{f(1)+f(3)+f(0)}{f(2)+f(-1)+fleft(frac{1}{2}right)} ) is?
A ( cdot frac{32}{5} )
в. ( frac{32}{25} )
c. ( frac{5}{32} )
D. ( frac{25}{32} )
12
116416. 17/11, 0) –> (2, 6) is given by A(1) = 1 +- then ”(x) equals
(a) (Vx?-)/2 (b) 1/(1+22) (20015)
12
1165Give examples of two functions ( boldsymbol{f} )
( N rightarrow Z ) and ( g: Z rightarrow Z ) such that ( g circ f )
is injective but ( g ) is not injective.
12
1166The function ( boldsymbol{f}: boldsymbol{N} rightarrow boldsymbol{N} ) defined by
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}-mathbf{5}left[frac{boldsymbol{x}}{mathbf{5}}right], ) where ( mathrm{N} ) is the set of
natural numbers and ( [x] ) denotes the
greatest integer less than or equal to ( x )
is
A. one-one but not onto
B. neither one-one nor onto
c. onto but not one-one
D. one-one and onto
12
1167( y=sin ^{2} x+4 sin x+5 . ) Then which of
the following statements is correct? ( (D: text { Domain, } R: text { Range }) )
A ( D=R )
( R )
B . ( D:[2,10] )
c. ( D:[2,10]-{0} )
D. ( R:[2,10] )
12
1168Let ( boldsymbol{A}={mathbf{1}, mathbf{2}} ) and ( boldsymbol{B}={mathbf{2}, mathbf{3}, mathbf{4}}, ) then
( boldsymbol{A} times boldsymbol{B}= )
{(1,2),(1,3),(1,4),(2,2),(2,3),(2,4)}
If true enter 1 , or else enter 0 .
12
1169Let ( boldsymbol{f}: mathbb{N} rightarrow mathbb{N} ) be defined by ( boldsymbol{f}(boldsymbol{n})= )
( left{begin{array}{ll}frac{n+1}{2}, & text { if } n text { is odd } \ frac{n}{2}, & text { if } n text { is even }end{array} n in mathbb{N}right. )
State whether the function ( f ) is
bijective. Justify your answer.
12
1170If ( f(x)=x+7 ) and ( g(x)=x-7, x in R )
find ( (f o g)(7) . ) function ( f: R rightarrow R )
defined by ( f(x)=frac{3 x+5}{2} ) is an invertible
function, find ( boldsymbol{f}^{-1} )
12
1171If ( boldsymbol{f}left(frac{boldsymbol{x}-boldsymbol{4}}{boldsymbol{x}+mathbf{2}}right)=mathbf{2} boldsymbol{x}+mathbf{1},(boldsymbol{x} in boldsymbol{R}-mathbf{1},-mathbf{2}) )
then ( int f(x) d x ) is equal to:
(where ( C ) is constant of integration)
A ( cdot 12 log _{e}|1-x|+3 x+C )
B. ( -12 log _{e}|1-x|-3 x+C )
C. ( 12 log _{e}|1-x|-3 x+C )
D. ( -12 log _{e}|1-x|+3 x+C )
12
1172Let ( boldsymbol{f}:{mathbf{1}, mathbf{3}, mathbf{4}} rightarrow{mathbf{1}, mathbf{2}, mathbf{5}} ) and ( boldsymbol{g}: )
( {1, mathbf{2}, mathbf{5}} rightarrow{1,3} ) be given ( boldsymbol{f}= )
( {(mathbf{1}, mathbf{2}),(mathbf{3}, mathbf{5}),(mathbf{4}, mathbf{1})} ) and ( boldsymbol{g}= )
( {(mathbf{1}, mathbf{3}),(mathbf{2}, mathbf{3}),(mathbf{5}, mathbf{1})} . ) Write down ( boldsymbol{g} boldsymbol{o} boldsymbol{f} )
12
1173( f ; A rightarrow B ) defined by ( f(x)=2 x+3 )
and if ( boldsymbol{A}={-mathbf{2},-mathbf{1}, mathbf{0}, mathbf{1}, mathbf{2}}, boldsymbol{B}= )
( {-1,1,3,5,7}, ) then which of function
is f?
A. One-one
B. onto
c. Bijection
D. constant
12
1174If ( a-b=0.9 ) and ( a b=0.36 ; ) find ( a^{2}- )
( boldsymbol{b}^{2} )
( mathbf{A} cdot pm 7.65 )
B . ±1.35
( mathrm{c} .pm 1.5 )
D. ±4.5
11
1175Show that the binary operation ( * ) on ( Z ) defined by ( a * b=3 a+7 b ) is not
commutative.
12
1176If ( boldsymbol{A}=left{boldsymbol{x} epsilon boldsymbol{R} / frac{boldsymbol{pi}}{boldsymbol{4}} leq boldsymbol{x} leq frac{boldsymbol{pi}}{boldsymbol{3}}right} ) and ( boldsymbol{f}(boldsymbol{x})= )
( sin x-x, ) then ( f(A)=: )
A. ( left[frac{sqrt{3}}{2}-frac{pi}{3}, frac{1}{sqrt{2}}-frac{pi}{4}right] )
B. ( left[-frac{1}{sqrt{2}}-frac{pi}{4}, frac{sqrt{3}}{2}-frac{pi}{3}right] )
( ^{mathbf{C}} cdotleft[-frac{pi}{3}, frac{-pi}{4}right] )
D. ( left[frac{pi}{4}, frac{pi}{3}right] )
12
1177Let ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x}) boldsymbol{f}(boldsymbol{y}) forall boldsymbol{x}, boldsymbol{y} in )
( boldsymbol{R}, boldsymbol{f}(mathbf{5})=mathbf{2}, boldsymbol{f}^{prime}(mathbf{0})=mathbf{3}, ) then ( boldsymbol{f}^{prime}(mathbf{5}) )
equals:
A .4
B.
( c cdot frac{1}{2} )
( D )
11
1178If ( f(x)=2, g(x)=x^{2}, h(x)=2 x, forall x in )
( boldsymbol{R} ) then find ( (boldsymbol{f}(boldsymbol{g}(boldsymbol{h}(boldsymbol{x})))) )
12
1179[2009
16. For real x, let )- +5x+1, then
(a) Sis onto R but not one-one
(b) is one one and onto R
(c) fis neither one-one nor onto R
(d) is one-one but not onto R
12
1180( operatorname{Let} A={1,2,3}, B={4,5,6,7} ) and
let ( boldsymbol{f}={(mathbf{1}, boldsymbol{4}),(mathbf{2}, mathbf{5}),(mathbf{3}, mathbf{6})} ) be a
function from ( A ) to ( B ). State whether ( f ) is
one-one or not.
12
1181The function ( boldsymbol{f}:(-infty,-1) rightarrowleft(mathbf{0}, e^{5}right] )
defined by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{e}^{boldsymbol{x}^{3}-boldsymbol{3} boldsymbol{x}+boldsymbol{2}} ) is
A. many-one and onto
B. many-one and into
c. one-one and onto
D. one-one and into
12
1182If ( A={1,3,5,7} ) and ( B={1,2,3,4,5,6,7, )
83, then the number of one-to-one functions from A into B is
A ( cdot 1340 )
B. 1860
( c cdot 1430 )
D. 1880
E . 1680
12
1183Given ( boldsymbol{A}={mathbf{5}, mathbf{6}, mathbf{7}} ) and ( boldsymbol{B}={mathbf{3}, mathbf{4}} )
Form all possible ordered pairs and write the total number of ordered pairs formed. So that :
Both the components are from A
Find the total number of such pairs.
12
1184( f(x)=2 x+1, ) then ( fleft(frac{x}{2}right)=? )12
1185Let ( * ) be a binary operation on ( Z ) defined by ( a * b=a+b-4 ) for all ( a, b in Z )
Find the identity element in ( Z )
12
1186( boldsymbol{p}=mathbf{3 . 8 2}, boldsymbol{q}=-mathbf{4 . 3 9} )
( [p+q] ) is equal to
A . -2
B . 2
( c cdot 0 )
D. -1
11
1187ff ( boldsymbol{x} times boldsymbol{y}=boldsymbol{x}^{2}+boldsymbol{y}^{2}-boldsymbol{x} boldsymbol{y} ) then the value
of ( 9 times 11 ) is :
A . 93
B. 103
( c cdot 113 )
D. 121
12
1188If the function ( boldsymbol{E}: boldsymbol{R} rightarrow ) defines by
( f(x) frac{3^{x}+3^{-x}}{2} ) than show that ( f(x+y)+ )
( boldsymbol{f}(boldsymbol{x}-boldsymbol{y})=2 boldsymbol{f}(boldsymbol{x}) boldsymbol{f}(boldsymbol{y}) )
12
1189Find ( f circ g ) and ( g circ f, ) if ( f(x)=sin ^{-1} x )
( boldsymbol{g}(boldsymbol{x})=boldsymbol{x}^{2} )
12
11907.
Let S={1,2,3,4}. The total number of unordered pairs of
disjoint subsets of S is equal to
(2010)
(a) 25 (6) 34 (6) 42 (d) 41
11
1191Find the domain and range of each of the following real value functions:
( boldsymbol{f}(boldsymbol{x})=sqrt{boldsymbol{x}-1} )
11
1192If ( boldsymbol{A}=[-mathbf{1}, mathbf{1}] ) then ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{A} ) defined
by ( f(x)=sin pi x ) is
A. one-one but not onto
B. onto but not one-one
c. neither one – one nor onto
D. bijective
12
1193Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a function defined by
( f(x)=frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} ) then
A. ( f ) is a bijection
B. ( f ) is an injection only
c. ( f ) is a surjection
D. ( f ) is neither an injection nor a surjection
12
1194If ( boldsymbol{x}=-1, ) then find ( frac{boldsymbol{x}^{4}-boldsymbol{x}^{3}+boldsymbol{x}^{2}}{boldsymbol{x}-mathbf{1}} )
A. ( -frac{3}{2} )
в. ( -frac{1}{2} )
c. 0
D. ( frac{1}{2} )
E ( cdot frac{3}{2} )
12
1195If ( f(x)=3 x-2 ) and ( (g o f)^{-1}(x)=x- )
2, then find the function ( g(x) )
12
1196Which of the following function is one-
one?
A ( . f: R rightarrow R ) given by ( f(x)=|x-1| ) for all ( x in R )
B. ( g:left[-frac{pi}{2}, frac{pi}{2}right] rightarrow R ) given by ( g(x)=|sin x| ) for all ( x in )
( left[frac{-pi}{2}, frac{pi}{2}right] )
( mathbf{c} cdot_{h}:left[frac{-pi}{2}, frac{pi}{2}right] in R operatorname{given} operatorname{by} h(x)=sin x ) for all ( x in )
( left[frac{-pi}{2}, frac{pi}{2}right] )
( mathbf{D} cdot phi: R rightarrow R ) given by ( f(x)=x^{2}-4 ) for all ( x in R )
12
1197Let ( T ) be the set of all triangles in the
Euclidean plane, and let a relation ( R ) on ( T ) be defined as ( a R b, ) if ( a ) is congruent to
( b ) for all ( a, b in T . ) Then, ( R ) is
A. reflexive but not symmetric
B. transitive but not symmetric
c. equivalence
D. none of these
12
1198( operatorname{Let} boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R}, boldsymbol{g}(boldsymbol{x})=boldsymbol{f}(boldsymbol{x})+mathbf{3} boldsymbol{x}-mathbf{1} )
then the least value of function ( y= )
( boldsymbol{g}(|boldsymbol{x}|) ) is
A. ( -frac{9}{4} )
B. ( -frac{5}{4} )
c. -2
D. –
12
1199Find ( boldsymbol{f} circ boldsymbol{g} ) and ( boldsymbol{g} ) o ( boldsymbol{f}, ) if ( boldsymbol{f}(boldsymbol{x})= )
( boldsymbol{x}^{2}, boldsymbol{g}(boldsymbol{x})=cos boldsymbol{x} )
12
1200The relation “is congruent to” on the set of all triangles in a plane is
A. Reflexive only
B. Symmetric only
C. Transitive only
D. Equivalence only
12
1201For a real number r let ( [r] ) denote the
largest integer less than or equal to ( r )
Let ( a>1 ) be a real number which is not
an integer, and let k be the smallest positive integer such that ( left[boldsymbol{a}^{k}right]>[boldsymbol{a}]^{k} )
Then which of the following statements is always true?
A ( cdot k leq(2[a]+1)^{2} )
B . ( k leq([a]+1)^{4} )
c. ( k<2^{[a]+1} )
D. ( _{k leq frac{1}{a-[a]}}+1 )
11
1202Show that the relation ( R ) in the ( operatorname{set} A= )
{1,2,3,4,5} given by ( R= )
( {(a, b):|a-b| text { is even }}, ) is an
equivalence relation.
12
1203( m ) is said to be related to ( n ), if ( m ) and ( n )
are integers and ( m-n ) is divisible by
13. Does this define an equivalence
relation?
12
1204Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be such that ( boldsymbol{f}(boldsymbol{x})=boldsymbol{2}^{boldsymbol{x}} )
Determine
(i) Range of ( boldsymbol{f} )
(ii) ( {x: f(x)=1} )
(iii) whether ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x}) boldsymbol{f}(boldsymbol{y}) )
holds.
12
1205Given ( boldsymbol{A}={boldsymbol{x} in boldsymbol{W}: mathbf{7} leq boldsymbol{x} leq mathbf{1 0}}, ) find
( boldsymbol{A} times boldsymbol{A} )
12
1206The p.d.f. of a continuous r.v. x is ( f(x)=frac{7}{9} x^{2}-2, quad 0<x<3 )
( =mathbf{0} )
otherwise
( boldsymbol{P}(boldsymbol{X} leq-1) ) is?
( mathbf{A} cdot mathbf{0} )
B. ( 1 / 2 )
( c cdot 1 / 3 )
D. ( 1 / 4 )
12
1207( boldsymbol{v}=frac{mathbf{2} boldsymbol{pi} boldsymbol{r}}{boldsymbol{T}} )
In Physics, Uniform Circular Motion is used to describe the motion of an object traveling at a constant speed in a circle. The speed of the object, also called tangential velocity, can be calculated using the formula above. Here, ( r )
represents the radius of the circle, ( boldsymbol{T} ) the
time it takes for the object to make one complete revolution, called a period. Determine which of the following formulae could be used to find the
length of one period if one knows the tangential velocity and the radius of the circle?
A ( cdot T=frac{v}{2 pi r} )
B・ ( T=frac{2 pi r}{v} )
c. ( T=2 pi r v )
D. ( T=frac{1}{2 pi r v} )
12
1208Given ( boldsymbol{A}={mathbf{5}, mathbf{6}, mathbf{7}} ) and ( boldsymbol{B}={mathbf{3}, mathbf{4}} )
Form all possible ordered pairs and write the total number of
ordered pairs formed so that both the
components are from B.
12
1209Let ( R ) be a relation on the set ( A ) of
ordered pairs of non-zero integers defined by ( (x, y) R(u, v) . ) If ( x v=y u )
then show that ( R ) is an equivalence
relation.
12
1210If
( [x] ) denotes the integral part of ( x, ) then domain of the function
( boldsymbol{f}(boldsymbol{x})=frac{sqrt{3-x}}{(x-1)(x-2)(x-3)}+sin ^{-1}left[frac{3 x-2}{2}right] ) is
A . [0,2]
в. ( [0,2)-{1} )
C ( .(-infty, 3)-{1,2} )
D. None of these
12
1211Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{x}+mathbf{1} ) and ( boldsymbol{g}(boldsymbol{x})=sin boldsymbol{x} )
Show that ( boldsymbol{f} circ boldsymbol{g} neq boldsymbol{g} circ boldsymbol{f} )
12
1212Find the domain and range of the following function:
( frac{boldsymbol{x}+boldsymbol{4}}{|boldsymbol{x}+mathbf{4}|} )
12
1213The relation ( R ) defined on the ( operatorname{set} A= )
{1,2,3,4,5} by ( R= )
( left{(x, y):left|x^{2}-y^{2}right|<16right} ) is given by
A ( cdot{(1,1),(2,1),(3,1),(4,1),(2,3)} )
B – {(2,2),(3,2),(4,2),(2,4)}
C ( cdot{(3,3),(4,3),(5,4),(3,4)} )
D. None of these
12
1214( boldsymbol{f}(boldsymbol{x})=boldsymbol{6} boldsymbol{x}+mathbf{5} ) find ( boldsymbol{f}(-mathbf{1}) )
( A cdot-1 )
B.
( c cdot c )
D. None of these
12
1215If ( g(x)=frac{5 x-3}{2 x^{2}-11-6}, ) what is the
sum of all the real numbers that are not
in the domain of ( g(x) ? )
A . -2
B. 0.5
( c cdot 2 )
D. 5.5
E . 6.5
12
1216Suppose that ( f(n) ) is a real valued function whose domain is the set of
positive integers and that ( boldsymbol{f}(boldsymbol{n}) ) satisfies the following two properties:
( boldsymbol{f}(1)=23 ) and ( boldsymbol{f}(boldsymbol{n}+1)=boldsymbol{8}+boldsymbol{3} cdot boldsymbol{f}(boldsymbol{n}) )
for ( n geq 1 )
It follows that there are constant ( boldsymbol{p}, boldsymbol{q} )
and ( r ) such that ( f(n)=p cdot q^{n}-r, ) for
( boldsymbol{n}=mathbf{1}, mathbf{2}, ldots ldots ) then the value of ( boldsymbol{p}+boldsymbol{q}+boldsymbol{r} )
is:
A . 16
B. 17
c. 20
D. 26
E . 31
12
1217Find the maximum value of ( g(f(x)) ) if ( f(x)=x+4 ) and
( boldsymbol{g}(boldsymbol{x})=boldsymbol{6}-boldsymbol{x}^{2} )
A . -6
B. -4
( c cdot 2 )
D. 4
E . 6
12
1218Determine the range of the function ( boldsymbol{f}(boldsymbol{x}) ) defined as ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{R}, boldsymbol{f}(boldsymbol{x})= )
( boldsymbol{x}^{3}+mathbf{1}, ) where ( boldsymbol{A}={-mathbf{1}, mathbf{0}, boldsymbol{3}, mathbf{7}, boldsymbol{9}} )
12
1219If ( boldsymbol{A}={mathbf{2}, mathbf{3}} ) and ( boldsymbol{B}={1,2, mathbf{3}, 4}, ) then
which of the following is not a subset of ( boldsymbol{A} times boldsymbol{B} )
A ( cdot{(2,3),(2,4),(3,3),(3,4)} )
B – {(2,2),(3,1),(3,4),(2,3)}
c. {(2,1),(3,2)}
D cdot {(1,2),(2,3)}
12
1220The function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}-[boldsymbol{x}]+cos boldsymbol{x} )
where ( [x] ) is the greatest integer ( leq n ), is
A. a periodic function of period ( 2 pi )
B. a periodic function of period 1
c. a periodic function of indeterminate period
D. a non-periodic function
11
122172. Let S={XER:x>0 and
21 VX-31+vX(Vx-6) +6=0. Then S: [JEE M 2018
(a) contains exactly one element.
(6) contains exactly two elements.
(C) contains exactly four elements.
(d) is an empty set.
12
1222A function ( boldsymbol{f}:(-mathbf{3}, mathbf{7}) rightarrow boldsymbol{R} ) is defined as
follows:
( f(x)=left{begin{array}{cc}4 x^{2}-1 ; & -3 leq x<2 \ 3 x-2 ; & 2 leq x leq 4 \ 2 x-3 ; & 4<x leq 6end{array}right. )
Find: ( boldsymbol{f}(boldsymbol{5})+boldsymbol{f}(boldsymbol{6}) )
12
1223Let L be the set of all lines in XY-plane and ( mathrm{R} ) be the relation in L defined as
( boldsymbol{R}=left{left(boldsymbol{L}_{1}, boldsymbol{L}_{2}right): boldsymbol{L}_{1} text { is parallel to } boldsymbol{L}_{2}right} )
Show that ( R ) is an equivalence relation. Find the set of all lines related to the
line ( y=2 x )
12
1224Let ( A={1,2,3, ldots, 14} . ) Define a relation ( boldsymbol{R} ) from ( boldsymbol{A} ) to ( boldsymbol{A} ) by ( boldsymbol{R}={(boldsymbol{x}, boldsymbol{y}) )
( mathbf{3} boldsymbol{x}-boldsymbol{y}=mathbf{0} text { where } boldsymbol{x}, boldsymbol{y} in boldsymbol{A}} . ) Write down
its domain, co-domain and range.
12
1225If a function ( boldsymbol{f}:(2, infty) rightarrow boldsymbol{B} ) defined by
( f(x)=x^{2}-4 x+5 ) is a bijection, then
( boldsymbol{B}= )
A. ( R )
B. ( [1, infty) )
c. (0,1]
D. [0,1]
12
1226The domain of definition of the function
( boldsymbol{y}(boldsymbol{x}) ) given by ( mathbf{2}^{boldsymbol{x}}+mathbf{2}^{boldsymbol{y}}=mathbf{2} ) is
12
1227If ar? +- 2c for all positive x where a>0 and b>0 show
that 27ab2 > 4c3.
(1982 – 2 Marks)
12
1228( operatorname{Let} phi(boldsymbol{x})=frac{boldsymbol{b}(boldsymbol{x}-boldsymbol{a})}{boldsymbol{b}-boldsymbol{a}}+frac{boldsymbol{a}(boldsymbol{x}-boldsymbol{b})}{boldsymbol{a}-boldsymbol{b}}, ) where
( x in R ) and ( a ) and ( b ) are fixed real numbers
with ( a neq b . ) Then ( phi(a+b) ) is equal to:
A. ( phi(a b) )
(i)
в. ( phi(-a b) )
( mathbf{c} cdot phi(a) mid phi(b) )
D. ( phi(a-b) )
E ( . phi(0) )
11
1229Assertion
The relation ( boldsymbol{R} ) given by
( boldsymbol{R}={(mathbf{1}, mathbf{3}),(mathbf{4}, mathbf{2}),(mathbf{2}, mathbf{4}),(mathbf{2}, mathbf{3}),(mathbf{3}, mathbf{1})} )
on a set ( A={1,2,3,4} ) is not
symmetric.
Reason
For symmetric relation ( boldsymbol{R}=boldsymbol{R}^{-1} )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
1230f ( x ) is real, then find the solution set of
( sqrt{x+1}+sqrt{x-1}=1 )
11
1231( begin{array}{cc}boldsymbol{x} & boldsymbol{f}(boldsymbol{x}) \ 1 & 1 \ 2 & 3 \ 3 & 5 \ 4 & 7 \ 5 & 9 \ 6 & 11end{array} ) For the function in the table above,
calculate ( boldsymbol{f}(boldsymbol{3}) )
A . 5
B. 4
( c .3 )
D. 6
12
1232The ( _{–}-_{-}- ) product of two sets is the set of all possible ordered pairs whose first component is a member of the first
set and whose second component is a member of the second set.
A. cartesian
B. coordinate
c. simple
D. discrete
12
1233Let ( R ) be arelation on the set ( A ), then ( R )
is symmetric. Prove that ( boldsymbol{R}^{-1} ) is symmetric.
12
1234The largest set of values of ‘x’ for which
( x operatorname{in} x>x-1 ) is?
A ( .(0,1) )
B. ( (1, infty) )
c. (0,1)( cup(1, infty) )
D. None
11
1235What is the distinction between a
relation and function and when do you call a relation reflexive, symmetric and transitive?
12
1236If ( boldsymbol{y}^{2}=boldsymbol{x}^{2}-boldsymbol{x}+mathbf{1} ) and ( quad boldsymbol{I}_{boldsymbol{n}}=int frac{boldsymbol{x}^{boldsymbol{n}}}{boldsymbol{y}} boldsymbol{d} boldsymbol{x} )
and ( boldsymbol{A I}_{3}+boldsymbol{B I}_{2}+boldsymbol{C I}_{1}=boldsymbol{x}^{2} boldsymbol{y} ) then
ordered triplet ( A, B, C ) is
( ^{A} cdotleft(frac{1}{2},-frac{1}{2}, 1right) )
В . (3,1,0)
( mathbf{c} cdot(1,-1,2) )
D. ( left(3,-frac{5}{2}, 2right) )
12
1237If ( A ) and ( B ) are two sets containing four and two elements, respectively. Then the
number of subsets of the set ( boldsymbol{A} times boldsymbol{B} )
each having at least three elements is
A .219
в. 256
c. 275
D. 510
12
1238If ( f(x)=frac{x+1}{x-1}, ) what is the value of
( boldsymbol{f}left(boldsymbol{f}left(boldsymbol{f}left(boldsymbol{f}left(left(frac{3}{5}right)right)right)right)right) ? )
A . –
B.
( c .0 .6 )
D. 1.3
E. 7
11
1239Solve the absolute value of inequation ( left|frac{2 x-1}{3}right| leq 5 )11
1240The domain of ( frac{1}{sqrt{x-x^{2}}}+sqrt{3 x-1-2 x^{2}} )
is:
12
1241A function ( boldsymbol{f}:[-mathbf{3}, mathbf{7}) rightarrow boldsymbol{R} ) is defined
as follows
( f(x)=left{begin{array}{cc}4 x^{2}-1 ; & -3 leq x<2 \ 3 x-2 ; & 2 leq x leq 4 \ 2 x-3 ; & 4<x<7end{array}right. )
(i) ( boldsymbol{f}(mathbf{5})+boldsymbol{f}(boldsymbol{6}) )
(ii) ( boldsymbol{f}(mathbf{1})-boldsymbol{f}(mathbf{3}) )
(iii) ( boldsymbol{f}(-mathbf{2})-boldsymbol{f}(-mathbf{4}) )
(iv) ( frac{boldsymbol{f}(mathbf{3})+boldsymbol{f}(mathbf{1})}{mathbf{2} boldsymbol{f}(boldsymbol{6})-boldsymbol{f}(mathbf{1})} )
11
1242Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a function defined by
( boldsymbol{f}(boldsymbol{x})=(mathbf{1}-boldsymbol{x})^{frac{1}{3}} ) is :
A. one-one and onto
B. many one and onto
c. one-one and into
D. many one and into
12
1243The following relations are defined on the set of real numbers check them for
( mathrm{R}, mathrm{S}, mathrm{T} )
( operatorname{arbiff}|a-b|>0 )
12
1244A function ( f ) is defined by ( f(x)=2 x^{3}- )
( mathbf{3} boldsymbol{x}, boldsymbol{x} boldsymbol{epsilon} boldsymbol{R} ). What is the value of ( boldsymbol{f}(boldsymbol{4}) ? )
12
1245Let ( boldsymbol{f}(boldsymbol{x})=cos left[boldsymbol{pi}^{2}right] boldsymbol{x}+cos left[-boldsymbol{pi}^{2}right] boldsymbol{x} )
where ( [x] ) is the greatest integer
function, then find ( fleft(frac{pi}{2}right) ) and ( f(pi) )
11
1246Solve: ( frac{|boldsymbol{x}|-mathbf{1}}{|boldsymbol{x}|-mathbf{3}} geq mathbf{0}, boldsymbol{x} in boldsymbol{R}, boldsymbol{x} neq pm mathbf{3} )11
1247Let ( A ) and ( B ) be two sets such that
( boldsymbol{n}(boldsymbol{A})=mathbf{5} ) and ( boldsymbol{n}(boldsymbol{B})=2 . ) If ( boldsymbol{a}, boldsymbol{b}, boldsymbol{c}, boldsymbol{d}, boldsymbol{e} )
are distinct and
( (a, 2),(b, 3),(c, 2),(d, 3),(e, 2) ) are in
( A times B, ) find ( A ) and ( B )
12
1248If ( mathbf{f}:[mathbf{0}, infty) rightarrow[mathbf{0}, infty), ) and ( mathbf{f}(mathbf{x})=frac{mathbf{x}}{mathbf{1}+mathbf{x}} )
then ( boldsymbol{f} ) is
A. one – one and onto
B. one – one but not onto
c. onto but not one – one
D. neither one – one nor onto
12
1249( A ) and ( B ) are two sets having 3 and 5
elements respectively and having 2
elements in common. Then the number
of elements in ( boldsymbol{A} times boldsymbol{B} ) is
( A cdot 6 )
B . 36
c. 15
D. none of these
12
12503.
Let f(x)= |x-1|. Then
(a) f(x2)=(f(x))2
(c) f(xD)=f(x)]
(1983 – 1 Mark)
(b) f(x+y)=f(x) + f(y)
(d) None of these
12
1251a ( mathrm{R} ) b if “a and b are animals in different
zoological parks” then R is
A. only reflexive
B. only symmetric
c. only transitive
D. equivalence
12
1252Range of the function ( boldsymbol{f}(boldsymbol{x})= ) ( frac{sec ^{2} x-tan x}{sec ^{2} x+tan x}-frac{pi}{2}<x<frac{pi}{2}, ) is
( A cdot R )
в. ( R-left(frac{1}{3}, 3right) )
c. ( left[frac{1}{3}, 3right] )
D. ( left[-1, frac{5}{3}right. )
12
1253( boldsymbol{f}(boldsymbol{x})=left(1-boldsymbol{x}^{3}right)^{frac{1}{3}}, ) then find ( boldsymbol{f o f}(boldsymbol{2}) )11
1254If ( * ) is a binary operation in ( boldsymbol{A} ) then
A. ( A ) is closed under
B. ( A ) is not closed under
c. ( A ) is not closed under ( + )
D. ( A ) is closed under –
12
1255If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a differentiable
function, such that ( boldsymbol{f}(boldsymbol{x}+mathbf{2} boldsymbol{y})= )
( boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{2} boldsymbol{y})+boldsymbol{4} boldsymbol{x} boldsymbol{y} ) for all ( boldsymbol{x}, boldsymbol{y} in boldsymbol{R} )
then
A ( cdot f^{prime}(1)=f^{prime}(0)+1 )
B . ( f^{prime}(1)=f^{prime}(0)-1 )
C ( cdot f^{prime}(0)=f^{prime}(1)+2 )
D. ( f^{prime}(1)=f^{prime}(0)+2 )
11
1256f ( x, y in{1,2,3,4} ) then check whether
( boldsymbol{f}={(boldsymbol{x}, boldsymbol{y}): boldsymbol{y}=boldsymbol{x}+1} ) is a function?
12
1257Let ( boldsymbol{R}=left{left(boldsymbol{a}, frac{mathbf{1}}{boldsymbol{a}}right): boldsymbol{a} in boldsymbol{N} text { and } mathbf{1}<boldsymbol{a}<right. )
5 ( } . ) Find the doman and range of ( R )
12
1258If ( f(x)=x+5 ) and ( g(x)=sqrt{x^{2}-9} ) then find the domain of ( operatorname{gof}(x) )
A ( cdot(-8,-2) )
В ( cdot(-infty,-8) cup(-2, infty) )
c. ( (-infty,-8] cup[-2, infty) )
D. ( (-(-infty,-8] cup[-2, infty) )
12
1259Let for ( boldsymbol{a} neq boldsymbol{a}_{1} neq mathbf{0}, boldsymbol{f}(boldsymbol{x})=boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+ )
( c, g(x)=a_{1} x^{2}+b_{1} x+c_{1} ) and ( p(x)= )
( f(x)-g(x) ). If ( p(x)=0 ) only for ( x=-1 )
and ( p(-2)=2, ) then the value of ( p(2) ) is
A . 3
B. 9
( c cdot 6 )
D. 18
11
1260If 1 is a zero of the polynomial ( p(x)= )
( a x^{2}-3(a-1) x-1, ) then find the
value of ( boldsymbol{a} )
12
1261Find ( boldsymbol{y} ) if ( boldsymbol{x}=mathbf{4} ) when ( boldsymbol{y}=boldsymbol{x}^{3} )12
1262If the ordered pairs ( (a-3, a+2 b) ) and
( (3 a-1,3) ) are equal, find the value of
( a+b )
12
1263If ( f(x)=log left[frac{1+x}{1-x}right] ) and ( -1_{1}, x_{2}<1, ) then ( fleft(x_{1}right) )
( mathbf{f}left(mathbf{x}_{2}right) ) is equal to:
( mathbf{A} cdot fleft[left(x_{1}-x_{2}right) / 1+x_{1} x_{2}right] )
B ( cdot fleft[left(x_{1}-x_{2}right) / 1-x_{1} x_{2}right] )
( mathbf{c} cdot fleft[left(x_{1}+x_{2}right) / 1-x_{1} x_{2}right] )
( mathbf{D} cdot fleft[left(x_{1}+x_{2}right) / 1+x_{1} x_{2}right] )
11
1264( operatorname{Let} boldsymbol{f}={(1, boldsymbol{a}),(boldsymbol{2}, boldsymbol{c}),(boldsymbol{4}, boldsymbol{d}),(boldsymbol{3}, boldsymbol{b})} ) and
( boldsymbol{g}^{-1}={(boldsymbol{2}, boldsymbol{a}),(boldsymbol{4}, boldsymbol{b}),(1, c),(boldsymbol{3}, boldsymbol{d})}, ) then
show that ( (boldsymbol{g} boldsymbol{o} boldsymbol{f})^{-1}=boldsymbol{f}^{-1} boldsymbol{o g}^{-1} )
12
1265Let ( A={1,2,3} ) and ( R= )
{(1,2),(2,3),(1,3)} be a relation on ( A )
Then ( boldsymbol{R} ) is
A. neither reflexive nor transitive
B. neither symmetric nor transitive
c. transitive
D. none of these
12
1266x, where is a
6. Letf: (0, 1) R be defined by f(x)=
constant such that 0 <b<1. Then
(a) f is not invertible on (0,1)
(6) ff – on (0, 1) and f'(b) =
b f'O
0
© f=f-1 on (0,1) and f' (b) = f'O)
(d) f- is differentiable (0,1)
12
1267If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) defined by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2} ) is
A. one-one but not onto
B. onto but not one-one
c. bijective
D. neither one-one nor onto
12
1268Let ( boldsymbol{f}(boldsymbol{x})=(boldsymbol{x}+mathbf{1})^{2}-mathbf{1},(boldsymbol{x} geq-mathbf{1}) . ) Then
the ( operatorname{set} S=left{x: f(x)=f^{-1}(x)right} ) is
12
1269If ( f(x)=x^{2}+frac{1}{x^{2}}, ) then find the
minimum value of the function ( g(x)= ) ( boldsymbol{f}(boldsymbol{x})+boldsymbol{f}left(frac{1}{boldsymbol{x}}right) )
12
1270If the binary operation on the set of integers ( Z, ) defined by ( a times b=a+3 b^{2} )
then find the value of ( 8 times 3 )
12
1271( operatorname{Given} M=(0,1,2) ) and ( N=(1,2,3) )
Find ( (M cup N) times(M-N) )
12
1272The number of continuous and
derivable function(s) ( f(x) ) such that
( f(1)=-1, f(4)=7 ) and ( f(x)>3 ) for
all ( boldsymbol{x} in boldsymbol{R} ) is/are
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. infinite
12
1273If a function satisfies ( (boldsymbol{x}-boldsymbol{y}) boldsymbol{f}(boldsymbol{x}+ )
( boldsymbol{y})-(boldsymbol{x}+boldsymbol{y}) boldsymbol{f}(boldsymbol{x}-boldsymbol{y})=boldsymbol{2}left(boldsymbol{x}^{2} boldsymbol{y}-right. )
( left.boldsymbol{y}^{3}right), forall boldsymbol{x}, boldsymbol{y} in boldsymbol{R} ) and ( boldsymbol{f}(1)=2, ) then
This question has multiple correct options
A. ( f(x) ) must be polynomial function
В. ( f(3)=12 )
c. ( f(0)=0 )
D. ( f(x) ) may not be differentiable
11
1274Given ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}+mathbf{2}}{mathbf{5}} ) for ( operatorname{gof}(boldsymbol{x})=boldsymbol{x}-mathbf{3} )
find ( boldsymbol{g} )
12
1275( F(n) ) be a polynomial function
satisfying ( boldsymbol{f}(boldsymbol{x}) cdot boldsymbol{f}left(frac{1}{x}right)=boldsymbol{f}(boldsymbol{x})+ )
( fleft(frac{1}{x}right) forall x in R-{0} ) and ( f(3)=-26 )
Determine ( boldsymbol{f}(mathbf{1}) )
11
1276If ( boldsymbol{A}=left{boldsymbol{x} in boldsymbol{R}: boldsymbol{x}^{2}-mathbf{5}|boldsymbol{x}|+boldsymbol{6}=mathbf{0}right} )
( operatorname{then} n(A)= )
( A cdot 2 )
B.
( c cdot 1 )
D.
11
1277Find the domain of definition of the
following function. ( boldsymbol{f}(boldsymbol{x}),=sqrt{mathbf{2}-boldsymbol{x}}+sqrt{mathbf{1}+boldsymbol{x}} )
12
1278Given that ( x ) and ( y ) satisfy the relation. ( boldsymbol{y}=mathbf{3}[boldsymbol{x}]+mathbf{7} )
( boldsymbol{y}=mathbf{4}[boldsymbol{x}-mathbf{3}]+mathbf{4} ) then find ( [boldsymbol{x}+boldsymbol{y}] )
11
1279What is the domain of function ( g )
defined by a set of ordered pairs as
follows: ( boldsymbol{g}= )
{(3,2),(5,9),(6,9),(11,0)}
( mathbf{A} cdot{2,9,9,0} )
B ( cdot{3,5,6,11} )
c. {3,9,9,11}
D ( cdot{2,5,6,11} )
12
1280Let ( boldsymbol{f}(boldsymbol{x})=sqrt{log left(2 boldsymbol{x}-boldsymbol{x}^{2}right)} . ) Then, dom
( (f)=? )
A ( .(0,2) )
B. [1,2]
( c cdot(-infty, 1] )
D. None of these
12
1281( boldsymbol{f}:left(-frac{boldsymbol{pi}}{2}, frac{boldsymbol{pi}}{2}right) rightarrow(-infty, infty) ) defined by
( boldsymbol{f}(boldsymbol{x})=mathbf{1}+mathbf{3} boldsymbol{x} ) is
A. one-one but not onto
B. onto but not one-one
c. neither one – one nor onto
D. bijective
12
1282Find range of ( boldsymbol{y}=frac{boldsymbol{x}}{mathbf{1}+|boldsymbol{x}|} )12
1283Find the domain of ( f(x)=cot x+ )
( cot ^{-1} x )
12
1284Show that the relation R defined on the
( operatorname{set} A={1,2,3,4,5}, ) given by ( R= )
( {(a, b):|a-b| text { is even }} ) is an
equivalence relation
12
1285The function ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R}, boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2} ) is
A. injective but not surjective
B. surjective but not injective
c. injective as well as surjective
D. neither injective nor surjective
12
1286If ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+mathbf{5}, ) then ( boldsymbol{f}(-mathbf{4}) ) is:
A . 26
B. 21
c. 20
D. -20
12
1287Number of points of focal minima of the functions ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+frac{mathbf{1}}{boldsymbol{x}^{2}}-mathbf{8} boldsymbol{x}-frac{boldsymbol{8}}{boldsymbol{x}}+ )
7 are
( mathbf{A} cdot mathbf{1} )
B . 2
( c .3 )
D.
12
1288Define a symmetric relation.12
1289f ( f(x)=4-3 x, ) find ( f(-5) ) and ( f(13) )11
1290If ( y=sin ^{-1} frac{x-1}{x+1}+log (2-x), ) then
its domain is:
A. (1,2)
(i) 5,(1,2)
в. (-1,2)
D. none
12
1291f ( A={a, b} ) and ( B={1,2,3} ) find
( A times B ) and ( B times A )
12
1292Let ( A=R times R ) and ( * ) be the binary
operation on A defined by ( (a, b) * ) ( (c, d)=(a+b, b+d) . ) Prove that ( * ) is
commutative and associative find the
identity element for ( * ) on ( boldsymbol{A} )
12
1293Let ( A ) be a finite set containing ( n )
distinct elements. The number of
relations that can be defined on A is:
A ( cdot 2^{n} )
B ( cdot n^{2} )
( c cdot 2^{n^{2}} )
D. 2n
12
1294When ( x geq 2, ) then function ( f(x)= )
( mathbf{2}|boldsymbol{x}-mathbf{2}|-|boldsymbol{x}+mathbf{1}|+boldsymbol{x} ) is reduced to
A. ( f(x)=-2 x+3 )
в. ( f(x)=2 x-5 )
( mathbf{c} cdot f(x)=5 )
D. ( f(x)=-1 )
E. ( f(x)=2 x+4 )
11
1295Show that if ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{B} ) and ( boldsymbol{g}: boldsymbol{B} rightarrow boldsymbol{C} )
are one-one, then ( g circ f: A rightarrow C ) is also
one-one.
12
1296The range of the function
( f(x)=sin cos left(ln left(frac{x^{2}+1}{x^{2}+e}right)right) ) is
A ( cdot[sin cos 1,1) )
B. ( [sin cos 1, cos 1) )
c. ( [sin cos 1, sin 1) )
D. ( (sin cos 1, sin 1] )
12
1297Range of ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+frac{mathbf{1}}{boldsymbol{x}^{2}+boldsymbol{4}} )12
1298Let ( A ) be a finite set, lf ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{A} ) is
onto. Show that ( f ) is one-one
12
1299Let ( O ) be the origin. We define a relation
between two points ( P ) and ( Q ) in a plane, if ( O P=O Q . ) Show that the relation, so
defined is an equivalence relation.
12
1300If ( f(x)=x+2, g(x)=x^{2}-x-2 )
then find ( frac{g(1)+g(2)+g(3)}{f(-4)+f(-2)+f(2)} )
11
1301Let ( R ) be the set of real numbers and the
functions ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) and ( boldsymbol{g}: boldsymbol{R} rightarrow boldsymbol{R} ) be
defined by ( f(x)=x^{2}+2 x-3 ) and
( g(x)=x+1 . ) Then the value of ( x ) for
which ( f(g(x))=g(f(x)) ) is
A . -1
B.
c. 1
( D )
12
1302Find ( g o f ) and ( f o g, ) if ( (i) f(x)=|x| ) and
( g(x)=|5 x-2| )
ii) ( f(x)=8 x^{3} ) and ( g(x)=x^{frac{1}{3}} )
12
1303If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) defined by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}, ) if
( boldsymbol{x}>2 ; boldsymbol{f}(boldsymbol{x})=mathbf{5} boldsymbol{x}-boldsymbol{2} ) if ( boldsymbol{x} leq 2 ) then ( boldsymbol{f} ) is
A. an identity function
B. one one
c. onto
D. one one and onto
12
1304If ( alpha ) is a real number for which ( f(x)= )
( cos ^{-1} x ) is defined then a possible value
of ( [alpha] ) is (where ( [.] ) denotes greatest
integer function)
This question has multiple correct options
( A cdot 2 )
B.
( c cdot-1 )
( D cdot-2 )
11
1305The range of the function ( boldsymbol{f}(boldsymbol{x})=sin left(boldsymbol{x} e^{[x]}+boldsymbol{x}^{2}-boldsymbol{x}right), forall boldsymbol{x} epsilon(-1, infty) )
where ( [x] ) denotes the greatest integer function is :
( A cdot phi )
B. [0,1]
c. [-1,1]
D. ( R )
12
1306Let ( A={-1,1} ) then find ( A times A )12
1307If ( boldsymbol{A}={mathbf{2}, boldsymbol{4}} ) and ( boldsymbol{B}={mathbf{3}, mathbf{4}, mathbf{5}} ) then
( (boldsymbol{A} cap boldsymbol{B}) times(boldsymbol{A} cup boldsymbol{B}) ) is
A ( cdot{(2,2),(3,4),(4,2),(5,4)} )
в. {(2,3),(4,3),(4,5)}
( mathbf{c} cdot{(2,4),(3,4),(4,4),(4,5)} )
D. {(4,2),(4,3),(4,4),(4,5)}
12
1308The domain of the function ( f(x)= ) ( sqrt{sec ^{-1}left{frac{1-|x|}{2}right}} )
A ( cdot(-infty,-3] cup[3,+infty) )
B. ( [3,+infty) )
( c . Phi )
D. ( R )
12
1309Find the range of each of the following
functions ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}-boldsymbol{6} boldsymbol{x}+mathbf{1 1}, ) all ( boldsymbol{x} in )
( boldsymbol{R} )
12
1310Let ( mathbf{f}: boldsymbol{N} rightarrow boldsymbol{N}, ) where ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}+ )
( (-1)^{x-1}, ) then
A. ( f(x) ) is symmetric about ( x=1 )
B. ( f(x) ) is symmetric about ( y=x )
c. ( f(x) ) is not symmetric about ( y=x )
D. ( f(x) ) is symmetric about ( x=a, a in N )
11
1311If ( g(x)=1+sqrt{x} ) and ( f(g(x))=3+ )
( 2 sqrt{x}+x, ) then ( f(x)= )
A ( cdot 1+2 x^{2} )
B ( cdot 2+x^{2} )
c. ( 1+x )
D. ( 2+x )
12
1312If ( f(x)=sin ^{2} x+sin ^{2}left(x+frac{pi}{3}right)+ )
( cos x cos left(x+frac{pi}{3}right) ) and ( gleft(frac{5}{4}right)=1 )
( boldsymbol{g}(1)=0 ) then ( (boldsymbol{g} boldsymbol{o} boldsymbol{f})(boldsymbol{x})= )
( A )
в.
( c cdot sin x )
D. Data is insufficient
12
1313Determine domain of ( frac{sqrt{4-x^{2}}}{sin ^{-1}(2-x)} )
A ( .(4,5) )
B. [1,2]
c. both a and ( b )
D. none of the above
12
1314If ( boldsymbol{A} times boldsymbol{B}= )
( {(3, a),(3,-1),(3,0),(5, a),(5,-1),(5 )
find ( boldsymbol{A} )
A ( cdot{a, 5} )
в. ( {a,-1} )
c. {0,5}
D. {3,5}
12
1315Check whether ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a function defined by ( f(x)=frac{x^{2}+2 x+5}{x^{2}+x+1} ) is one
one or not.
12
13162.
The function f(x) = log x + Vx+1), is
[2003]
(a) neither an even nor an odd function
(b) an even function
(c) an odd function
(d) a periodic function.
12
1317Let ( L ) be the set of all straight lines in
the Euclidean plane. Two lines ( l_{1} ) and ( l_{2} )
are said to be related by the relation ( boldsymbol{R} ) if
( l_{1} ) is parallel to ( l_{2} ). Then the relation ( R ) is This question has multiple correct options
A. Reflexive
B. Symmetric
c. Transitive
D. Equivalence
12
1318An object launched straight up into the air is said to have parabolic motion (because it goes up, reaches a maximum height, and then comes back down). The height (h) of a projectile at time ( t ) is given by the equation ( h= ) ( frac{1}{2} g t^{2}+u t+h_{0}, ) where ( g ) is the
acceleration due to gravity and ( u ) and ( h_{0} ) are the object’s initial velocity and initial height, respectively. Which of the following equations correctly represents the object’s acceleration due to gravity in terms of the other variables?
A ( cdot g=frac{h-u t-h_{0}}{t} )
B. ( g=frac{h-u t-h_{0}}{2 t^{2}} )
c. ( g=frac{2left(h-u t-h_{0}right)}{t^{2}} )
D. ( g=t sqrt{2left(h-u t-h_{0}right)} )
12
1319Let, ( f(x+1)=2 x^{2}-3 x-1, ) then find
the value of ( boldsymbol{f}(mathbf{0}) ) and ( boldsymbol{f}(boldsymbol{x}+mathbf{2}) )
11
1320Let ( A={1,2,3,4,6} ) and ( R ) be the relation on ( boldsymbol{A} ) defined by ( {(boldsymbol{a}, boldsymbol{b}): boldsymbol{a}, boldsymbol{b} in )
( A, b text { is exactly divisible by } a} )
(i) Write ( R ) in roster form
(ii) Find the domain of ( boldsymbol{R} )
(iii) Find the range of ( boldsymbol{R} )
12
1321f ( A={1,2,3} ) and ( B={3,4} ) and
( boldsymbol{C}={mathbf{1}, mathbf{3}, mathbf{5}} . ) Find ( boldsymbol{A} times(boldsymbol{B} cup boldsymbol{C}) )
12
1322Let ( A={1,2} ) and ( B={3,4} . ) Write
( A times B . ) How many subsets will ( A times B )
have?
12
1323The number of solution of the equation ( boldsymbol{a}^{f(boldsymbol{x})}+boldsymbol{g}(boldsymbol{x})=mathbf{0}, ) where ( boldsymbol{a}>mathbf{0}, boldsymbol{g}(boldsymbol{x}) neq )
0 and ( g(x) ) has minimum value ( frac{1}{4}, ) is
A. one
B. two
c. infinitely many
D. zero
11
132480. The number of solutions of [sin x + cos x] = 3 + [-sinx] +
[-cosx] (where [-] denotes the greatest integer function),
xe [0, 21], is
a. 0
Sb. 41
c. infinite
d .1
12
132521. Let function f:R→ R be defined by f(x) = 2x + sin x for
XER, then fis
(2002)
(a) one-to-one and onto
(b) one-to-one but NOT onto
(C) onto but NOT one-to-one
(d) neither one-to-one nor onto
12
1326Classify the following function as injection, surjection or bijection:
( boldsymbol{f}: boldsymbol{Z} rightarrow boldsymbol{Z} ) given by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{boldsymbol{3}} )
12
1327Special relativity is a branch of physics that deals with the relationship between space and time. The Lorentz
term, is the factor by which time, length and relativistic mass change for an object while that object is moving, is given by the following formula: ( gamma= )
( sqrt{1-frac{v^{2}}{c^{2}}} )
where, v is the relative velocity of the object ( c ) is the speed of light in a vacuum Which of the following equations
correctly represents the relative velocity
A ( cdot v=c sqrt{frac{1}{gamma^{2}}-1} )
B . ( v=c sqrt{1-gamma^{2}} )
c. ( v=cleft(1-frac{1}{gamma^{2}}right) )
D. ( v=c sqrt{1-frac{1}{gamma^{2}}} )
12
13286.
The domain of definition of the function
1
y =
– +Vx+2 is
(1983 – 1 Mark)
log10 (1 – x)
(a) (-3,-2) excluding -2.5 (b) [0, 1] excluding 0.5
(c) [-2, 1) excluding 0 (d) none of these
12
1329( g(x)=log left(frac{2+x}{2-x}right) ) for ( 0<x<2 )
find the value of ( frac{1}{2} gleft[frac{8 x}{4+x^{2}}right] )
12
1330Find the range of the following
functions
( sqrt{x-5} )
12
1331For all positive integers ( mathbf{w} ) and ( mathbf{y}, ) where ( mathbf{w}>mathbf{y}, ) let the operation ( otimes ) be defined by ( mathbf{w} otimes mathbf{y}=frac{mathbf{2}^{w+y}}{mathbf{2}^{w-y}} . ) For how many positive
integers ( mathbf{w} ) is ( mathbf{w} otimes mathbf{1} ) equal to ( mathbf{4} ) ?
A. None
B. One
c. Two
D. Four
E. More than four
12
1332Let ( ^{prime} *^{prime} ) be a binary operation on set ( Q ) of rational number defined as ( a * b=frac{a b}{5} )
Write the identity for ( ^{prime} *^{prime}, ) if any.
12
1333The union of two equivalence relations
¡s:
A. always an equivalence relation
B. reflexive and symmetric but needn’t be transitive
C. reflexive and transitive but need not be symmetric
D. None of these
12
1334( boldsymbol{V}=boldsymbol{pi}left(boldsymbol{R}^{2}-boldsymbol{r}^{2}right) boldsymbol{h} ; ) make ‘r’ the subject
of formula.
A ( cdot r=sqrt{R^{2}+frac{V}{pi h}} )
в. ( _{r}=sqrt{R-frac{V}{pi h}} )
( mathbf{c} cdot_{r}=sqrt{R^{2}-frac{V}{pi h}} )
D. ( r=sqrt{R^{2}-frac{V}{pi h^{2}}} )
11
1335If ( f: R^{+} rightarrow R^{+}, f(x)=x^{2}+2 ) and ( g: )
( boldsymbol{R}^{+} rightarrow boldsymbol{R}^{+}, boldsymbol{g}(boldsymbol{x})=sqrt{boldsymbol{x}+mathbf{1}} )
then ( (boldsymbol{f}+boldsymbol{g})(boldsymbol{x}) ) equals
A. ( sqrt{x^{2}+3} )
B. ( x+3 )
c. ( sqrt{x^{2}+2}+(x+1) )
D. ( x^{2}+2+sqrt{(x+1)} )
11
1336Show that the Signum Function ( boldsymbol{f} )
( R rightarrow R, ) is neither one one nor onto?
[
f(x)=left{begin{array}{l}
1, text { if } x>0 \
0, text { if } x=0 \
-1, text { if } x<0
end{array}right.
]
12
1337Show that the Modulus Function ( boldsymbol{f} ) ( boldsymbol{R} rightarrow boldsymbol{R}, ) given by ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}|, ) is neither
one-one nor onto, where ( |x| ) is ( x, ) if ( x ) is
positive or 0 and ( |x| ) is ( -x, ) if ( x ) is
negative.
12
1338If ( boldsymbol{f}(boldsymbol{x}+mathbf{2} boldsymbol{y}, boldsymbol{x}-mathbf{2} boldsymbol{y})=boldsymbol{x} boldsymbol{y}, ) then ( boldsymbol{f}(boldsymbol{x}, boldsymbol{y}) )
is equal to
A ( cdot frac{1}{4} x y )
B. ( frac{1}{4}left(x^{2}-y^{2}right) )
c. ( frac{1}{8}left(x^{2}-y^{2}right) )
D. ( frac{1}{2}left(x^{2}+y^{2}right) )
12
1339The number of real solutions of the
equation ( 1-x=[cos x] ) is
( mathbf{A} cdot mathbf{1} )
B. 2
( c .3 )
D.
12
1340Sum of all integral values of ( boldsymbol{a} in[mathbf{1}, mathbf{5 0 0}] )
for which the equation ( [x]^{3}+x-a=0 )
has a solution ( ( [.] ) denotes the greatest integer function) is
A .1342
в. 512
c. 784
D. 812
11
1341If ( boldsymbol{A} times boldsymbol{B}= )
( (2,4),(2, a),(2,5),(1,4),(1, a),(1,5) )
find ( B )
A ( cdot{4,2,5} )
в. ( {4, a, 5} )
c. {4,1,5}
D. ( {2, a, 5} )
12
1342f ( boldsymbol{A}={mathbf{2}, boldsymbol{4}, boldsymbol{5}}, boldsymbol{B}={boldsymbol{7}, boldsymbol{8}, boldsymbol{9}} ) then
( n(A times B) ) is equal to-
( A cdot 6 )
B. 9
( c .3 )
( D )
12
1343Given ( f(x)=5 x-1 . ) then ( f(-3)=-16 )
(Enter 1 if true or 0 otherwise)
12
1344which of the following function(s) is/are Transcendental?
This question has multiple correct options
( mathbf{A} cdot f(x)=5 sin sqrt{x} )
B. ( f(x)=frac{2 sin 3 x}{x^{2}+2 x-1} )
C. ( f(x)=sqrt{x^{2}+2 x-1} )
D. ( f(x)=left(x^{2}+3right) 2^{x} )
12
1345If ( f(x)=log _{e}{log (x)}, ) then ( f^{prime}(x) ) at
( boldsymbol{x}=boldsymbol{e} ) is :
( A )
B. ( -c )
( c cdot e^{2} )
D. ( e^{-1} )
11
1346The function ( f ) is continuous and has
the property ( f(f(x))=1-x, ) then the
value of ( boldsymbol{f}left(frac{1}{4}right)+boldsymbol{f}left(frac{3}{4}right) ) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot frac{-1}{2} )
D. -1
12
1347Let ( A={5,6,7,8} ; B={-11,4,7,-10,-7,-9, )
-13} and ( boldsymbol{f}= )
( {(x, y): y=3-2 x, x epsilon A, y epsilon B} )
(i) Write down the elements of ( f . ) (ii)
What is the co-domain?
(iii) What is the range?
(iv) Identify the type of function.
12
1348Let ( f: X rightarrow Y ) be a function defined by
[
f(x)=a sin left(x+frac{pi}{4}right)+b cos x+c text { If }
]
is both one-one and onto, then find the
( operatorname{sets} X ) and ( Y )
A ( x inleft[-frac{pi}{2}-alpha, frac{pi}{2}-alpharight] ) and ( Y in[c-r, c+r] ) where ( alpha )
[
tan ^{-1}left(frac{a+b sqrt{2}}{a}right) text { and } r=sqrt{a^{2}+sqrt{2} a b+b^{2}}
]
в. ( X inleft[-frac{pi}{1}-alpha, frac{pi}{1}-alpharight] ) and ( Y in[c-r, c+r] ) where ( alpha )
[
tan ^{-1}left(frac{a+b sqrt{2}}{a}right) text { and } r=sqrt{a^{2}+sqrt{2} a b+b^{2}}
]
c. ( x inleft[-frac{pi}{4}-alpha, frac{pi}{4}-alpharight] ) and ( Y in[c-r, c+r] ) where ( alpha= )
[
tan ^{-1}left(frac{a+b sqrt{2}}{a}right) text { and } r=sqrt{a^{2}+sqrt{2} a b+b^{2}}
]
D. ( X inleft[-frac{pi}{8}-alpha, frac{pi}{8}-alpharight] ) and ( Y in[c-r, c+r] ) where ( alpha= )
[
tan ^{-1}left(frac{a+b sqrt{2}}{a}right) text { and } r=sqrt{a^{2}+sqrt{2} a b+b^{2}}
]
12
1349Let ( boldsymbol{A}=boldsymbol{x} in boldsymbol{R}: boldsymbol{x} leq mathbf{1} ) and ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{A} )
be defined as ( f(x)=x(2-x), ) then
( f^{-1}(x) ) is
A. ( 1+sqrt{1-x} )
B. ( 1-sqrt{1-x} )
c. ( sqrt{1-x} )
D. ( 1 pm sqrt{1-x} )
12
1350If ( h(x)=2 x, g(x)=x^{2}, f(x)=2 ), then
find ( (f circ g circ h)(x) )
12
1351The set of all solution of the inequality ( (1 / 2)^{x^{2}-2 x}<1 / 4 ) contains the set:
( A cdot(-infty, 0) )
B ( cdot(-infty, 1) )
( mathbf{c} cdot(1, infty) )
D cdot ( (3, infty) )
12
1352A function ( f ) has domain [-1,2] and range ( [0,1] . ) The domain and range respectively of the function ( g ) defined by ( boldsymbol{g}(boldsymbol{x})=mathbf{1}-boldsymbol{f}(boldsymbol{x}+mathbf{1}) ) is?
A ( cdot[-1,1] ;[-1,0] )
B cdot [-2,1]( ;[0,1] )
c. [0,2]( ;[-1,0] )
D cdot [1,3]( ;[-1,0] )
12
1353If ( f(x)=frac{x}{sqrt{1-x^{2}}}, ) then ( (f o f)(x)= )
A ( cdot frac{x}{sqrt{1-x^{2}}} )
в. ( frac{x}{sqrt{1-2 x^{2}}} )
c. ( frac{x}{sqrt{1-3 x^{2}}} )
D.
12
1354Cartesian product of sets ( A ) and ( B ) is
denoted by
A ( . A times B )
в. ( B times A )
c. ( A times A )
D. ( B times B )
12
1355For the function ( f(x) ) which value of ( a ) is
not possible
A ( cdot frac{1}{4} )
B. ( -frac{1}{4} )
( c cdot frac{1}{2} )
D. ( -frac{1}{2} )
12
1356The value of ( [sin x]+[1+sin x]+2+ )
( [sin x]+3+[sin x] ) in ( left(pi, frac{3 pi}{2}right] ) is
( mathbf{A} cdot mathbf{0} )
B. 1
( c cdot 2 )
D. 3
11
1357If ( (x, y)=(3,5) ; ) then values of ( x ) and ( y )
are
A. 3 and
B. 4 and 7
c. -1 and 17
D. 2 and 4
12
1358Let ( boldsymbol{f}=left{left(boldsymbol{x}, frac{boldsymbol{x}^{2}}{mathbf{1}+boldsymbol{x}^{2}}right): boldsymbol{x} in boldsymbol{R}right} ) be a
function from ( R ) into ( R ) Determine the
range of ( boldsymbol{f} )
12
1359Let ( boldsymbol{A}=boldsymbol{R}-{mathbf{3}} ) and ( boldsymbol{B}=boldsymbol{R}-{mathbf{1}} )
Then, ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{B}: boldsymbol{f}(boldsymbol{x})=frac{(boldsymbol{x}-boldsymbol{2})}{(boldsymbol{x}-boldsymbol{3})} ) is?
A. One-one and into
B. One-one and onto
c. Many-one and into
D. Many-one and onto
12
1360S.TF :Q defined by ( f(x)=2 x+3 / x-3 )
is one -one but not onto
12
1361Which of the following functions is/are
constant?
A ( cdot f(x)=x^{2}+2 )
в. ( f(x)=x+frac{1}{x} )
( mathbf{c} cdot f(x)=7 )
D. ( f(x)=6+x )
11
1362table shows the number of students
studying one or more of the following
subjects in this class:
Number of Subject
Mathematics
Physics
Chemistry
Mathematics and Physics 30
Mathematics and Chemistry 28
Physics and Chemistry 23
Mathematics, Physics and 18 Chemistry
How many students are enrolled in
Mathematics alone, Physics alone and
Chemistry alone ( ? ) Are there students who have not offered any of these three
subjects ?
12
1363If ( n(P times Q)=0 ) then ( n(P) ) can possibly
be
A .
B. 10
c. 20
D. Any value
12
136428. The set of all real numbers x for which x2-x+21+x>0, is
(2002)
(a) (-00,-2) U (2,00) 6 (-0, J2) (12,00)
© (- 0,-1) U (1,00) (d) (12,00)
12
1365Which of following is unary operation?
A. addition
B. multiplication
c. square root
D. None of Above
12
1366The set of all ( x ) in ( (-pi, pi) ) satisfying ( |4 sin x-1|<sqrt{5} ) is
( boldsymbol{x} boldsymbol{epsilon}left(-boldsymbol{pi},-frac{boldsymbol{k} boldsymbol{pi}}{mathbf{1 0}}right) cupleft(-frac{boldsymbol{pi}}{mathbf{1 0}}, frac{boldsymbol{pi}}{mathbf{1 0}}right) cup )
( left(frac{k pi}{10}, piright) . ) Find the value of ( k )
11
1367( boldsymbol{A}={boldsymbol{a}, boldsymbol{b}, boldsymbol{c}} ) and ( boldsymbol{B}={mathbf{5}, mathbf{7}, boldsymbol{9}} . ) The
relation from ( B ) to ( A ) is
This question has multiple correct options
A ( cdot{(a, 5),(a, 7),(b, 7),(c, 9)} )
B . {(5,7),(9,9),(7,5)}
c. ( {(5, a),(5, b),(5, c)} )
D cdot ( {(5, b),(7, c),(7, a),(9, b)} )
12
1368( f * ) is a binary operation on set ( N ),of
natural no defined as ( a * b=H C F ) of
( (a, b) . ) Evaluate ( 3 *(2 * 5) )
12
1369The following relation is defined on the set of real numbers. ( boldsymbol{a} boldsymbol{R} boldsymbol{b} ) if ( |boldsymbol{a}-boldsymbol{b}|>mathbf{0} )12
1370If ( A ) and ( B ) are two non-empty sets
having n elements in common, then what is the number of common
elements in the sets ( A times B ) and ( B times A ? )
( A cdot n )
B ( cdot n^{2} )
( c cdot 2 n )
D. zero
12
1371f ( boldsymbol{R}={(boldsymbol{x}, boldsymbol{y}): boldsymbol{x}+mathbf{2} boldsymbol{y}=mathbf{8}} ) is a
relation on ( N, ) then write the range of ( R )
12
1372Find the domain and range of the
following real functions:
¡) ( boldsymbol{f}(boldsymbol{x})=-|boldsymbol{x}| )
ii) ( boldsymbol{f}(boldsymbol{x})=sqrt{mathbf{9}-boldsymbol{x}^{2}} )
12
1373Assertion
Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R}, boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+boldsymbol{x}^{2}+ )
( 100 x+5 sin x, ) then ( f(x) ) is bijective.
Reason
( mathbf{3} boldsymbol{x}^{mathbf{2}}+mathbf{2} boldsymbol{x}+mathbf{9 5}>mathbf{0} boldsymbol{x} in boldsymbol{R} )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
1374Let ( mathbb{Z} ) be the set of integers and ( boldsymbol{f}: mathbb{Z} rightarrow )
( mathbb{Z} ) be defined as ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}, boldsymbol{x} in mathbb{Z} )
then function is
A. bijection
B. injection
c. surjection
D. none of these
12
1375If ( (x+3,4-y)=(1,7), ) then ( (x- )
( mathbf{3}, mathbf{4}+boldsymbol{y}) ) is equal to
A ( cdot(-2,-3) )
в. (-5,1)
c. (3,4)
(年. ( (3,4)) )
D. None of these
11
1376Let ( boldsymbol{A}={mathbf{1}, mathbf{2}, mathbf{3}, mathbf{4}, mathbf{5}, mathbf{6}} ) and let ( boldsymbol{R}= )
( {(a, b): a, b in A text { and } b=a+1} . ) Show
that ( R ) is not reflexive.
12
1377ff ( boldsymbol{f}: boldsymbol{R}-{mathbf{0}} rightarrow boldsymbol{R} ) is defined by
( f(x)=x^{3}-frac{1}{x^{3}} ) then show that ( f(x)+ )
( boldsymbol{f}left(frac{1}{x}right)=mathbf{0} )
11
1378Check the commutativity and associativity of the following binary operation:
( ^{prime} *^{prime} ) on ( Q ) defined by ( a * b=frac{a b}{4} ) for all
( boldsymbol{a}, boldsymbol{b} in boldsymbol{Q} )
12
1379Consider ( boldsymbol{f}:{1,2,3} rightarrow{boldsymbol{a}, boldsymbol{b}, boldsymbol{c}} ) and
( boldsymbol{g}:{boldsymbol{a}, boldsymbol{b}, boldsymbol{c}} rightarrow{text { apple,ball,cat }} ) defined
as ( boldsymbol{f}(mathbf{1})=boldsymbol{a}, boldsymbol{f}(mathbf{2})=boldsymbol{b}, boldsymbol{f}(mathbf{3})=boldsymbol{c}, boldsymbol{g}(boldsymbol{a})= )
apple, ( g(b)= ) ball and ( g(c)= ) cat. Show
that ( f, g ) and ( g circ f ) are invertible. Find
( f^{-1}, g^{-1} ) and ( (g circ f)^{-1} ) and show that ( (g circ f)^{-1}=f^{-1} circ g^{-1} )
12
1380Let ( A ) be a set of ( n ) distinct elements.
Then, the total number of distinct
functions from ( A ) to ( A ) is ( ldots ) and out of
these ( ldots . ) are onto functions.
12
1381ff ( P={a, b} ) and ( Q={x, y, z}, ) show
that ( boldsymbol{P} times boldsymbol{Q} neq boldsymbol{Q} times boldsymbol{P} )
12
1382( boldsymbol{F}=frac{9}{5} boldsymbol{C}+mathbf{3 2} ; ) make ‘C’ the subject of
formula.
A ( cdot C=frac{5}{9}(F-32) )
B. ( C=frac{9}{5}(F-32) )
c. ( c=frac{5}{9}(F+32) )
D. none of the above
11
1383The number of real roots of the equation
( mathbf{5}+left|mathbf{2}^{x}-mathbf{1}right|=mathbf{2}^{x}left(mathbf{2}^{x}-mathbf{2}right) ) is:
( A cdot 2 )
B. 3
( c cdot 4 )
( D )
12
1384Assertion
if ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-mathbf{2}|+|boldsymbol{x}-mathbf{4}|+|boldsymbol{x}-mathbf{6}| )
an identify function for ( 4<x<6 )
Reason
( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{A}, ) defined by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} ) is an
identify function
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
11
1385Show that the function ( f: R rightarrow R ) given
by ( f(x)=x^{3} ) is injective
12
1386( n(A)=m ) and ( n(B)=n ; ) then
( A cdot n(A)+n(B)=n(A+B) )
B. ( n(A)-n(B)=n(A+B) )
C. ( A times B=m n )
( D cdot n(A) times n(B=n(A times B) )
12
1387In a closed lab, experiment is conducted by heating of water. When water is heated, the vapour pressure increased slowly then rapidly increases. As the water reaches to its boiling point, i.e.
( mathbf{1 0 0}^{0} boldsymbol{C}, ) the vapour pressure reaches 1 atm. Which of the following functions best describes the increase in vapour
pressure as water is heated to its boiling point?
A. Linear
B. Quadratic
c. Polynomial
D. Exponential
12
1388Find ( g circ f ) and ( f circ g ) when ( f: R rightarrow R ) and ( g: R rightarrow R ) are defined by ( f(x)= )
( mathbf{2} boldsymbol{x}+boldsymbol{x}^{2} ) and ( boldsymbol{g}(boldsymbol{x})=boldsymbol{x}^{3} )
11
1389Which pair of functions is identical?
( A cdot sin ^{-1}(sin x) ) and ( sin left(sin ^{-1} xright) )
B ( cdot log _{e} e^{x}, log ^{log _{x} x} )
( mathbf{c} cdot log _{e} x^{2}, 2 log _{e} x )
D. none of these
12
1390For each set of ordered pairs below, state whether it is a function or not give
reasons
( (i){(3,2),(4,2),(5,2)} )
if a function enter 1 else 0
12
1391( R ) is a relation on ( N ) given by ( R= )
( {(x, y) mid 4 x+3 y=20} . ) Which of the
following doesnot belong to ( R ) ?
A ( .(-4,12) )
B . (5,0)
c. (3,4)
(年. ( (3,4)) )
D. (2,4)
12
1392What positive value(s) of ( x, ) less than
( 360^{circ}, ) will give a minimum value for ( 4-2 ) ( sin x cos x ? )
( A cdot frac{pi}{4} ) only
B . ( frac{5 pi}{4} ) only
( mathbf{c} cdot frac{pi}{2} ) and ( frac{5 pi}{2} )
D. ( frac{3 pi}{2 pi} )
E ( cdot frac{pi}{4} ) and ( frac{5 pi}{4} )
11
1393The domain of the function ( boldsymbol{f}(boldsymbol{x})= ) ( frac{1}{sqrt{1-e^{frac{1}{x}-1}}} ) is
B ( cdot(-infty, infty) )
C . ( (-infty, 0] cup[1, infty) )
D. none of these
12
1394Find ( x ) and ( y, ) if ( (x+3,5)=(6,2 x+y) )
A. ( x=3, y=-1 )
B . ( x=6, y=2 )
c. ( x=2, y=3 )
D. none of these
12
1395The number of ordered pairs ( (a, b) ) of positive integers such that ( frac{mathbf{2 a – 1}}{mathbf{b}} ) and ( frac{2 b-1}{a} ) are both integers is
A . 1
B . 2
( c .3 )
D. more than 3
12
1396Let ( S ) be a set containing ( n ) elements. Then the total number of binary
operations on ( boldsymbol{S} ) is
( mathbf{A} cdot n^{n} )
B ( .2^{n} )
c. ( n^{n^{2}} )
D. ( n^{2} )
12
1397In
( Z, ) the set of all integers, the inverse
of -7 w.r.t. defined by ( a times b=a+b+7 )
for all ( a, b, in Z ) is :
A . -14
B. 7
c. 14
D. –
12
1398If ( A ) and ( B ) have ( n ) elements in
common,then the number of elements
common to ( boldsymbol{A} times boldsymbol{B} ) and ( boldsymbol{B} times boldsymbol{A} ) is:
12
1399If ( X={2,3,5,7,11} ) and ( Y={4,6,8,9,10} ) then find the number of one-one functions
from ( X ) to ( Y )
A . 720
B. 120
( c cdot 24 )
( D cdot 12 )
12
1400The range of the function ( f(x)= )
( left(8^{x}+4^{x}+8^{-x}+4^{-x}+5right) ) is
( A cdotleft(frac{7}{4}, inftyright) )
B. ( left[frac{7}{4}, inftyright) )
c. ( (9, infty) )
D ( cdot[9, infty) )
12
1401( boldsymbol{R}= )
( {(a, b): a, b epsilon N, a neq b text { and } a text { divided } b} )
is ( R ) reflexive. Give reason
12
1402If ( x ) co-ordinate of a point is 2 and ( y ) co-
ordinate is ( 0, ) then ordered pair for its
coordinate on ( X Y ) plane is
A ( .(0,0) )
в. (2,2)
D. (2,0)
12
1403If ( f(x)=l x ) and ( a, b, c ) are in A.P. ( k, l ) are
constants. Then ( boldsymbol{f}(boldsymbol{a}-boldsymbol{k}), boldsymbol{f}(boldsymbol{b}-boldsymbol{k}), ) and
( f(c-k) ) are in
A . A.
в. с.
c. н.
D. A.G.P
12
1404If ( boldsymbol{a} * boldsymbol{b}=|boldsymbol{a}-boldsymbol{b}| ) then ( boldsymbol{6} * boldsymbol{8} ) will be
A . -2
B. 14
( c cdot 2 )
D. cannot be determined
12
1405Show that the function ( boldsymbol{f}: boldsymbol{R}_{*} rightarrow boldsymbol{R}_{*} )
defined by ( f(x)=frac{1}{x} ) is one-one, where
( R_{*} ) is the set of all non-zero real
numbers. Is the result true, if the
domain ( R_{*} ) is replaced by ( N ) with ( c 0 )
domain being same as ( R_{*} ? )
12
1406f ( boldsymbol{R}={(boldsymbol{x}, boldsymbol{y})|boldsymbol{y}=boldsymbol{2} boldsymbol{x}+boldsymbol{7}| text { , where } boldsymbol{x} in boldsymbol{R} )
and ( -5 leq x leq 5} ) is relation, find the
domain of ( boldsymbol{R} )
12
1407If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) is defined by ( boldsymbol{f}(boldsymbol{x})=mathbf{3} boldsymbol{x}- )
2 then ( (f o f)(x)+2= )
A ( . f(x) )
B. ( 2 f(x) )
c. ( 3 f(x) )
D. ( -f(x) )
12
1408Let ( f: R rightarrow R ) be defined by ( f(x)=cos (5 x+2) )
Is f invertible?
if yes enter 1 else enter 0
12
1409For ( boldsymbol{x} in boldsymbol{R}, ) let ( [boldsymbol{x}] ) denotes the greatest integer ( leq x ), then the value of ( left[-frac{1}{3}right]+ ) ( left[-frac{1}{3}-frac{2}{100}right]+ldots+left[-frac{1}{3}-frac{99}{100}right] ) is
A. -100
B. -123
c. -135
D. -153
11
1410Let ( boldsymbol{f}: boldsymbol{N} rightarrow boldsymbol{N}(boldsymbol{N} ) being the set of
positive integers) be a function defined by ( f(x)= ) the biggest positive integer
obtained by reshuffling the digits of ( x ) For example, ( boldsymbol{f}(mathbf{2 9 6})=mathbf{9 6 2} )
( f ) is
A. One-one and onto
B. One-one and into
c. Many-one and onto
D. Many-one and into
12
1411If ( {(x, 2),(4, y)} ) represents an identity function, then ( (x, y) ) is :
( A cdot(2,4) )
B. (4,2)
c. (2,2)
( D cdot(4,4) )
11
1412If ( f(x)=x+frac{1}{x} ) find ( f(5) )12
1413A telephone company charges ( x ) cents for the first minute of a call and charges for any additional time at the rate of ( y ) cents per minute. If a certain call costs ( $ 5.55 ) and lasts more than 1 minute, which of the following expressions represents the length of that call, in minutes?
( ^{text {A. }} frac{555-x}{y} )
B. ( frac{555+x-y}{y} )
c. ( frac{555-x+y}{y} )
D. ( frac{555-x-y}{y} )
E ( frac{555}{x+y} )
12
1414Let ( R ) be a reflexive on a finite ( operatorname{set} A )
having ( n ) elements, and let there be ( m )
ordered pairs in ( R ) Then
A ( . m geq n )
в. ( m leq n )
c. ( m=n )
D. None of these
12
1415Check if the relation ( R ) in the set ( R ) of
real numbers defined as
( boldsymbol{R}={(boldsymbol{a}, boldsymbol{b}): boldsymbol{a}<boldsymbol{b}} ) is
(i) symmetric;
(ii) transitive
12
1416If ( f: R rightarrow R ) defined by ( f(x)=frac{x-1}{2} )
find ( (f o f)(x) )
12
1417The number of real solutions of the
equation ( e^{|x|}-|x|=0 ) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. None of these
11
1418The range of ( boldsymbol{f}(boldsymbol{x})=mathbf{1 0}+|boldsymbol{x}+mathbf{4}| ) is
A. ( (0, infty) )
)
B. ( [10, infty) )
c. ( (-infty, 10] )
( D cdot(-infty, infty) )
12
1419( f(x)=3 x^{4}+17 x^{3}+9 x^{2}-7 x- )
( 10 ; g(x)=x+5 )
12
1420Find the value of ( x ) for which following is
true –
( operatorname{sgn}left(x^{2}-2 x-8right)=-1 )
11
1421Find the number of ordered pairs in
RoR?
12
1422If ( log x-5 log 3=-2, ) then ( x ) equals
A . 1.25
B. 0.81
c. 2.43
D. 0.8
E. either 0.8 or 1.25
11
1423The range of the function ( f(x)= )
( log _{e}left(3 x^{2}-4 x+5right) )
( ^{A} cdotleft(-infty, log _{e} frac{11}{3}right] )
B. ( quadleft[log _{e} frac{11}{3}, inftyright) )
c. ( left[-log _{e} frac{11}{3}, log _{e} frac{11}{3}right. )
D. None of these
12
1424Let ( S ) be the set of all rational numbers
except 1 and ( * ) be defined on ( S ) by ( a * )
( b=a+b-a b, ) for all ( a, b in S . ) Prove
that ( * ) is a binary operation on ( S )
12
1425Let ( ^{prime} *^{prime} ) be a binary operation on ( N ) defined by ( a * b=mathrm{LCM}(a, b) ) for all
( boldsymbol{a}, boldsymbol{b} in boldsymbol{N} . ) Find ( boldsymbol{2} * boldsymbol{4}, boldsymbol{3} * boldsymbol{5}, boldsymbol{1} * boldsymbol{6} )
12
1426Find ( (f . g)(2) ) when ( f(x)=x-1 ) and
( boldsymbol{g}(boldsymbol{x})=-mathbf{5} boldsymbol{x}^{2}+mathbf{1 4} boldsymbol{x}+mathbf{7} )
A . 45
B. -39
c. 39
D. 15
11
1427Find the domain of definition of the
following function. ( boldsymbol{y}=sqrt{boldsymbol{x}-boldsymbol{x}^{2}} )
12
1428Write the identity relation on ( operatorname{set} A= )
( {a, b, c} )
12
1429If ( a ) and ( b ) are two variables and ( (a, b)= )
( (b, a), ) then
( mathbf{A} cdot a=0 )
В. ( b=0 )
( mathbf{c} cdot a=b )
( mathbf{D} cdot a pm b )
12
1430Let ( boldsymbol{A}={-1,1}, ) Then discuss whether
the given functions from ( A ) to itself are one-one onto or bijective:
(i) ( f(x)=frac{x}{2} )
(ii) ( g(x)=|x| )
(iii) ( h(x)=X^{2} )
12
1431( x^{2}=x y ) is a relation (defined on set ( R ) )
which is
A. Symmetric
B. Reflexive
c. Transitive
D. None of these
12
1432Which one of the following relations on ( mathbf{Z} ) is equivalence relation?
A ( . x R_{1} y Leftrightarrow|x|=|y| )
В. ( x R_{2} y Leftrightarrow x geq y )
c. ( x R_{3} y Leftrightarrow frac{x}{y} )
D. ( x R_{4} y Leftrightarrow x<y )
12
1433If ( f: R rightarrow R ) be defined by ( f(x)=e^{x} )
and ( g: R rightarrow R ) be defined by ( g(x)= )
( boldsymbol{x}^{2} . ) The mapping ( boldsymbol{g} circ boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be
defined by ( (boldsymbol{g} circ boldsymbol{f}(boldsymbol{x}))=boldsymbol{g}(boldsymbol{f}(boldsymbol{x})) forall boldsymbol{x} in boldsymbol{R} )
Then
A. ( g circ f ) is injective but ( f ) is not injective
B. ( g circ f ) is injective and ( g ) is injective
c. ( g circ f ) is injective but ( g ) is not injective
D. ( g circ f ) is surjective and ( g ) is surjective
12
1434The relation ( R ) define on the set of
natural numbers as ( {(a, b): a ) differs
from b by ( 3} ) is given.
A ( cdot{(1,4),(2,5),(3,6), ldots ldots . .} )
в. ( {(4,1),(5,2),(6,3), ldots ldots .} )
c. ( {(1,3),(2,6),(3,9), ldots ldots .} )
D. None of above
12
1435If ( boldsymbol{A}={mathbf{1}, mathbf{2}, mathbf{3}} ) and ( boldsymbol{B}={mathbf{3}, mathbf{8}}, ) then
( (boldsymbol{A} cup boldsymbol{B}) times(boldsymbol{A} cap boldsymbol{B}) ) is
A ( cdot{(3,1),(3,2),(3,3),(3,8)} )
B・ {(1,3),(2,3),(3,3),(8,3)}
c. {(1,2),(2,2),(3,3),(8,8)}
D ( cdot{(8,3),(8,2),(8,1),(8,8)} )
12
1436Find the domain of the real valued
square root function ( f ) given below ( boldsymbol{f}(boldsymbol{x})=sqrt{(mathbf{2} boldsymbol{x})} )
12
1437Check the commutativity and associativity of the following binary operation:
( ^{prime} *^{prime} ) on ( N, ) defined by ( a * b=a^{b} ) for all
( boldsymbol{a}, boldsymbol{b} in boldsymbol{N} )
12

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