# Application Of Derivatives Questions

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

#### List of application of derivatives Questions

Question NoQuestionsClass
1The normal to the curve ( x^{2}=4 y )
passing through (1,2) is
A. ( x+y=3 )
в. ( x-y=3 )
c. ( x+y=1 )
D. ( x-y=1 )
12
2Water is running into an underground right circular conical reservoir, which is 10 ( m ) deep and radius of its base is 5 m.
If the rate of change in the volume of water in the reservoir is ( frac{3}{2} pi m^{3} / m i n quad, ) then the rate ( ( ) in ( mathrm{m} / mathrm{min} ) ) at which water rises in it, when
the water level is ( 4 boldsymbol{m}, ) is:
( A cdot frac{3}{8} )
в. ( frac{1}{8} )
c. ( frac{1}{4} )
D. ( frac{3}{2} )
12
33. The minimum value of the expression sin a + sin ß +
sin y, where a, b, y are real numbers satisfying a + B+ y
= it is
a. positive
b. zero
c. negative
d. -3 (IIT-JEE 1995)
12
4The approximate value of ( (33)^{1 / 5} ) is
A . 2.0125
B. 2.1
c. 2.0
D. none of these
12
5Find the greatest value of ( boldsymbol{f}(boldsymbol{x})= )
( x^{2} log x ) on ( [1, e] )
( mathbf{A} cdot mathbf{0} )
B ( cdot e^{2} )
c. ( e / 2 )
D.
12
6Illustration 2.39 If A = 4 sin 0 + cos²e, then which of the
following is not true?
a. Maximum value of A is 5.
b. Minimum value of A is – 4.
c. Maximum value of A occurs when sin 0= 1/2.
d. Minimum value of A occurs when sin 0= 1.
12
7Find the absolute maximum and
minimum values of function given by
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}-mathbf{4} boldsymbol{x}+mathbf{8} ) in the interval ( [mathbf{1}, mathbf{5}] )
12
8The sides of two squares are ( x ) and ( y )
respectively, such that ( y=x+x^{2} ). The
rate of change of area of second square with respect to area of first square is
A. ( x^{2}+3 x-1 )
B . ( 2 x^{2}-3 x+1 )
c. ( 2 x^{2}+3 x+1 )
D. ( 1+2 x )
12
9The maximum value of ( 3 cos theta+4 sin theta )
is
A . -5
B. 5
c. 25
D. None of these
12
10Maximum value of ( frac{log x}{x} ) is
( A cdot frac{2}{e} )
B.
( c cdot frac{1}{e} )
D.
12
11The number of stationary points of ( mathbf{f}(mathbf{x})=sin mathbf{x} operatorname{in}[mathbf{0}, mathbf{2} boldsymbol{pi}] ) are
A
B. 2
( c cdot 3 )
D.
12
12( operatorname{Given} f(x)left{begin{array}{ll}|x-2|+lambda & text { if } x leq 2 \ x^{2}+1 & text { if } x>2end{array}right. )
If ( f(x) ) has a local minimum at ( x=2 )
then which of the following is most appropriate
A. ( lambda leq 5 )
в. ( lambda geq 5 )
c. ( lambda leq 0 )
D. ( lambda geq 0 )
12
13If a ball is thrown vertically upwards and the height ‘s’ reached in time ‘t’ is
given by ( s=22 t-11 t^{2}, ) then the total
distance travelled by the ball is
A. 44 units
B. 33 units
c. 11 units
D. 22 units
12
14I: If ( mathrm{f}^{prime}(mathrm{a})<0 ) then the function ( mathrm{f} ) is
decreasing at ( mathbf{x}=mathbf{a} )
Il: If ( f ) is decreasing at ( x=a ) then
( mathbf{f}^{prime}(mathbf{a})<mathbf{0} )
Which of the above statements are true
( ? )
A. only
B. only II
c. both land II
12
15Let the function ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be defined
by ( boldsymbol{f}(boldsymbol{x})=mathbf{2} boldsymbol{x}+cos boldsymbol{x} ) then ( boldsymbol{f} )
A. Has maximum at ( x=0 )
B. Has minimum at ( x=pi )
c. Is an increasing function
D. Is a decreasing
12
16A cube of ice melts without changing its shape at the uniform rate of
( 4 mathrm{cm}^{3} / ) min. The rate of change of the surface area of the cube, in ( c m^{2} / )min
when the volume of the cube is ( 125 mathrm{cm}^{3} )
is
A. -4
в. ( -frac{16}{5} )
c. ( -frac{16}{6} )
D. ( -frac{8}{15} )
12
17Calculate the maximum value of
( boldsymbol{f}(boldsymbol{x})=boldsymbol{3}-(boldsymbol{x}+boldsymbol{2})^{2} )
A . -3
B. –
c. 1
D. 3
E. 5
12
18If a particle moves according to the law, ( s=6 t^{2}-frac{t^{3}}{2}, ) then the time at which it
is momentarily at rest
A. ( t=0 ) only
B. ( t=8 ) only
c. ( t=0,8 )
D. none of these
12
19The equation of tangent to the curve
( boldsymbol{y}=3 x^{2}-x+1 ) at the point (1,3) is
A. ( y=5 x+2 )
в. ( y=5 x-2 )
c. ( _{y=frac{1}{5} x+2} )
D. ( _{y=frac{1}{5} x-2} )
12
20The maximum value of ( sin ^{2} x cos ^{3} x ) is
A ( cdot frac{6 sqrt{3}}{25 sqrt{5}} )
В. ( frac{9 sqrt{3}}{25 sqrt{5}} )
c. ( frac{9 sqrt{2}}{6 sqrt{5} / 5} )
D. ( frac{sqrt{2}}{sqrt{5}} )
12
21If ( f(x)=x^{x}, ) then ( f(x) ) is increasing in
interval:
A . ( [0, e] )
B. ( left(frac{1}{e}, inftyright) )
c. [0,1]
D. None of these
12
22The coordinates of the point(s) on the
graph of the function ( f(x)= ) ( frac{x^{3}}{3}-frac{5 x^{2}}{2}+7 x-4 ) where the tangent
drawn cut off intercepts from the
coordinate axes which are equal in
magnitude but opposite in sign is
This question has multiple correct options
( mathbf{A} cdotleft(2, frac{8}{3}right) )
B. ( left(3, frac{7}{2}right) )
( mathbf{c} cdotleft(1, frac{5}{6}right) )
D. none
12
23Find the point of inflexion of ( boldsymbol{f}(boldsymbol{x})= ) ( 3 x^{4}-4 x^{3} ) and hence draw the graph of
( boldsymbol{f}(boldsymbol{x}) )
12
24Maximum value of ( r ) where ( frac{c^{2}}{r^{2}}= ) ( frac{mathbf{a}^{2}}{sin ^{2} theta}+frac{b^{2}}{cos ^{2} theta} ) where ( c, a, b )
are constants is
A. ( frac{c}{a+b} )
B. ( frac{c}{a-b} )
c. ( frac{a^{2}+b^{2}}{c^{2}} )
D. ( frac{c^{2}}{a^{2}+b^{2}} )
12
25The altitude of a cone is ( 20 mathrm{cm} ) and its
semi-vertical angle is ( 30^{0} . ) If the semivertical angle is increasing at the rate
of ( 2^{0} ) per second, then the radius of the base is increasing at the rate of
A. ( 30 mathrm{cm} / mathrm{sec} )
в. ( frac{160}{3} mathrm{cm} / mathrm{sec} )
c. ( 10 mathrm{cm} / mathrm{sec} )
D. ( 160 mathrm{cm} / mathrm{sec} )
12
26Initially, equation of ellipse was ( 3 x^{2}+ )
( 4 y^{2}=12 . ) Keeping major axis constant
ellipse is bulge to form circle (with major axis as diameter). Its eccentricity
changes at a rate ( 0.1 / ) sec. Time taken to form this circle is
A . 2 sec
B. 3 sec
c. 5 sec
D. 7 sec
12
27( mathbf{f}(mathbf{x})=|mathbf{x}| ) is minimum at ( mathbf{x}= )
( mathbf{A} cdot mathbf{1} )
B.
( c cdot-1 )
D. 2
12
28The values of ‘a’ for which ( y=x^{2}+ )
( a x+25 ) touches ( x ) -axis are
( mathbf{A} cdot pm 10 )
B. ±2
( c .pm 1 )
D.
12
29Find the maxima of function ( 8-7 x^{2} )12
30The function ( f(x)=x^{2} ) is decreasing in
( A cdot(-infty, infty) )
в. ( (-infty, 0) )
( c cdot(0, infty) )
D. ( (-2, infty) )
12
31The function ( f(x)=frac{sin x}{x} ) is decreasing
in the interval
( ^{mathbf{A}} cdotleft(-frac{pi}{2}, 0right) )
B. ( left(0, frac{pi}{2}right) )
( ^{mathbf{c}} cdotleft(-frac{pi}{4}, 0right) )
D. None of these
12
32If ( x ) and ( y ) are sides of two squares such
that ( y=x-x^{2} ). Find the rate of change
of area of second square (side ( y ) ) with respect to area of the first square (side
( x) ) when ( x=1 mathrm{cm} )
12
33The slope of tangent to the curve ( x= ) ( boldsymbol{t}^{2}+mathbf{3} boldsymbol{t}-mathbf{8}, boldsymbol{y}=mathbf{2} boldsymbol{t}^{2}-boldsymbol{2} boldsymbol{t}-mathbf{5} ) at the
point (2,-1) is :
A ( cdot frac{22}{7} )
B. ( frac{6}{7} )
c. -6
D. None of these
12
34Suppose ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) is differentiable
function satisfying ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x})+ )
( boldsymbol{f}(boldsymbol{y})+boldsymbol{x} boldsymbol{y}(boldsymbol{x}+boldsymbol{y}) ) for every ( boldsymbol{x}, boldsymbol{y} in boldsymbol{R} ) if
( f^{prime}(0)=0, ) then which of the following hold(s) good?
This question has multiple correct options
( A . f ) is an odd function
B. ( f ) is a bijective mapping
c. ( f ) has a minima but no maxima
D. ( f ) has an inflection point
12
35A particle starts moving from rest from a fixed point in a fixed direction. The distance s from the fixed point at a time ( t ) is given by ( s=t^{2}+a t-b+17, ) where
( a ) and ( b ) are real numbers. If the particle
comes to rest after ( 5 s ) at a distance of
( s=25 ) units from the fixed point, then
value of ( a ) and ( b ) are respectively.
A .10,-33
B. -10,-33
c. -8,33
D. -10,33
12
36( boldsymbol{f}:(mathbf{0}, infty) rightarrowleft(-frac{boldsymbol{pi}}{2}, frac{boldsymbol{pi}}{2}right) ) be defined as
( f(x)=arctan (x) )
The above function can be classified as
A. injective but not surjective
B. surjective but not injective
c. neither injective nor surjective
D. both injective as well as surjective
12
37Find the equation of the tangent and the normal to the following curve at the indicated point. ( c^{2}left(x^{2}+y^{2}right)=x^{2} y^{2} ) at ( left(frac{x}{cos theta}, frac{c}{sin theta}right) )12
38A particle moves along a straight line
according to the law ( s=16-2 t+3 t^{3} )
where ( s ) metres is the distance of the
particle from a fixed point at the end of ( t ) second. The acceleration of the particle
at the end of 2 s is
A ( cdot 3.6 m / s^{2} )
B. ( 36 m / s^{2} )
c. ( 36 k m / s^{2} )
D. ( 360 m / s^{2} )
12
39A function ( f ) such that ( f^{prime}(2)=f^{prime prime}(2)= )
0 and ( f ) has a local maximum of -17 at ( x )
( =2 ) is :
A ( cdot(x-2)^{4} )
B . ( 3-(x-2)^{4} )
c. ( -17-(x-2)^{4} )
D. None of these
12
40The radius of a sphere is given by ( r=2 t )
The rate of change of surface area at ( t= )
1 is equal to
( A cdot 8 pi )
B. ( 32 pi )
c. ( 16 pi )
D. ( 4 pi )
12
41When the temperature of medium is
( 20^{circ} mathrm{C} ) a certain substance cools from
( 100^{circ} mathrm{C} ) to ( 60^{circ} mathrm{C} ) in 10 minutes. Its
temperature after 40 minutes from the beginning is
A ( cdot 15^{circ} mathrm{C} )
B . ( 20^{circ} mathrm{C} )
( mathbf{c} cdot 25^{circ} C )
D. ( 30^{circ} mathrm{C} )
12
42The radius of a circle is increasing uniformly at the rate of ( 5 mathrm{cm} / mathrm{sec} ). Find
the rate at which the area of the circle is
increasing when the radius is ( 6 mathrm{cm} )
12
43The difference between the greatest and
the least values of the function ( f(x)= ) ( int_{0}^{x}left(a t^{2}+1+cos tright) d t, a>0, x in[2,3] )
is:
A ( cdot frac{19}{3} a+1+sin 3-sin 2 )
в. ( frac{18}{3} a+1+2 sin 3 )
c. ( frac{18}{3} a-1+2 sin 3 )
D. none of these
12
44Let ( f(x)=7 e^{sin ^{2} x}-e^{cos ^{2} x}+2, ) then the
value of ( sqrt{7 f_{m i n}+f_{m a x},} ) is
12
45The tangent to the curve ( x= ) ( a sqrt{cos 2 theta} cos theta, y=a sqrt{cos 2 theta} sin theta ) at
the point corresponding to ( boldsymbol{theta}=boldsymbol{pi} / boldsymbol{6} ) is
A. parallel to the ( x ) -axis
B. parallel to the ( y ) -axis
c. parallel to line ( y=x )
D. none of these
12
46A curve ( C ) has the property that if the tangent drawn at any point ‘P’ on C meets the coordinate axes at ( A ) and ( B )
and ( P ) is midpoint of ( A B . ) If the curve passes through the point (1,1) then the equation of the curve is?
A ( . x y=2 )
в. ( x y=3 )
c. ( x y=1 )
D. ( x y=4 )
12
47The function ( f(x)=x^{9}+3 x^{7}+64 ) is
increasing on
A. ( R )
в. ( (-infty, 0) )
( c cdot(0, infty) )
D. ( R_{0} )
12
483. If the function f(x) = 2x} – 9ax² +12a+x+1, where
a>0, attains its maximum and minimum at p and a
respectively such that p2 = q , then a equals [2003]
(a) –
(b) 3
(c) 1
(d) 2
.
12
49Find the equation of the normal to the curve ( x^{2}=4 y ) which passes through
the point (1,2)
12
50A quadratc function in x has the value
10 when ( x=1 ) and has minimum value 1
when ( x=-2 ) the function is
( A cdot 2 x^{2}+3 x+5 )
B. ( 3 x^{2}+2 x+5 )
c. ( x^{2}+3 x+6 )
D. ( x^{2}+4 x+5 )
12
51A balloon, which always remains
spherical, has a variable diameter ( frac{3}{2}(2 x+1) . ) Find the rate of change of its volume with respect to ( x )
12
52The surface area of a sphere when its volume is increasing at the same rate as its radius is
( mathbf{A} cdot mathbf{1} )
B. ( frac{1}{2 sqrt{pi}} )
c. ( 4 pi )
D. ( frac{4 pi}{3} )
12
5310.
A spherical iron ball 10 cm in radius is coated with a layer of
ice of uniform thickness that melts at a rate of 50 cm /min.
When the thickness of ice is 5 cm, then the rate at which the
thickness of ice decreases is
[2005]
(a)

36
cm/min.
(b)
o
18 TC
cm/min.
c)
– cm/min.
547
cm/min.
(d)
cm/min
6T
cm/min
12
54Find an angle ( theta, 0<theta<frac{pi}{2}, ) which
increases twice as fast as its sine.
12
55The surface area of a solid sphere is
increased by ( 21 % ) without changing its shape.Find the percentage increase in its :
(ii)volume
12
56The graph of the curve ( x^{2}=3 x-y-2 )
is
A ( cdot ) between the lines ( x=1 ) and ( x=frac{3}{2} )
B. between the lines ( x=1 ) and ( x=2 )
C. strictly below the line ( 4 y=1 )
D. none of these
12
57The angle which the tangent to a curve at any point ( (x, y) ) on it makes with axis
of ( x ) is ( tan ^{-1}left(x^{2}-2 xright) ) for all values of ( x )
and it passes through the point (2,0) Determine the point on it whose
ordinate is maximum.
A ( .(2,8 / 3) )
в. ( (0,4 / 3) )
c. ( (1,2 / 3) )
D. ( (-1,4 / 3) )
12
586.
Find the shortest distance of the point (0,c) from the parabola
y=x2 where 0<c< 5.
(1982 – 2 Marks
12
59( f(x)=left{begin{array}{l}k-2 x, text { if } x leq-1 \ 2 x+3, text { if } x>-1end{array}right}, ) if ( f ) has
a local minimum at ( x=-1, ) then ( k= )
12
60Find the angle of intersection of the
following curve:
( x^{2}+y^{2}=2 x ) and ( y^{2}=x )
12
61Consider a real valued function ( boldsymbol{f} ) ( boldsymbol{R} rightarrow boldsymbol{R} ) satisfying ( boldsymbol{f}left(frac{boldsymbol{2} boldsymbol{x}+boldsymbol{3} boldsymbol{y}}{mathbf{5}}right)= )
( frac{2 f(x)+3 f(y)}{5} forall x, y in R quad ) and ( f(0)= )
( mathbf{2} ; boldsymbol{f}^{prime}(mathbf{0})=mathbf{1} )
Minimum distance of a point on graph
of ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) ) from origin is less than or
equal to
A. ( sqrt{2} ) units
B. ( frac{1}{2} ) units
c. ( frac{1}{sqrt{5}} ) units
D. ( frac{1}{sqrt{7}} ) units
12
62If ( f(x)=(x-a)^{2 n}(x-b)^{2 m+1} )
where ( m cdot n in N, ) then
A. ( x=a ) is a point of minimum
B. ( x=a ) is a point of maximum
c. ( x=a ) is not a point of maximum or minimum
D. No value of k satisfies the requirement
12
63Maximum value of ( 1+ )
( 8 sin ^{2}left(x^{2}right) cos ^{2}left(x^{2}right) ) is
A. 3
B. –
( c cdot-8 )
D.
12
64Find points on the curve ( frac{x^{2}}{9}+frac{y^{2}}{16}=1 )
at which the tangents are

Parallel to ( x ) -axis are ( (a, pm b) ).Find ( a+b )

12
65If ( boldsymbol{x} sqrt{mathbf{1}+boldsymbol{y}}+boldsymbol{y} sqrt{mathbf{1}+boldsymbol{x}}=mathbf{0} ) and ( boldsymbol{x} neq boldsymbol{y} )
then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
A. ( frac{1}{1+x} )
B. ( frac{1}{(1+x)^{2}} )
c. ( frac{-1}{(1+x)^{2}} )
D. ( frac{-1}{1+x} )
12
66Find the point on the curve ( y^{2}=2 x )
which is at a minimum distance from
the point (1,4)
12
67If ( boldsymbol{f}(mathbf{0})=mathbf{0} ) and ( boldsymbol{f}^{prime prime}(boldsymbol{x})>mathbf{0} ) for all ( boldsymbol{x}>mathbf{0} )
then ( frac{boldsymbol{f}(boldsymbol{x})}{boldsymbol{x}} )
A ( cdot ) decreases on ( (0, infty) )
B . increases on ( (0, infty) )
c. decreases on ( (1, infty) )
D. neither increases nor decreases on ( (0, infty) )
12
68If ( x ) be real then the minimum value of
( 40-12 x+x^{2} ) is?
A . 28
B. 4
( c cdot-4 )
D.
12
69Find the value of ( a ) if ( f(x)=2 e^{x}- )
( a e^{-x}+(2 a+1) x-3 ) is increasing for
all values of ( x )
12
70Find the value of ( frac{d y}{d x} ) at ( theta=frac{pi}{4} ) if ( x= ) ( boldsymbol{a} e^{theta}(sin theta-cos theta) ) and ( y=a e^{theta}(sin theta+ )
( cos theta) )
12
71An equation for the line that passes
through (10,-1) and is perpendicular to ( y=frac{x^{2}}{4}-2 ) is
A. ( 4 x+y=39 )
в. ( 2 x+y=19 )
c. ( x+y=9 )
D. ( x+2 y=8 )
12
72If ( log _{10}left(x^{3}+y^{3}right) )
( log _{10}left(x^{2}+y^{2}-x yright) leq 2, ) then the
maximum value of ( x y, ) for all ( x geq )
( mathbf{0}, boldsymbol{y} geq mathbf{0}, ) is
A . 2500
в. 3000
c. 1200
D. 3500
12
73Consider the function ( boldsymbol{f}(boldsymbol{x})= )
[
begin{array}{cl}
sqrt{boldsymbol{x}} ln boldsymbol{x} text { when } & boldsymbol{x}>mathbf{0} \
mathbf{0} text { for } & boldsymbol{x}=mathbf{0}
end{array}
]
Let the inflection points of the graph of
[
boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) text { be } boldsymbol{x}=boldsymbol{k} text { .Find } boldsymbol{k} ?
]
12
74The lines tangent to the curves ( y^{3}- ) ( x^{2} y+5 y-2 x=0 ) and ( x^{4}-x^{3} y^{2}+ )
( 5 x+2 y=0 ) at the origin intersect at
an angle ( theta ) equal to
A ( cdot frac{pi}{6} )
B.
c.
D.
12
75If the cube equation ( x^{3}-p x+q ) has
three distinct real roots, where ( p>0 )
and ( boldsymbol{q}>mathbf{0} )
Then, which one of the following is
correct?
A ( cdot ) the cubic has maxima at both ( sqrt{frac{p}{3}} ) and ( -sqrt{frac{p}{3}} )
B. The cubic has minima at ( sqrt{frac{p}{3}} ) and maxima at ( -sqrt{frac{p}{3}} )
c. The cubic has minima at ( -sqrt{frac{p}{3}} ) and maxima at ( sqrt{frac{p}{3}} )
D. The cubic has minima at both ( sqrt{frac{p}{3}} ) and ( -sqrt{frac{p}{3}} )
12
76( left(frac{2 a x}{a^{2}+x^{2}}right)+frac{1}{3}left(frac{2 a x}{a^{2}+x^{2}}right)^{3}+ )
( frac{1}{5}left(frac{2 a x}{a^{2}+x^{2}}right)^{5}+dots= )
( A cdot log (a+x) )
B. ( log (a-x) )
( ^{mathbf{C}} cdot log left(frac{a+x}{a-x}right) )
( D )
12
773x(x +1)
38.
Prove that for
sin x + 2x >
. Explain
the identity if any used in the proof.
(2004 – 4 Marks)
12
78Find the intervals in which the function
( f ) given by ( f(x)=x^{3}+frac{1}{x^{3}}, x neq 0 ) is
(i) increasing
(ii) decreasing.
12
79fthe rate of change in ( y=frac{x^{3}}{3}-2 x^{2}+ )
( 5 x+7 ) is two times the rate of change
in ( boldsymbol{x}, ) then ( boldsymbol{x}= )
A .2,3
в. 1,3
c. 1,2
D. 2,5
12
8032. Iff:R
→ R is a twice differentiable function such that
f”(x) > 0 for all x e R, and f f
f(1)=1, then
(a) f'(1)50
(b) 0<f'l)s
* <f'(1) 1
12
81Find minimum value ( 4 x+frac{9}{x} quad x>0 )12
82( text { If }boldsymbol{f}(boldsymbol{x})=boldsymbol{a} sec boldsymbol{x}-boldsymbol{b} tan boldsymbol{x}, boldsymbol{a}rangle boldsymbol{b}rangle boldsymbol{0}, ) then
the minimum value of ( f(x) ) is
A ( cdot sqrt{a^{2}+b^{2}} )
B. ( 2 sqrt{a^{2}-b^{2}} )
c. ( sqrt{a^{2}-b^{2}} )
D. ( a-b )
12
83If ( 1^{0}=alpha ) radians then the approximate
value of ( cos 60^{0} 1^{prime} ) is
A ( cdot frac{1}{2}+frac{alpha sqrt{3}}{120} )
в. ( frac{1}{2}-frac{alpha}{120} )
( ^{mathrm{C}} cdot frac{1}{2}-frac{alpha sqrt{3}}{120} )
D. none of these
12
84A particle moves along a straight line and its velocity at a distance ‘x’ from the origin is ( k sqrt{a^{2}-x^{2}} . ) Then acceleration of the particle is
( A cdot k )
B. ( -k^{2} )
c. ( k x )
( mathbf{D} cdot-k^{2} x )
12
8511. Let f:R → R be given by fire
be given by f (x) = (x – 1) (x – 2) (x – 5).
Define F(x) = S f (t)dt,x>0.
0
Then when of the following options is/are correct?
(a) Fhas a local maximum at x=2
(b) Fhas a local minimum at x=1
(c) Fhas two local maxima and one local minimum in (0,00)
(d) F(x) O for all x € (0,5)
12
86Two trains are running in opposite directions with the same speed. If the length of each train is 120 metres and they cross each other in 12 seconds, then the speed of each train (in ( mathrm{km} / mathrm{hr} ) )
is
A . 10
B. 18
( c . ) 36
D. 72
12
87Sum of the maximum and minimum
values of ( 12 cos ^{2} x-6 sin x cos x+ )
( 2 sin ^{2} x ) is
( mathbf{A} cdot mathbf{0} )
B. 7
c. 14
D. 15
12
88If ( y=frac{a x+b}{(x-1)(x-4)} ) has a turning
value at (2,-1) find ( a & b )
A ( . a=0, b=0 )
В. ( a=1, b=0 )
c. ( a=0, b=1 )
D. ( a=1, b=1 )
12
8911. The maximum and minimum values of x3 – 18×2 +96x in
interval (0,9) are
(a) 160,0
(b) 60,0
(c) 160, 128
(d) 120, 28
12
906.
The value of a for which the sum of the squares of
he sum of the squares of the roots
of the equation x
value is
(a) I (b) 0
(a-2) x-a-1=0 assume the least
[2005]
(c) 3 (d) 2
12
91A mans walks along a straight path at a speed of ( 4 f t / ) sec.A scarch light is
located on the ground ( 20 f t ) from the path and is kept focused on man. At what rate is the scarch light rotating when the man is ( 15 f t ) from the point on
the path closest to the search light.
c. ( 0.8 mathrm{rad} / mathrm{sec} )
D. ( 0.24 mathrm{rad} / mathrm{sec} )
12
92The temperature has dropped by 15 degree Celsius in the last 30 days. If the rate of temperature drop remains the same, how much more will the
temperature drop in the next 10 days?
12
93Show that ( boldsymbol{f}(boldsymbol{x})= )
( frac{x}{1+x tan x}, x epsilonleft(0, frac{pi}{2}right) ) is maximum
when ( boldsymbol{x}=cos boldsymbol{x} )
12
9422.
The curve y = ax3 + bx2 + cx + 5, touches the x-axis at
P(-2, 0) and cuts the y axis at a point Q, where its gradient
is 3. Find a, b, c.
(1994 – 5 Marks)
12
95Find the point on the curve ( y^{2}=8 x ) for
which the abscissa and ordinate
change at the same rate.
12
96If ( g(x)=7 x^{2} e^{-x^{2}} forall x in R, ) then ( g(x) ) has
This question has multiple correct options
A. local maximum at ( x=0 )
B. local minima at ( x=0 )
c. local maximum at ( x=1 )
D. two local maxima and one local minima
12
97Let ( f ) be the function ( f(x)=cos x- )
( left(1-frac{x^{2}}{2}right), ) then find the interval in which ( f(x) ) is strictly increasing.
( A cdot(-infty, infty) )
в. ( (-2, infty) )
c. ( [0, infty) )
()
D. ( (0, infty) )
12
98The function f(x) = 2x + x +21-1x + 21-2 xhas a local
minimum or a local maximum atx= (JEE Adv. 2013)
12
99( frac{1}{2}-frac{1}{2^{2} cdot 2}+frac{1}{2^{3} cdot 3}-frac{1}{2^{4} cdot 4}+ldots infty= )
A ( . log 2 )
B. ( log 3 )
( mathrm{c} cdot log frac{3}{2} )
D. ( log _{3} frac{2}{3} )
12
100( operatorname{Let} g^{prime}(x)>0 ) and ( f^{prime}(x)f(f(x-1)) )
в. ( f(g(x-1))>f(g(x+1)) )
( mathbf{c} . g(f(x+1))>g(f(x-1)) )
D. ( g(g(x+1))>g(g(x-1)) )
12
101Let ( f(x)=x^{3} e^{-3 x}, x>0 . ) Then the
maximum value of ( f(x) ) is
A ( cdot e^{-3} )
B. ( 3 e^{-3} )
( mathbf{c} cdot 27 e^{-9} )
D. ( infty )
12
102Let the interval ( I[-1,4] ) and ( f: I rightarrow R )
be a function such that ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{boldsymbol{3}}- )
( 3 x ). Then the range of the function is
A . [2,52]
в. [-2,2]
( mathbf{c} cdot[-2,52] )
D. none of these
12
103Find the interval of increase and
decrease of the following functions. ( f(x)=frac{x}{ln x} )
12
104If the radius of a sphere is measured as ( 7 m ) with an error of ( 0.02 m, ) then find the approximate error in calculating its volume12
105For what value of ( x, 8 x^{2}-7 x+2 ) has
the minimum value?
12
106To examine the function ( f(x)=2 x^{3}- )
( 15 x^{2}+36 x+10 ) for maxima and
minima, if any. Also find the maximum and minimum value.
12
107Assertion
The points on the curve ( y^{2}=x+sin x )
at which tangent is parallel to x-axis lies on a straight line.
Reason Tangent is parallel to ( x ) -axis then ( frac{d y}{d x}= )
0 or ( frac{d x}{d y}=infty )
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
108The angle made by the tangent line at (1
3) on the curve ( y=4 x-x^{2} ) with ( O X ) is
A ( cdot tan ^{-1}(2) )
B. ( tan ^{-1}(1 / 3) )
c. ( tan ^{-1}(3) )
D. ( pi / 4 )
12
109Show that ( f(x)=x^{3}-15 x^{2}+75 x- )
50 is an increasing function for all ( x in )
( boldsymbol{R} )
12
110The maximum value of ( frac{17}{3}-left(x-frac{4}{5}right)^{2} ) is
A ( cdot 4 / 5 )
B. ( -4 / 5 )
c. ( 17 / 3 )
D. ( -17 / 3 )
12
111Equation of a straight line passing
through (1,4) if the sum of its positive intercepts on the coordinate axis is the smallest is
A. ( 2 x+y-6=0 )
B . ( x+2 y-9=0 )
c. ( y+2 x+6=0 )
D. none
12
11224. Let (h, k) be a fixed point, where h>0,k>0. A straight line
passing through this point cuts the positive direction of the
coordinate axes at the points P and Q. Find the minimum
area of the triangle OPO, O being the origin.(1995-5 Marks)
12
113A man is moving away from a tower 41.6 ( mathrm{m} ) high at a rate of ( 2 mathrm{m} / mathrm{s} ). If the eye level of the man is ( 1.6 mathrm{m} ) above the ground,
then the rate at which the angle of elevation of the top of the tower changes, when he is at a distance of 30
( mathrm{m} ) from the foot of the tower, is
( A cdot-frac{4}{125} mathrm{rad} / mathrm{s} )
D. none of these
12
114( f(x)=frac{x}{log x}-frac{log 5}{5} ) is decreasing in
( A cdot(e, infty) )
B. (0,1) ( mathrm{U}(1, mathrm{e}) )
( c cdot(0,1) )
D. (1, e)
12
115If y=a In x + bx2 + x has its extremum values at x=-1 and
x = 2, then
(1983 – 1 Mark)
(a)
a=2, b=-1
(b) a=2,b= –
(c) a=- 2, b= 1
(d) none of these
12
116Find the minimum value of
( e^{(2 x-2 x+1) sin ^{2} x} )
12
117For ( boldsymbol{f}(boldsymbol{x})=sin ^{2} boldsymbol{x}, boldsymbol{x} in(mathbf{0}, boldsymbol{pi}) ) point of
inflection is
A ( cdot frac{pi}{6} )
в. ( frac{2 pi}{4} )
( c cdot frac{3 pi}{4} )
D. ( frac{4 pi}{3} )
12
118The volume of a cube is increasing at
the rate of ( 8 c m^{3} / s . ) How fast is
the surface area increasing when the
length of an edge is ( 12 mathrm{cm} ? )
12
119The total cost ( C(x) ) associated with the
production of x units of an item is given
by ( C(x)=0.05 x^{3}-0.02 x^{2}+30 x+ )
5000. Find the marginal cost when 3 units are produced, where by marginal
cost we mean the instantaneous rate of
change of total cost at any level of
output.
12
120The contentment obtained after
drinking ( x- ) units of a new drink at a
trial function is given by the function
( C(x)=2 x^{3}-x^{2}+7 x+2 . ) If the
marginal contentment is defined as rate of change of ( C ) with respect to the number of units consumed at an
instant, then find the marginal contentment when 4 units of drink are
consumed
12
121Write the set of values of ( a ) for which the
function ( f(x)=a x+b ) is decreasing
for all ( boldsymbol{x} in boldsymbol{R} )
12
122Illustration 2.39 If A = 4 sin 0 + cos²e, then which of the
following is not true?
a. Maximum value of A is 5.
b. Minimum value of A is – 4.
c. Maximum value of A occurs when sin 0= 1/2.
d. Minimum value of A occurs when sin 0= 1.
12
123Equation of tangent at that point of the curve ( boldsymbol{y}=mathbf{1}-boldsymbol{e}^{frac{x}{2}}, ) where it meets ( mathbf{y} ) -axis
( mathbf{A} cdot x+2 y=0 )
в. ( 2 x+y=0 )
c. ( x-y=2 )
D. None of these
12
124A closed conical vessel is filled with
water fully and is placed with its vertex down. The water is let out at a constant
speed. After 21 minutes, it was found that the height of the water column is half of the original height. How much more time in minutes does it empty the
vessel?
A . 21
B. 14
( c cdot 7 )
D. 3
12
125The equation of the tangent to the
curves ( x=t cos t ) and ( y=t sin t ) at the
origin is?
( mathbf{A} cdot x=0 )
в. ( y=0 )
c. ( x+y=0 )
D. ( x-y=0 )
12
126Find the points of maxima, minima and the intervals of monotonicity of the following function:
( boldsymbol{y}=boldsymbol{x}^{4}+boldsymbol{4} boldsymbol{x}^{3}-boldsymbol{8} boldsymbol{x}^{2}+boldsymbol{3} )
12
127( operatorname{Let} mathbf{h}(mathbf{x})=mathbf{f}(mathbf{x})-[mathbf{f}(mathbf{x})]^{2}+[mathbf{f}(mathbf{x})]^{3} ) for
every real number x then
A. h is increasing whenever fis increasing
B. h is increasing whenever f is decreasing
c. ( h ) is decreasing whenever ( f ) is increasing
D. Nothing can be said in general
12
12823.
The circle x2 + y2 = 1 cuts the x-axis at P and Q. Another
circle with centre at Q and variable radius intersects the first
circle at R above the x-axis and the line segment PQ at S.
Find the maximum area of the triangle OSR. (1994 – 5 Marks)
(
11)
12
12926.
J Bulleenlulaule alivu
Let f:R → R be a positi
→ R be a positive increasing function with
R
(3x)
lim
2= 1. Then lim
[2010]
=
*-700 f(x)
(a) ſ
6)
f(x)
3
(d) 1
12
130Determine the intervals over which the
function is decreasing, increasing, and
constant
A. Increasing ( [3, infty) ; ) Decreasing ( (-infty, 3] )
B. Increasing ( (-infty, 3] ); Decreasing ( [3, infty) )
c. Increasing ( (-infty, 3] ; ) Decreasing ( (-infty, 3 )
D. Increasing [3
( infty) ; ) Decreasing ( [3, infty) )
12
131If there is an error of ( 0.1 % ) in the
measurement of the radius of a sphere, find approximately the percentage error in the calculation of the volume of the
sphere
12
132( frac{a-b}{a}+frac{1}{2}left(frac{a-b}{a}right)^{2}+frac{1}{3}left(frac{a-b}{a}right)^{3}+= )
A ( cdot log _{e}left(frac{a}{b}right) )
( ^{mathrm{B}} cdot log _{e}left(frac{b}{a}right) )
( mathbf{c} cdot log _{e}(a+b) )
( mathbf{D} cdot log _{e}(a-b) )
12
133Find the angle of intersection of the
following curve:
( x^{2}=27 y ) and ( y^{2}=8 x )
12
134Find the absolute maximum and the
absolute minimum value of the
following function in the given
intervals.
( f(x)=3 x^{4}-8 x^{3}+12 x^{2}-48 x+25 )
in ( [mathbf{0}, mathbf{3}] )
12
135Show that ( frac{log x}{x} ) has a maximum value
at ( boldsymbol{x}=boldsymbol{e} )
12
136Two measurements of a cylinder are varying in such a way that the volume is kept constant. If the rates of change of the radius ( (r) ) and height ( (h) ) are equal in magnitude but opposite in sign, then
A ( . r=2 h )
B . ( h=2 r )
c. ( h=r )
D. ( h=4 r )
12
137If ( f(x)=frac{x^{3}}{3}-frac{5 x^{2}}{2}+6 x+7 ) increas
in the interval ( T ) and decreases in the
interval ( S ), then which one of the
following is correct?
A ( . T=(-infty, 2) cup(3 infty) ) and ( S=(2,3) )
в. ( T=Phi ) and ( S=(-infty, infty) )
c. ( T=(-infty, infty) ) and ( S=Phi )
D. ( T=(2,3) ) and ( S=(-infty, 2) cup(3, infty) )
12
138The function ( f(x)=frac{1}{x} ; x>0 ) on its domain is
A. increasing
B. decreasing
c. constant
D. information insufficient
12
139A body travels a distance ( s ) in ( t ) seconds.
It starts from rest and ends at rest. In
the first part of the journey, it move with
constant acceleration ( f ) and in the
seconds part with constant retardation
( r . ) The value of ( t ) given by
( ^{mathrm{A}} cdot_{2 s}left(frac{1}{f}+frac{1}{r}right) )
в. ( frac{2 s}{frac{1}{f}+frac{1}{r}} )
c. ( sqrt{2 s(f+r)} )
D. ( sqrt{2 sleft(frac{1}{f+r}right)} )
12
140Approximate value of ( tan ^{-1}(0.999) ) is
A . 0.7847
B. 0.748
c. 0.787
D. 0.847
12
141The total ( operatorname{cost} C_{(x)} ) and the total revenue ( R_{(x)} ) associated with production and
sale of ( x ) units of an item are given by
( C(x)=0.1 x^{2}+30 x+1000 ) and
( R(x)=0.2 x^{2}+36 x-100 . ) Find the
marginal cost and the marginal revenue when ( x=20 )
12
142The absolute maximum and minimum
value of ( boldsymbol{f}(boldsymbol{x})=sin boldsymbol{x}+cos boldsymbol{x}, boldsymbol{x} in[mathbf{0}, boldsymbol{pi}] )
are respectively
A ( cdot sqrt{2},-1 )
B. ( sqrt{2}, 1 )
c. ( sqrt{2},-sqrt{2} )
D. ( sqrt{3}, sqrt{2} )
12
143A boat goes ( 60 k m ) upstream and downs
stream in ( 6 h r ) and ( 2 h r ) respectively. Determine the product of speed of the
stream and that of the boat in still
water
12
144Consider the following in respect of the function ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}2+boldsymbol{x}, & boldsymbol{x} geq 0 \ 2-boldsymbol{x}, & boldsymbol{x}<0end{array}right. )
1. ( lim _{x rightarrow 1} f(x) ) does not exist.
2. ( f(x) ) is differentiable ( a t x=0 )
3. ( f(x) ) is continuous at ( x=0 )
Which of the above statements is/are
correct?
A. 1 only
B. 3 only
c. 2 and 3 only
D. 1 and 3 only
12
145Show that ( y=log (1+x)-frac{2 x}{2+x}, x> )
( -1, ) is an increasing function of ( x )
throughout its domain.
12
146Table12
147If errors of ( 1 % ) each are made in the
base radius and height of a cylinder, then the percentage error in its volume is
A . ( 1 % )
B. 2%
c. ( 3 % )
D. none of these
12
148A man 2 metres tall walks away from a lamp post 5 metres height at the rate of ( 4.8 mathrm{km} / mathrm{hr} . ) The rate of increase of the length of his shadow is
A. ( 1.6 mathrm{km} / mathrm{hr} )
B. ( 6.3 mathrm{km} / mathrm{hr} )
c. ( 5 mathrm{km} / mathrm{hr} )
D. 3.2 km/hr
12
149Let ( S ) be a square with sides of length ( x )
If we approximate the change in size of the area of ( S ) by ( left.h cdot frac{d A}{d x}right|_{x=x_{0}}, ) when the sides are changed from ( x_{0} ) to ( x_{o}+h )
then the absolute value of the error in
our approximation, is
( mathbf{A} cdot h^{2} )
B. ( 2 h x_{0} )
c. ( x_{0}^{2} )
D. ( h )
12
150The least value of ( 5^{sin x-1}+5^{-sin x-1} ) is
A . 10
в. ( frac{5}{2} )
( c cdot frac{2}{5} )
D.
12
151A water tank has the shape of a right circular cone with its vertex down. Its
altitude is ( 10 mathrm{cm} ) and the radius of the
base is ( 15 mathrm{cm} ). Water leaks out of the
bottom at a constant rate of
1 ( c u ) cm / sec. Water is poured into the
( operatorname{tank} ) at a constant rate of ( C ) cu.cm / sec.
Compute ( C ) so that the water level will
be rising at the rate of ( 4 mathrm{cm} / ) sec at the
instant when the water is ( 2 mathrm{cm} ) deep.
12
152The condition that ( mathbf{f}(mathbf{x})=mathbf{x}^{mathbf{3}}+mathbf{a x}^{mathbf{2}}+ )
( mathbf{b x}+mathbf{c} ) is an increasing function for al
real values of ( mathbf{x} ) is
A ( cdot a^{2}<12 b )
B . a ( ^{2}<3 b )
( c cdot a^{2}<4 b )
D. ( a^{2}<16 b )
12
153Assertion (A): The points on the curve
( y=x^{3}-3 x ) at which the tangent is
parallel to ( x ) -axis are (1,-2) and (-1,2)
Reason (R): The tangent at ( left(x_{1}, y_{1}right) ) on the curve ( y=f(x) ) is vertical then ( frac{d y}{d x} )
( operatorname{at}left(x_{1}, y_{1}right) ) is not defined
A. Both A and R are true and R is the correct explanation for A
B. Both A and R are true but R is not the correct explanation for ( A )
C. A is true but R is false
D. A is false but R is true
12
154Two trains of equal lengths take 10 seconds and 15 seconds respectively to cross a telegraph post. If the length of each train is 120 metres, in how much time (in seconds) will they cross each
other if travelling in opposite direction?
A . 10
B. 12
c. 15
D. 20
12
155The maximum value of ( 4 sin ^{2} x- )
( 12 sin x+7 ) is.
A . 25
B. 4
c. Does not exist
D. None of these
12
156The equation of the normal to the curve ( x=a cos ^{3} theta, y=a sin ^{3} theta ) at the point
( boldsymbol{theta}=frac{boldsymbol{pi}}{boldsymbol{4}} ) is
( mathbf{A} cdot x=0 )
В. ( y=0 )
c. ( x=y )
D. ( x+y=a )
12
157The percentage error in the surface area of a cube with edge ( x mathrm{cm}, ) when the edge is increased by ( 11 % ) is
A . 11
B . 22
c. 10
D. 44
12
1584. Given A = {x: 55 x 55 and
3
f(x) = cos x-x(1+x); find f(A).
(1980)
12
159Consider the function ( left{begin{array}{c}x sin frac{pi}{x} text { for } x>0 \ text { 0 for } x=0end{array} ) then the number right.
of points in (0,1) where the derivative ( f^{prime}(x) ) vanishes, is
A . 0
B. 1
( c cdot 2 )
D. infinite
12
160If the velocity of a body moving in a straight line is proportional to the square root of the distance traversed,
then it moves with
A. variable force
B. constant force
c. zero force
D. zero acceleration
12
161The maximum value of function ( boldsymbol{f}(boldsymbol{x})= ) ( 3 x^{3}-18 x^{2}+27 x-40 ) on the ( operatorname{set} S= )
( left{boldsymbol{x} in boldsymbol{R}: boldsymbol{x}^{2}+mathbf{3 0} leq mathbf{1 1} boldsymbol{x}right} ) is:
A . 122
B . -222
c. -122
D. 222
12
162Match the maxima of the functions ( f(x) )
on L.H.S
12
163If ( f ) and ( g ) are two increasing function such that ( f o g ) is defined, then fog will
be
A. increasing function
B. decreasing function
c. neither increasing nor decreasing
D. None of these
12
164The pressure ( P ) and volume ( V ) of a gas
are connected by the relation ( boldsymbol{P} boldsymbol{V}^{1 / 4}= )
constant. The percentage increase in
the pressure corresponding to a deminition of ( frac{1}{2} % ) in the volume is
A ( cdot frac{1}{2} % )
B. ( frac{1}{4} % )
c. ( frac{1}{8} % )
D. none of these
12
165Attempt the following:
The price P for demand D is given as
( P=183+120 D-3 D^{2} . ) Find D for
which the price is increasing.
12
16637.
The eccentricity of an ellipse whose centre is at the origin
If one of its directices is x=-4, then the equation of
the normal to it at
[JEE M 2017||
(a)
(c)
x+2y=4
4x – 2y=1
(b) 2y-x=2
(d) 4x +2y=7
12
16741.
If q denotes the acute angle between the curves,
y=10- x2 and y=2 + x2 at a point of their intersection, then
tan 0 is equal to:
JEE M 2019-9 Jan (M)
(b) 15
tla na
12
168Find the rate of change of the area of a
circle with respect to its radius ( r, ) when
( boldsymbol{r}=mathbf{5} mathrm{cm} )
12
169Find the intervals in which the function
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}-boldsymbol{4} boldsymbol{x}+boldsymbol{6} ) is strictly
decreasing.
12
170Example 2.5 Water pours out at the rate of Q from a tap,
into a cylindrical vessel of radius r. Find the rate at which the
height of water level rises when the height is h.
12
171If ( a ) and ( b ) are the non-zero distinct roots
of ( x^{2}+a x+b=0, ) then the least value
of ( x^{2}+a x+b ) is
A ( cdot frac{3}{2} )
B. 9
c. ( -frac{9}{4} )
D.
12
172The maximum value of ( boldsymbol{f}(boldsymbol{x})= ) ( frac{log x}{x}(x neq 0, x neq 1) ) is
( A )
B.
( c cdot e^{2} )
D. ( frac{1}{e^{2}} )
12
173The function ( f(x)=int_{-1}^{x} tleft(e^{t}-1right)(t- )
1) ( (t-2)^{3}(t-3)^{5} d t ) has a local
( operatorname{maximum} operatorname{at} x= )
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. 3
12
174Two candles of the same height are lighted at the same time. The first is
consumed in 4 hours and the second in
3 hours. Assuming that each candle burns at a constant rate, in how many
hours after being lighted was the first candle twice the height of the second?
A ( cdot frac{3}{4} ) hr.
B. ( frac{11}{2} ) hr.
c. ( 2 h r )
D. ( frac{12}{5} ) h.
E ( cdot frac{21}{2} ) hr.
12
175Solve: ( frac{d y}{d x}+2 y tan x=sin x, ) given that
( y=0, ) when ( x=frac{pi}{3} . ) Show that maximum value of ( y ) is ( frac{1}{8} )
12
176Using differentials the approximate value of ( sqrt{401} ) is
A . 20.100
B. 20.025
c. 20.030
D. 20.125
12
177A point on the parabola ( y^{2}=18 x ) at
which the ordinate increases at twice
the rate of the abscissa is
A ( cdot(2,4) )
В ( cdot(2,-4) )
( ^{mathrm{c}} cdotleft(-frac{9}{8}, frac{9}{2}right) )
D. ( left(frac{9}{8}, frac{9}{2}right) )
12
178Two towns ( A ) and ( B ) are ( 60 mathrm{km} ) apart. ( A )
school is to be built to serve 150
students in town ( A ) and 50 students in
town B. If the total distance to be
travelled by all 200 students is to be as small as possible, then the school be built is
A. town B
B. ( 45 mathrm{km} ) from town
c. town A
D. ( 45 mathrm{km} ) from town ( mathrm{B} )
12
179The population of a village increases continuously at the rate proportional to the number of its inhabitants present at
any time. If the population of the village was 20,000 in 1999 and 25000 in the year ( 2004, ) what will be the population of the village in ( 2009 ? )
12
180A particle is moving in a straight line such that its distance at any time t is given by ( mathbf{S}=frac{boldsymbol{t}^{4}}{mathbf{4}}-mathbf{2} boldsymbol{t}^{mathbf{3}}+mathbf{4} boldsymbol{t}^{2}+mathbf{7} ) then
its acceleration is minimum at ( t= )
( A )
B. 2
( c cdot 1 / 2 )
D. 3/2
12
1817. If from a wire of length 36 metre a rectangle of greatest
(a) 6, 12
(b) 9,9
(C) 10,8
(d) 13,5
12
182Find the approximate value of ( boldsymbol{f}(mathbf{2 . 0 1}) )
where ( boldsymbol{f}(boldsymbol{x})=mathbf{4} boldsymbol{x}^{2}+mathbf{5} boldsymbol{x}+mathbf{2} )
12
183Assertion (A): Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a
function such that ( boldsymbol{f}(boldsymbol{X})=boldsymbol{X}^{3}+boldsymbol{X}^{2}+ )
( 3 X+sin X, ) then ( f ) is one to one
Reason ( (R): f(x) ) is neither increasing nor decreasing function.
A. Both (A) and (R) are true and (R) is the correct explanation of (A)
B. Both (A) and (R) are true and (R) is not the correct explanation of (A).
c. (A) is true but (R) is false.
D. (A) is false but (R) is true
12
184Set up an equation of a tangent to the graph of the following function. ( boldsymbol{y}=boldsymbol{3}^{x}+boldsymbol{3}^{-2 x} ) at the points with
abscissa ( boldsymbol{x}=mathbf{1} )
12
185The greatest value of the function ( mathbf{f}(mathbf{x})=sin left{boldsymbol{x}[boldsymbol{x}]+mathbf{e}^{[x]}+frac{boldsymbol{pi}}{2}-1right} ) for all
( boldsymbol{x} in[mathbf{O}, infty) ) is
( A cdot-1 )
B.
( c )
D.
12
18615. The minimum value of (x2+299) is
(a) 75 (b) 50 (c) 25
(d) 55
12
187The slope of the tangent to the curve
( boldsymbol{x}=boldsymbol{t}^{2}+boldsymbol{3} boldsymbol{t}-mathbf{8}, boldsymbol{y}=boldsymbol{2} boldsymbol{t}^{2}-boldsymbol{2} boldsymbol{t}-mathbf{5} ) at the
point (2,-1) is
A ( cdot frac{22}{7} )
B. ( frac{6}{7} )
( c cdot frac{7}{6} )
D. ( -frac{6}{7} )
12
188A man 1.5 m tall walks away from a
lamp post ( 4.5 ~ m ) high at a rate fo ( 4 k m / h r . ) How fast is the farther end of shadow moving on the pavement?
A. ( 4 mathrm{km} / mathrm{hr} )
B. ( 2 mathrm{km} / mathrm{hr} )
( mathrm{c} .6 mathrm{km} / mathrm{hr} )
D. ( 5 mathrm{km} / mathrm{hr} )
12
189If there is an error of ( a % ) in measuring
the edge of a cube, then the percentage error in its surface area is
( mathbf{A} cdot 2 a )
B. ( frac{a}{2} )
( c .3 a )
D. None of the above
12
190Write the differential equation representing the family of curves ( y= )
( m x, ) where ( m ) is an arbitrary constant.
12
191The point on the curve ( x y^{2}=1 ) which is
nearest to the origin is
A. (1,1)
) 1
в. (1,-1)
c. ( left(4, frac{1}{2}right) )
D. ( left(frac{1}{2^{frac{1}{3}}}, 2^{frac{1}{6}}right) )
12
192Find the equation of the tangent to the curve ( boldsymbol{y}=sqrt{mathbf{3} boldsymbol{x}-mathbf{2}} ) which is parallel to
the line ( 4 x-2 y+5=0 )
12
193The radius of a circular disc is given as
( 24 c m ) with a maximum error in
measurement of ( 0.02 mathrm{cm} . ) Estimate the
maximum error in the calculated area
of the disc and compute the relative error by using differentials.
12
194Answer the following question in one word or one sentence or as per exact requirement of the question.

Write the maximum value of ( f(x)= ) ( boldsymbol{x}+frac{mathbf{1}}{boldsymbol{x}}, boldsymbol{x}<mathbf{0} )

12
195If the real-valued function ( f(x)=x^{3}+ )
( 3left(a^{2}-1right) x+1 ) be invertible then the
set of possible real values of a is
A ( cdot(-infty,-1) cup(1+infty) )
в. (-1,1)
c. [-1,1]
D ( cdot(-infty,-1] cup[1,+infty) )
12
196Find the maximum and minimum
values, if any of the following function given by:
( boldsymbol{g}(boldsymbol{x})=boldsymbol{x}^{boldsymbol{3}}+mathbf{1} )
12
197The velocity of a body varies with time
as ( V=3 t^{2}+2 . ) Find the instantaneous
acceleration ( t=3 ) sec
A ( cdot 31 m / s^{2} )
B . ( 18 mathrm{m} / mathrm{s}^{2} )
c. ( 3 m / s^{2} )
D . ( -2 / 3 mathrm{m} / mathrm{s}^{2} )
12
1989.
1+x
Find the coordinates of the point on the curve y=- 2
where the tangent to the curve has the greatest slope.
(1984 – 4 Marks)
12
199If ( boldsymbol{y}=boldsymbol{4}^{l o g_{2} sec x}-boldsymbol{9}^{l o g_{3} t a n x}, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
( mathbf{A} cdot mathbf{0} )
B.
c. 2
D. 3
12
200The function ( x^{x} ) decreases in the
interval-
A ( cdot(0, e) )
в. (0,1)
( ^{mathbf{c}} cdotleft(0, frac{1}{e}right) )
D. None of the above.
12
201All possible values of ( x ) for which the
function
( f(x)=x ln x-x+1 ) is positive is
A. ( (1, infty) )
(n)
B. ( left(frac{1}{e}, inftyright) )
c. ( [e, infty) )
D. ( (0, infty)-{1} )
12
2028. A spherical iron ball 10 cm in radius is coated with a
layer of ice of uniform thickness that melts at a rate of
50 cm /min. When the thickness of ice is 5 cm, then the
rate at which the thickness of ice decreases, is
(a) 5cm/min
– cm/min
547
(b)
cm/min
367 cm/min
cm/min
187
12
203The equation to the normal to the hyperbola ( frac{x^{2}}{16}-frac{y^{2}}{9}=1 ) at (-4,0) is
A ( .2 x-3 y=1 )
B. ( x=0 )
( mathbf{c} cdot x=1 )
D. ( y=0 )
12
204A cylindrical water tank of diameter
1.4 ( m ) and height 2.1 ( m ) is being fed by a pipe of diameter ( 3.5 mathrm{cm} ) through which
water flows at the rate of ( 2 m / s ) Calculate the time, it takes to fill the
( operatorname{tank} )
12
205The perimeter of a sector is ( P . ) The area of the sector is maximum when its radius is
( A cdot 1 / sqrt{P} )
B. ( P / 2 )
c. ( P / 4 )
D. ( sqrt{P} )
12
2063. The area of a rectangle will be maximum for the given
perimeter, when rectangle is a
(a) Parallelogram (b) Trapezium
(c) Square
(d) None of these
12
207Given ( f^{2}(x)+g^{2}(x)+h^{2}(x) leq 9 ) and
( boldsymbol{U}(boldsymbol{x})=boldsymbol{3} boldsymbol{f}(boldsymbol{x})+mathbf{4} boldsymbol{g}(boldsymbol{x})+mathbf{1 0 h}(boldsymbol{x}), ) where
( f(x) cdot g(x) ) and ( h(x) ) are continuous ( forall x in )
R. If maximum value of ( U(x) ) is ( sqrt{N} )
Then find ( N )
12
208( frac{1}{2.3}+frac{1}{4.5}+frac{1}{6.7}+ldots infty= )
A ( cdot 1+log _{e} 2 )
B . ( 1-log _{mathrm{e}} 2 )
c. ( 2+log _{mathrm{e}} 2 )
D. ( 2-log _{mathrm{e}} 2 )
12
209Find the set of values of ( a ) for which
( f(x)=x+cos x+a x+b ) is
increasing on ( boldsymbol{R} )
12
2108.
On the interval [0, 1] the function x25 (1 – x)75 takes its
maximum value at the point
(1995)
(a) 0
(6)
12
211Assume that a shperical randrop evaporates at a rate proportional to its a surface area. if it’s radius is originally 3 ( mathrm{mm}, ) and ( 1 mathrm{minute} ) later has been
reduced to ( 2 mathrm{mm} ). Find an expression for the radius of the raindrop at any
time.
12
21216. Ifx=-1 and x=2 are extreme points of
f(x) = a log|x+3x² + x then
JEEM 2014
(a) a=2,8 =-
(e) =-6,8 = 1
(b) a = 2, B=
(d) a = -6,8 =
12
213Find the interval of increase and
decrease of the following functions. ( f(x)=x^{2} e^{-x} )
12
214The slope of the tangent to the curve ( boldsymbol{x}=mathbf{3} boldsymbol{t}^{2}+mathbf{1}, boldsymbol{y}=boldsymbol{t}^{3}-mathbf{1} ) at ( boldsymbol{x}=mathbf{1} ) is
A ( cdot frac{1}{2} )
B.
( c cdot-2 )
( D cdot infty )
12
215Show that ( f(x)=tan ^{-1}(sin x+cos x) )
is a decreasing function on the interval ( left(frac{pi}{4}, frac{pi}{2}right) )
12
216Show that the function given by ( f(x)= ) ( 3 x+17 ) is strictly increasing on ( R )12
217The point at which the tangent to the
curve ( y=x^{3}+5 ) is perpendicular to
the line ( x+3 y=2 ) are
A. (6,1),(-1,4)
В. (6,1)(4,-1)
c. (1,6),(1,4)
D. (1,6),(-1,4)
12
218Write the angle between the curves ( y= )
( e^{-x} ) and ( y=e^{x} ) at their point of
intersection.
12
219xed*, x 50
27. Let f(x)= x + ax2 – x?, x>0
(1996 – 3 Marks)
Where a is a positive constant. Find the interval in which
f'(x) is increasing.
12
220Identify a local maxima for:
( boldsymbol{y}=boldsymbol{x}^{3}-boldsymbol{3} boldsymbol{x}+boldsymbol{2} )
( mathbf{A} cdot x=2 )
B. ( x=1 )
c. ( x=-2 )
D. ( x=-1 )
12
221The minimum of ( f(x)=frac{1+x+x^{2}}{1-x+x^{2}} )
occurs at ( x= )
( A cdot-1 )
B.
( c cdot 2 )
( D ldots-2 )
12
222If ( boldsymbol{alpha}=cos 10^{circ}-sin 10^{circ}, beta=cos 45^{circ}- )
( sin 45^{circ}, gamma=cos 70^{circ}-sin 70^{circ} ) then the
descending order of ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) is
( mathbf{A} cdot alpha, beta, gamma )
в. ( gamma, beta, alpha )
c. ( alpha, gamma, beta )
( mathbf{D} cdot beta, alpha, gamma )
12
223ABCD is a rectangle in which ( A B=10 ) ( mathrm{cms}, mathrm{BC}=8 mathrm{cms} . ) A point ( mathrm{P} ) is taken on
AB such that ( P A=x ). Then the minimum
value of ( P C^{2}+P D^{2} ) is obtained when ( x )
( = )
A . 10
B. 5
( c cdot 8 )
D. 4
12
224Water is flowing out at the rate of
( 6 m^{3} / ) min from a reservoir shaped like
a hemispherical bowl of radius ( R=13 )
( mathrm{m} ) The volume of water in the
hemispherical bowl is given by ( v= ) ( frac{pi}{3} cdot y^{2}(3 R-y) ) when the water is ( y )
meter deep Find at what rate is the water level changing when the water is
( 8 mathrm{m} ) deep.
( ^{mathbf{A}} cdot-frac{1}{12 pi} m / m i n )
B. ( -frac{1}{18 pi} m / ) min
c. ( -frac{1}{24 pi} m / ) min
D. ( -frac{1}{30 pi} m / ) min
12
225Function ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}}{mathbf{1}+|boldsymbol{x}|} ) is
A. increasing function
B. decreasing function
C. neither increasing nor decreasing
D. not differentiable at ( x=0 )
12
226If the function ( boldsymbol{f}(boldsymbol{x})= )
( left(a^{2}-3 a+2right) cos frac{x}{2}+(a-1) x )
possesses critical points, then ( a ) belongs to the interval
( mathbf{A} cdot(-infty, 0) cup(4, infty) )
B . ( (-infty, 0] cup[4, infty) )
( mathbf{c} cdot(-infty, 0] cup{1} cup[4, infty) )
D. None of these
12
227Show that ( f(x)=frac{1}{x} ) is decreasing
function on ( (mathbf{0}, infty) )
12
228If error in measuring the edge of a cube is ( k % ) then the percentage error in
estimating its volume is
( A cdot k )
B. ( 3 k )
( c cdot frac{k}{3} )
D. none of these
12
229The interval in which ( y=x^{2} e^{-x} ) is
increasing is:
( A cdot(-infty, infty) )
в. (-2,0)
c. ( (2, infty) )
D. (0,2)
12
230The volume of a ball increases at
( 4 pi c . c / )sec. The rate of increases of radius when the volume is ( 288 pi c . c . s ) is
A ( cdot frac{1}{6} mathrm{cm} / mathrm{sec} )
B ( cdot frac{1}{36} mathrm{cm} / mathrm{sec} )
c. ( frac{1}{9} mathrm{cm} / mathrm{sec} )
D. ( frac{1}{49} mathrm{cm} / mathrm{sec} )
12
231If ( f(x)=a-(x-3)^{89}, ) then greatest
value of ( boldsymbol{f}(boldsymbol{x}) ) is
( mathbf{A} cdot mathbf{3} )
B. ( a )
c. no maximum value
D. none of these
12
232Answer the following question in one
word or one sentence or as per exact requirement of the question.

Write the point where ( f(x)=x log _{e} x )
attains minimum value.

12
233Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. ( sqrt{49.5} )12
234Find the equation of the quadratic function ( boldsymbol{f} ) whose graph increases over the interval ( (-infty,-2) ) and decreases over the interval ( (-2,+infty), f(0)=23 )
and ( boldsymbol{f}(mathbf{1})=mathbf{8} )
A ( cdot f(x)=3(x+2)^{2}+35 )
B. ( f(x)=-3(x+2)^{2}-35 )
C. ( f(x)=-3(x-2)^{2}+35 )
D. ( f(x)=-3(x+2)^{2}+35 )
12
235The minimum value of ( f(x)=mid 2 x+ )
( mathbf{5} mid+mathbf{6} ) is:
A . 2
B. 3
c. 5
D. 6
E . 8
12
236The minimum & maximum value of ( f(x)=sin (cos x)+cos (sin x) forall-frac{pi}{2} leq )
( x leq frac{pi}{2} ) are respective
A. ( cos 1 ) and ( 1+sin 1 )
1
B. ( sin 1 ) and ( 1+cos 1 )
c. ( cos 1 ) and ( cos left(frac{1}{sqrt{2}}right)+sin left(frac{1}{sqrt{2}}right) )
D. None of these
12
2375. If the edge of a cube increases at the rate of 60 cm per
second, at what rate the volume is increasing when the
edge is 90 cm
(a) 486000 cu cm per sec
(b) 1458000 cu cm per sec
(c) 43740000 cu cm per sec
(d) None of these
12
238Show that ( f(x)=tan ^{-1} x-x ) is a
decreasing function on ( boldsymbol{R} )
12
239Find an angle ( theta, 0<theta<frac{pi}{2}, ) which
increases twice as fast as its sine.
A ( .60^{circ} )
B. ( 120^{circ} )
( c .90^{circ} )
D. ( 45^{circ} )
12
240The length ( x ) of rectangle is decreasing at the rate of ( 5 mathrm{cm} / mathrm{minute} ) and the
width ( y ) is increasing at the rate of 4 ( mathrm{cm} / mathrm{minute} . ) When ( boldsymbol{x}=mathbf{8} mathrm{cm} ) and ( boldsymbol{y}=mathbf{6} )
( mathrm{cm}, ) find the rates of change of the area of the rectangle.
12
241The curve ( y=a x^{3}+b x^{2}+c x+5 )
touches the ( x ) -axis at ( P(-2,0) ) and
cuts the ( y ) -axis at a point ( Q, ) where its
gradient is ( 3 . ) Find ( a, b, c )
A ( cdot a=-frac{1}{5}, b=1, c=3 )
B. ( a=-frac{1}{4}, b=-1, c=4 )
c. ( _{a=-frac{1}{4}, b=0, c=3} )
D. ( a=-frac{1}{3}, b=1, c=-3 )
12
242The least vlue of the function ( f(x)= ) ( boldsymbol{a x}+frac{boldsymbol{b}}{boldsymbol{x}}(boldsymbol{a}>mathbf{0}, boldsymbol{b}>mathbf{0}, boldsymbol{x}>mathbf{0}) ) is12
243What is the nature of the graph: ( y=frac{4}{x} )
A. Rectangular hyperbola in first and third quadrant
B. Rectangular hyperbola in first and second quadrant
C. Hyperbola but not rectangular hyperbola
D. Rectangular hyperbola in second and fourth quadrant
12
244The distance moved by a particle in
time ( t ) seconds is given by ( s=t^{3}- )
( 6 t^{2}-15 t+12 . ) The velocity of the
particle when acceleration becomes
zero is
A . 15
в. -27
( c cdot 6 / 5 )
D. none of these
12
245If the error committed in measuring the radius of the circle is ( 0.05 % ), then the
corresponding error in calculating the area is:
( mathbf{A} cdot 0.05 % )
B. ( 0.025 % )
c. ( 0.25 % )
D. ( 0.1 % )
12
246( f(x)=tan ^{-1} x-x ) is ( _{-1} )
( boldsymbol{R} )
A. increasing to ( R )
B. decreasing to ( R )
C . increasing on ( R^{+} )
D. increasing on ( (-infty, 0) )
12
247Assertion
The largest term in the sequence ( a_{n}= ) ( frac{n^{2}}{n^{3}+200}, n in N ) is the ( 7^{t h} ) term
Reason
The function ( f(x)=frac{x^{2}}{x^{3}+200} ) attains
local maxima at ( x=7 )
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
248The Point ( (s) ) on the cure ( y^{3}+3 x^{2}= )
( 12 y ) where the tangent is vertical
(parallel to y-axis), is/are.
( mathbf{A} cdotleft[pm frac{4}{sqrt{3}},-2right] )
B ( cdotleft(pm frac{sqrt{11}}{3}, 1right) )
D ( cdotleft(pm frac{4}{sqrt{3}}, 2right) )
12
249Using the differentials, the approximate
value of ( (627)^{1 / 4} ) is
A . 5.002
B. 5.003
c. 5.005
D. 5.004
12
250Find the greatest and the least values of the following functions. ( f(x)=sin x+cos 2 x ) on the interval
( [mathbf{0}, boldsymbol{pi}] )
12
251Find ( a ) for which ( f(x)=a(x+sin x)+ )
( a ) is increasing on ( boldsymbol{R} )
12
252Find the angle of intersection of the
following curve:
( x^{2}+4 y^{2}=8 ) and ( x^{2}-2 y^{2}=2 )
12
253The maximum value of ( f(x)=2 x^{3}- )
( 21 x^{2}+36 x+20 ) in ( 0 leq x leq 2 ) is
A . 37
B. 44
( c .32 )
D. 30
12
254Find the maximum or minimum value
of the quadratic expression ( 2 x-7- )
( 5 x^{2} )
12
255Approximate change in the volume V of a cube of side x metres caused by increasing the side by ( 3 % ) is?12
256If the tangent at any point on the curve ( x^{4}+y^{4}=c^{4} ) cuts off intercepts ( a ) and ( b )
on the coordinate axes, the value of
( boldsymbol{a}^{-mathbf{4} / mathbf{3}}+boldsymbol{b}^{-mathbf{4} / mathbf{3}} ) is
A ( cdot c^{-4 / 3} )
B ( cdot c^{-1 / 2} )
c. ( c^{1 / 2} )
D. ( c^{4 / 3} )
12
257For the curve ( y=3 sin theta cos theta, x= )
( e^{theta} sin theta, 0 leq theta leq pi ; ) the tangent is
parallel to ( x ) -axis when ( theta ) is
( mathbf{A} cdot mathbf{0} )
в. ( frac{pi}{2} )
c.
D.
12
258Find the value of a for which the
function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{a} boldsymbol{x}^{3}-boldsymbol{3}(boldsymbol{a}+boldsymbol{2}) boldsymbol{x}^{2}+ )
( 9(a+2) x-1 ) is decreasing for all ( x in )
( boldsymbol{R} )
12
25926. If(x) is differentiable and strictly increasing function, then
the value of lima
f(x²)-f(x)
(2004)
(2) 1 (b) 0 (0) 1 (d) 2.
F(x)-f(0) is
12
260A cylindrical gas container is closed at
the top and open at the bottom; if the iron plate of the top is ( 5 / 4 ) times as thick as the plate forming the cylindrical sides, find the radio of the radius to the height of the cylinder using minimum material for the same capacity.
A . 5 ( bar{A} )
в. ( frac{4}{5} )
( c cdot frac{5}{2} )
D. ( frac{2}{5} )
12
261Show that the function ( f(x)=2 x+ ) ( cot ^{-1} x-log {x+sqrt{left(1+x^{2}right)}} )
increasing ( forall mathrm{R} )
12
262An inverted cone has a depth of ( 40 mathrm{cm} )
and a base of radius ( 5 mathrm{cm} ). Water is
poured into it at a rate of 1.5 cubic centimetres per minute. Find the rate at
which the level of water in the cone is
rising when the depth is 4 cm.
12
263The radius of a circular plate is increasing at the ratio of 0.20 cm/sec.At what rate is the area
increasing when the radius of the plate
is ( 25 mathrm{cm} )
12
264A particle moves along the y-axis so that its position at time ( 0 leq t leq 20 ) is
( operatorname{given} operatorname{by} y(t)=5 t-frac{t^{2}}{3} cdot ) At what time
does the particle change direction?
A. 5 seconds
B. 7.5 seconds
c. 10 seconds
D. 15 seconds
E. 18 seconds
12
265( P(x, y) ) is a point on a straight line which makes intercepts a and b on the
( x, y ) axes respectively, then the minimum value of ( x^{2}+y^{2}= )
A ( cdot frac{a^{2}+b^{2}}{a^{2} b^{2}} )
B. ( frac{a^{2}-b^{2}}{a^{2}+b^{2}} )
c. ( frac{a^{2}-b^{2}}{2left(a^{2}+b^{2}right)} )
D. ( frac{mathrm{a}^{2} b^{2}}{left(mathrm{a}^{2}+b^{2}right)} )
12
266Find the max.value of the total surface
of a right circular cylinder which can be inscribed in a sphere of radius a.
A ( cdot pi a^{2}(1+1 / sqrt{5}) )
B ( cdot pi / 2 a^{2}(1+sqrt{5}) )
c. ( pi a^{2}(1+sqrt{5}) )
D cdot ( pi a^{2}(2+sqrt{5}) )
12
267Find the approximate change in the
surface area of a cube of side ( x ) metres
caused by decreasing the side by 1
percent
12
268If ( f(x)=x^{3}-6 x^{2}+9 x+3 ) then ( f(2) )
( A cdot 3 )
B. 4
( c cdot 5 )
D. None of these
12
269Match the points on the curve ( 2 y^{2}= )
( x+1 ) with the slope of normals at those
points and choose
A ( . i-b, i i-d, i i i-c, i v-a )
B . ( i-b, i i-a, i i i-d, i v-c )
c. ( i-b, i i-c, i i i-d, i v-a )
D. ( i-b, i i-d, i i i-a, i v-c )
12
270Find the tangents and normal to the
curve ( boldsymbol{y}(boldsymbol{x}-mathbf{2})(boldsymbol{x}-mathbf{3})-boldsymbol{x}+mathbf{7}=mathbf{0}, mathbf{a t} )
point (7,0) are
A. ( x-20 y-7=0,20 x+y-140=0 )
в. ( x+20 y-7=0,20 x-y-140=0 )
D. ( 7 x+20 y-1=0,20 x-7 y-100=0 )
12
271It is given that at ( x=1 ), the function
( x^{4}-62 x^{2}+a x+9 ) attains its
maximum value, on the interval [0,2].
Find the value of a.
12
272The length ( x ) of a rectangle is decreasing at the rate of ( 5 mathrm{cm} / mathrm{minute} ) and the width y is increasing at the rate of ( 4 mathrm{cm} / mathrm{minute} . ) When ( x=8 mathrm{cm} ) and
( y=6 mathrm{cm} ) find the rates of change of
perimeter, and
(b) the area of the
rectangle.
12
273Find the value of ( 64.75-75.97+36.82 )
A . 25.60
B. 23.98
c. 29.76
D. 21.96
12
274Let ( f(x) ) be a derivable function,
( f^{prime}(x)>f(x) ) and ( f(0)=0 . ) Then?
A. ( f(x)>0 ) for all ( x>0 )
B. ( f(x)0 )
c. No sign of ( f(x) ) can be ascertained
D. ( f(x) ) is a constant function
12
275For ( boldsymbol{x} in boldsymbol{R}-{boldsymbol{n} boldsymbol{pi}}, ) the graph is: ( boldsymbol{y}= )
( x+sin x )
A. Monotonically Increasing
B. Monotonically Decreasing
c. Either ( A ) or ( B )
D. None of these
12
276The maximum value of the function ( f(x)=sin left(x+frac{pi}{6}right)+cos left(x+frac{pi}{6}right) ) in
the interval ( left(0, frac{pi}{2}right) ) occurs at
A ( cdot frac{pi}{12} )
в.
( c cdot frac{pi}{4} )
D. ( frac{pi}{3} )
12
277f ( boldsymbol{y}=boldsymbol{a} log |boldsymbol{x}|+boldsymbol{b} boldsymbol{x}^{2}+boldsymbol{x} ) has its
extremum values at ( x=-1 ) and ( x=2 )
then the value of ( -2 a b ) is
12
278Answer the following question in one
word or one sentence or as per exact requirement of the question.

Write the minimum value of ( f(x)= ) ( boldsymbol{x}+frac{mathbf{1}}{boldsymbol{x}}, boldsymbol{x}>mathbf{0} )

12
279The shortest distance between line y-x=1 and curvex=y?
[2011]
is
12
280The circumference of a circle is
measured as ( 56 mathrm{cm} ) with an error 0.02
( mathrm{cm} . ) The percentage error in its area is
A ( cdot frac{1}{7} )
в. ( frac{1}{28} )
c. ( frac{1}{14} )
D. ( frac{1}{56} )
12
281The equation of the tangent to the curve ( y=x+frac{4}{x^{2}}, ) that is parallel to the ( x- )
axis, is
( mathbf{A} cdot mathbf{y}=1 )
В. ( mathrm{y}=2 )
( mathbf{c} cdot mathbf{y}=3 )
D. ( mathrm{y}=0 )
12
282At an extreme point of a function ( boldsymbol{f}(boldsymbol{x}) )
the tangent to the curve is
A. Parallel to the ( x ) -axis
B. Perpendicular to the ( x ) -axis
c. Inclined at an angle ( 45^{circ} ) to the ( x ) -axis
D. Inclined at an angle ( 60^{circ} ) to the ( x ) -axis
12
283Find the equation of normal to the curve ( x^{2}=4 y ) passing through the point
(1,2)
A ( . x+y=3 )
B. ( x-y=3 )
c. ( 2 x-y=4 )
D. ( 2 x-3 y=1 )
12
284The image of the interval [-1,3] under
the mapping ( f(x)=4 x^{3}-12 x ) is
A. [-2,0]
B. [-8, 72]
c. [-8,0]
D. [-8,-2]
12
285Illustration 2.34
Find the minimum value of 2 cos 0 +
sino + V2 tan ® in (0)
12
286The first and second order derivatives of
a function ( f(x) ) exist at all points in ( (a, b) ) with
( mathbf{f}^{prime}(mathbf{c})=mathbf{0}, ) where ( mathbf{a}<mathbf{c}<mathbf{b} . ) Further
more, if ( mathrm{f}^{prime}(mathrm{x})mathbf{0} ) for all
points on the immediate right of ( c ), then at ( mathbf{x}=mathbf{c}, mathbf{f}(mathbf{x}) ) has a
A. local maximum
B. local minimum
c. point of inflexion
D. none of these
12
28725.
A curve y=f(x) passes through the point P(1,1). The normal
to the curve at Pis a(y – 1) + (x – 1) = 0. If the slope of the
tangent at any point on the curve is proportional to the
ordinate of the point, determine the equation of the curve.
Also obtain the area bounded by the y-axis, the curve and
the normal to the curve at P.
(1996 – 5 Marks)
12
288If the function ( f(x)=frac{K sin x+2 cos x}{sin x+cos x} )
is strictly increasing for all values of ( x )
then
( mathbf{A} cdot K1 )
c. ( K2 )
12
289If the function ( f(x)=frac{a x+b}{(x-1)(x-4)} )
has a local maxima at ( (2,-1), ) then
A. ( b=1, a=0 )
В. ( a=1, b=0 )
c. ( b=-1, a=0 )
D. ( a=-1, b=0 )
12
290Minimum and maximum ( z=5 x+2 y )
subject to the following constraints:
( x-2 y leq 2 )
( 3 x+2 y leq 12 )
( -3 x+2 y leq 3 )
( boldsymbol{x} geq mathbf{0}, boldsymbol{y} geq mathbf{0} )
12
291The tangent to the curve ( boldsymbol{x}=boldsymbol{a}(boldsymbol{theta}- )
( sin theta) ; y=a(1+cos theta) ) at the points
( boldsymbol{theta}=(mathbf{2 k + 1}) boldsymbol{pi}, boldsymbol{k} in boldsymbol{Z} ) are parallel to
( mathbf{A} cdot y=x )
в. ( y=-x )
( mathbf{c} cdot y=0 )
D. ( x=0 )
12
292The point on the curve ( y=x^{2} ) which is
nearest to (3,0) is
A . (1,-1)
B. (-1,1)
c. (-1,-1)
D. (1,1)
12
293Find the condition tht the curves ( 2 x= )
( y^{2} ) and ( 2 x y=k ) intersect orthogonally.
12
294The curve ( y=a x^{3}+b x^{2}+c x+d ) has a
point on inflexion at ( x=1 ) then
( mathbf{A} cdot a+b=0 )
B . ( a+3 b=0 )
c. ( 3 a+b=0 )
D. ( 3 a+b=1 )
12
295The changes in a function y and the independent variable ( x ) are related as ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{x}^{2} . ) Find ( y ) as a function of ( boldsymbol{x} )12
296Consider the following statements:
( f(x)=ln x ) is an increasing function
on ( (0, infty) )
2. ( f(x)=e^{x}-x(ln x) ) is an increasing
function on ( (1, infty) ) Which of the above statements is lare
correct?
A. 1 only
B. 2 only
c. Both 1 and 2
D. Neither 1 nor 2
12
297ff ( boldsymbol{y}=boldsymbol{a} boldsymbol{l} boldsymbol{n}|boldsymbol{x}|+boldsymbol{b} boldsymbol{x}^{2}+boldsymbol{x} ) has its
extremum values at ( x=-1 ) and ( x=2 )
then
A ( . a=2, b=-1 )
в. ( a=2, b=-frac{1}{2} )
c. ( a=-2, b=frac{1}{2} )
D. none of these
12
298The maximum value of the function
( f(x)=x^{3}+2 x^{2}-4 x+6 ) exists at
A. ( x=-2 )
B. ( x=1 )
c. ( x=2 )
D. ( x=-1 )
12
299The interval(s) of decrease of of the
function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2} log 2 mathbf{7}- )
( 6 x log 27+ )
( left(3 x^{2}-18 x+24right) log left(x^{2}-6 x+8right) ) is
This question has multiple correct options
A ( cdot(3-sqrt{1+1 / 3 e}, 2) )
B. ( (4,3+sqrt{1+1 / 3 e}) )
c. ( (3,4+sqrt{1+1 / 3 e}) )
D. none of these
12
300Find the rate of change of the area of a circle with respect to its radius ( r ) when
(i) ( r=3 mathrm{cm} )
(ii) ( r=4 mathrm{cm} )
12
301The minimum value of the function
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} log boldsymbol{x} ) is
A. ( -frac{1}{e} )
B. ( -e )
( c cdot frac{1}{e} )
D.
12
302A ladder ( 13 mathrm{m} ) long leans against a wall. The foot of the ladder is pulled along the
ground away from the wall, at the rate
( 1.5 mathrm{m} / mathrm{sec} . ) How fast is the angle ( theta ) between the ladder and the ground is changing when the foot of the ladder is
( 12 mathrm{m} ) away from the wall.
12
303The points on the curve ( 12 y=x^{3} ) whose
ordinate and abscissa change at the
same rate, are
A ( cdot(-2,-2 / 3),(2,2 / 3) )
в. ( (-2,2 / 3),(2 / 3,2) )
c. ( (-2,-2 / 3) ) only
D. ( (2 / 3,2) ) only
12
30437. If a and b are positive quantities such that a > b, the
minimum value of a seco- b tan is
a. 2ab
b. Ja² – 6²
c. a-b
d. a² +6²
12
30514.
If the normal to the curve y=f(x) at the point (3,4) makes an
3 T
angle
with the positive x-axis, then f'(3) = (20008)
(6) 3 (c) 1 / 2 (d) 1
A
4
(a) –1
12
306Let ( y=frac{6 x^{3}-45 x^{2}+108 x+2}{2 x^{3}-15 x^{2}+36 x+1} )
( boldsymbol{x} in(mathbf{0}, mathbf{1 0}) )
Maximum value of ( y )
A . 3
B . 2
c. ( frac{86}{29} )
D. ( frac{82}{29} )
12
307Find the greatest and the least values of the following functions. ( y=x^{3}+3 x^{2}+3 x+2 ) on the
interval [-2, 2].
12
308The value of ‘a’ for which the function ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{cc}-boldsymbol{x}^{3}+cos boldsymbol{a} & boldsymbol{0}<boldsymbol{x}<1 \ boldsymbol{x}^{2} & boldsymbol{x} geq mathbf{1}end{array}right. )
has a local minimum at ( x=1 ), is
This question has multiple correct options
A . -1
B. 1
( c cdot 0 )
D. ( -frac{1}{2} )
12
309The point on the curve ( y=sqrt{x-1} )
where the tangent is perpendicular to the line ( 2 x+y-5=0 ) is
A. (2,-1)
в. (10,3)
c. (2,1)
()
D. (5,-2)
12
310Find the minimum value of ( boldsymbol{f}(boldsymbol{x})= ) ( frac{e^{x}}{[x+1]}, x geq 0 )12
31129. Let
: R
R given by
x<0;
x + 5×4 +10×3 +10x² +3x +1,
1 x²-x+1,
2x – 4x +7x-
f(x)
0x<1;
15x<3;
x23
(x-2) loge(x-2)-x+-
Then which of the following options is/are correct?
(a) f'has a local maximum atx=1
(b) fis increasing on (-0,0)
(c) f' is NOT differentiable at x=1
(d) f is onto
12
312The minimum value of ( 2 x^{3}-9 x^{2}+ )
( 12 x+4 ) is
A . 4
B. 5
( c .6 )
D.
E. 8
12
31312. Find the point on the curve 4×2 + a²y2 = 4a?, 4<a? <8
that is farthest from the point (0, -2). (1987 – 4 Marks)
12
314A police cruiser, approaching a rightangled intersection from the north, is chasing a speeding car that has turned the corner and is now moving straight
east. When the cruiser is ( 0.6 mathrm{km} ) north
of the intersection and the car is ( 0.8 mathrm{km} )
to the east, the police determine with radar that the distance of the car is
increasing at ( 20 mathrm{km} / mathrm{h} ). Suppose that the cruiser is moving at ( 60 mathrm{km} / mathrm{h} ) at the instant of measurement.
The speed of the car is (in ( mathrm{km} / mathrm{h}) )
A . 70
B. 80
c. 75
D. 60
12
315The maximum value of function ( x^{3}- )
( 12 x^{2}+36 x+17 ) in the interval [1,10]
is
A . 17
в. 17
c. 77
D. None of these
12
316Let ( x ) be a number which exceeds its
square by the greatest possible quantity, then ( x= )
A. ( 1 / 2 )
B . ( 1 / 4 )
c. ( 3 / 4 )
D. ( 1 / 3 )
12
317The maximum of ( mathbf{f}(mathbf{x})=frac{log mathbf{x}}{mathbf{x}^{2}}(mathbf{x}>mathbf{0}) )
occurs at ( mathbf{x}= )
( A )
B. ( sqrt{e} )
( c cdot frac{1}{e} )
D. ( frac{1}{sqrt{mathrm{e}}} )
12
318The value of ( a ) in order that ( f(x)= ) ( sqrt{3} sin x-cos x-2 a x+b ) decreases
for all real values of ( x, ) is given by
( mathbf{A} cdot a<1 )
в. ( a geq 1 )
( c cdot a geq sqrt{2} )
D. ( a<sqrt{2} )
12
319If a particle moves according to the law, ( s=6 t^{2}-frac{t^{3}}{2}, ) then the time at which it
is momentarily at rest
A. ( t=0 ) only
B. ( t=8 ) only
c. ( t=0,8 )
D. none of these
12
320( mathbf{A} mathbf{x}=mathbf{0}, mathbf{f}(mathbf{x})=cos mathbf{x}-1+frac{mathbf{x}^{2}}{mathbf{2}}-frac{mathbf{x}^{mathbf{3}}}{mathbf{6}} )
A. Has a minimum
B. Has a maximum
c. Does not have an extremum
D. Is not defined
12
321Find the least value of secºx + cosecx +
Illustration 2.36
secx coseco x.
12
32233. The maximum value of the expression
Usin’x + 2a’ – 12a – 1 – cos? x], where a and x are
real numbers, is
a. 13
b. √2
c. 1
bed. V5
12
323If ( f(x)=max left{sin x, cos ^{-1} xright}, ) then
A. ( f ) is differentiable everywhere
B. ( f ) is continuous everywhere but not differentiable
C tis discountinuous at ( x=frac{n pi}{2}, n in N )
D. ( f ) is non-differentiable at ( x=frac{n pi}{2}, n in N )
12
324The local minimum value of the
function ( f^{prime} ) given by ( f(x)=3+|x|, x epsilon R )
is.
( A cdot 3 )
B.
( c cdot-1 )
D.
12
325A particle is moving along the curve
( boldsymbol{x}=boldsymbol{a} boldsymbol{t}^{2}+boldsymbol{b} boldsymbol{t}+boldsymbol{c} ). If ( boldsymbol{a} boldsymbol{c}=boldsymbol{b}^{2}, ) then the
particle would be moving with uniform
A. rotation
B. velocity
c. acceleration
D. retardation
12
326The rate at which ice-ball melts is
proportional to the amount of ice in it. If half of it melts in 20 minutes, the
amount of ice left after 40 minutes compared to it original amount is
( ^{A} cdotleft(frac{1}{8}right) t h )
B. ( left(frac{1}{16}right) ) th
( ^{c} cdotleft(frac{1}{4}right) t h )
D. ( left(frac{1}{32}right) t h )
12
327Use differential to approximate ( sqrt{mathbf{1 0 1}} )
A .10 .00
B. 10.05
( c cdot 11.00 )
D. Non of these
12
328The maximum and minimum values of
the function ( y=x^{3}-3 x^{2}+6 ) are
A .2,0
в. 6,0
( c cdot 6,2 )
D. 4,2
12
329( boldsymbol{x}=boldsymbol{3} cos boldsymbol{Theta}, boldsymbol{y}=boldsymbol{3} sin boldsymbol{Theta}, boldsymbol{t h e n} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=? )
( A cdot-cot theta )
B. ( -sin theta )
( c cdot sin Theta )
D. ( tan theta )
12
330If ( f(x)=x e^{x(1-x)}, ) then ( f(x) ) is
A ( cdot ) increasing on ( left[-frac{1}{2}, 1right] )
B. decreasing on R
C. increasing on R
D. decreasing on ( left[-frac{1}{2}, 1right] )
12
331A balloon, which always remains
spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is ( 10 mathrm{cm} )
12
332=
+
9. If a € (0, 1), and f(a) = (a? – a + 1) + osina
91 Va? – a +1
27 cosec? a
=, then the least value of f(a)/2 is
ſa² –at1
12
333The radius of a circle is increasing at
the rate of ( 0.7 mathrm{cm} / mathrm{s} ). What is the rate of
increases of its circumference?
12
334The maximum value of the function
( 2 x^{3}-15 x^{2}+36 x+4 ) is attained at
( mathbf{A} cdot mathbf{0} )
B. 3
( c cdot 4 )
D. 2
E. 5
12
335f ( log _{e} 4=1.3868, ) then ( log _{e} 4.01= )
A . 1.3968
B . 1.3898
c. 1.3893
D. none of these
12
336Mark the correct alternative of the
following.
Let ( f(x)=2 x^{3}-3 x^{2}-12 x+5 ) on
( [-2,4] . ) The relative maximum occurs at
( boldsymbol{x}=? )
A . -2
B. – –
( c cdot 2 )
D. 4
12
337Prove that the function given by ( f(x)= )
( x^{3}-3 x^{2}+3 x-100 ) is inreasing in ( R )
12
338Find the local maxima and local
minima of the function ( f(x)=sin x- )
( cos x, 0<x<2 pi . ) Also find the local
maximum and local minimum values.
12
339The maximum value of ( left(frac{1}{x}right)^{x} ) is ( e^{1 / e} )
A. True
B. False
12
340The percentage error in the ( 11^{t h} ) root of
the number 28 is approximately times the
percentage error in 28
A ( cdot frac{1}{28} )
в. ( frac{1}{11} )
c. 11
D. 28
12
341Consider the cubic ( f(x)=8 x^{3}+ )
( 4 a x^{2}+2 b x+a, ) where ( a, b, in R )
For ( boldsymbol{a}=mathbf{1}, ) if ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) ) is strictly
increasing ( forall x in R, ) then maximum
range of values of ( b ) is
( ^{mathbf{A}} cdotleft(-infty, frac{1}{3}right) )
B. ( left(frac{1}{3}, inftyright) )
( ^{c} cdotleft[frac{1}{3}, inftyright) )
( D cdot(-infty, infty) )
12
342Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. ( (15)^{frac{1}{4}} )12
343If the radius of a sphere is measured as
( 7 mathrm{m} ) with an error of ( 0.02 mathrm{m}, ) find the
approximate error in calculating its volume
12
344If ( f(x)=3 x^{2}+15 x+5, ) then the
approximate value of ( boldsymbol{f}(mathbf{3} . mathbf{0 2}) ) is.
A . 47.66
B. 57.66
c. 67.66
D. 77.66
12
345Find the intervals in which the function
( f(x)=frac{x^{4}}{4}-x^{3}-5 x^{2}+24 x+12 ) is
(a) strictly increasing,
(b) strictly decreasing.
12
346A particle moves along the curve ( x^{2}= )
2y. At what point, ordinate increases at
the same rate as abscissa increases?
12
347Which of the following options is the only CORRECT combination?
A. ( (I I)(i i i)(S) )
В. ( (I)(i i)(R) )
c. ( (I I I)(i v)(P) )
D. ( (I V)(i)(S) )
12
348Find the values of ( x ) if ( f(x)=frac{x}{x^{2}+1} ) is
i) an increasing function
ii) a decreasing function
12
349An aeroplane at an altitude of
960 meters flying horizontally at ( mathbf{7} 20 k m / h r . ) passes directly over an observer. The rate at which it is
approaching the observers when it is
1600 meters directly away from him is
A. ( 576 mathrm{km} / mathrm{hr} )
B. ( 676 mathrm{km} / mathrm{hr} )
c. ( 720 mathrm{km} / mathrm{hr} )
D. ( 570 mathrm{km} / mathrm{hr} )
12
350( operatorname{Let} f(x)=frac{sin 4 pi[x]}{1+[x]^{2}}, ) where ( [x] ) is the
greatest integer less than or equal to ( x )
then
A. ( f(x) ) is not differentiable at some points
B. ( f(x) ) exists but is different from zero
c. ( L H D(a t x=0)=0 . R H D(a t x=1)=0 )
D. ( f(x)=0 ) but ( f ) is not a constant function
12
351Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. ( (401)^{frac{1}{2}} )12
352If ( T=2 pi sqrt{frac{l}{g}}, ) then relative errors in ( T ) and I are in the ratio
A . ( 1 / 2 )
B.
c. ( 1 / 2 pi )
D. none of these
12
353If ( f ) is a real-valued differentiable
function such that ( f(x) f^{prime}(x)<0 ) for all
real ( x, ) then
A. ( f(x) ) must be a increasing function
B. ( f(x) ) must be a decreasing function
c. ( |f(x)| ) must be a increasing function
D. ( |f(x)| ) must be a decreasing function
12
354A point is moving along the curve ( y^{3}= ) ( 27 x ). The interval in which the abscissa
changes at slower rate than ordinate, is
A ( cdot(-3,3) )
B ( cdot(-infty, infty) )
c. (-1,1)
D. ( (-infty,-3) cup(3, infty) )
12
355The two curves ( y=x^{2}-1 ) and ( y= )
( 8 x-x^{2}-9 ) at the point (2,3) have
common
A. tangent as ( 4 x-y-5=0 )
B. tangent as ( x+4 y-14=0 )
c. normal as ( 4 x+y=11 )
D. normal as ( x-4 y=10 )
12
356The value of ‘a’ for which the function
( f(x)=(a+2) x^{3}-3 a x^{2}+9 a x-1 )
decreases for all real values of ( x ) is
B ( cdot(-infty,-3) )
( c cdot(-infty,-2) )
D. ( (-infty,-3] cup[0, infty) )
12
357The angle at which the curve ( y=k e^{k x} )
intersects the ( y ) – axis is This question has multiple correct options
A ( cdot tan ^{-1} k^{2} )
B ( cdot cot ^{-1}left(k^{2}right) )
( mathrm{c} cdot_{sin }^{-1}left(frac{1}{sqrt{1+k^{4}}}right) )
D・sec- ( 1(sqrt{1+k^{4}}) )
12
358A man ( 1.5 ~ m ) tall walks away from a
lamp post ( 4.5 ~ m ) high at a rate of
( 4 k m / h r( ) i) How fast is his shadow lengthening?(ii)How fast is the farther end of shadow moving on the
pavement??
12
359Water flows at the rate of 10 m/minute
through a cylindrical pipe ( 5 m m ) in diameter. How long would it take to fill a
conical vessel whose diameter at the
base is ( 40 mathrm{cm} ) and depth ( 24 mathrm{cm} ) ?
12
360Let ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}|, ) then
( mathbf{A} cdot f^{prime}(0)=0 )
B. ( f(x) ) has a maximum at ( x=0 )
( mathbf{c} cdot f(x) ) has a minimum at ( x=0 )
D. ( f(x) ) has no maximum and not minimum
12
361The graph of the equation ( y=x^{2}- )
( 4 x+5 ) has its lowest point at
A ( .(2,1) )
в. (-2,1)
c. (-2,-1)
D. (2,-1)
12
36229.
The intercepts on x-axis made by tangents to the curve,
dt, x ER, which are parallel to the line y = 2x, are
equal to :
(a) #1
(6) #2
(c)
3
[JEE M 2013]
(d) 24
12
363( P(-2,3), Q(3,7) . ) The point ( A ) on the ( x ) axis for which ( mathbf{P A}+mathbf{A Q} ) is least is
( ^{mathrm{A}} cdotleft(frac{-1}{2}, 0right) )
в. ( left(frac{1}{2}, 0right) )
c. ( left(0, frac{-1}{2}right) )
D. ( left(0, frac{1}{2}right) )
12
364( frac{1}{3}+frac{1}{3}left(frac{1}{3}right)^{3}+frac{1}{5}left(frac{1}{3}right)^{5}+dots= )
A ( cdot frac{1}{2} log _{e} 2 )
B ( cdot 2 log _{e} 2 )
( mathbf{c} cdot log _{e} 2 )
D. ( log _{e} 3 )
12
36511.
1 where it is
A function is matched below against an interval where
I to be increasing. Which of the following pairs is
incorrectly matched?
[2005]
Interval
Function
(a) – 0,0)
x² – 3x² + 3x+3
(b) [2,00)
2×3 – 3×2 – 12x+6
3x² – 2x+1
12
366Answer the following question in one
word or one sentence or as per exact
requirement of the question. Find the least value of ( f(x)=a x+frac{b}{x} )
where ( a>0, b>0 ) and ( x>0 )
12
367If a point moves along the curve ( y^{2}=x )
At what point on the curve does the ( y )
coordinate change at the same rate as the ( x ) coordinate.
12
368Prove that the function ( f(x)=log _{e} x ) is
increasing on ( (mathbf{0}, infty) )
12
369Statement-I: The equation ( frac{x^{3}}{4}- ) ( sin pi x+3=2 frac{1}{2} ) has at least one
solution in [2,2]
Because
Statement-II : If ( boldsymbol{f}:[boldsymbol{a}, boldsymbol{b}] rightarrow boldsymbol{R} ) be a
function & let ( c ) be a number such that
( f(a)<c<f(b), ) then there is at least
one number ( n in(a, b) ) such that
( boldsymbol{f}(boldsymbol{n})=boldsymbol{c} )
A. Statement-l is true, Statement-ll is true ; Statement-I is correct explanation for Statement-
is NOT a correct explanation for statement-
c. Statement-I is true, Statement-II is false
D. Statement-I is false, Statement-II is true
12
370The function ( mathbf{f}(mathbf{x})=mathbf{2} log (mathbf{x}-mathbf{3})- )
( x^{2}+6 x+3 ) increases in the interval
A . (3,4)
в. ( (-infty, 2) )
( c cdot(3, infty) )
D. None ot these
12
371Function ( f(x)=log _{10} cos x ) is
function in ( left(0, frac{pi}{2}right) )
A. Decreasing
B. Increasing
c. constant
D. Increasing and decreasing
12
372The side of a square is increased by ( 20 % . ) Find the ( % ) change in its area.
A. ( 44 % ) increase
B. ( 40 % ) increase
c. No change
D. None of these
12
373The curve ( frac{x^{n}}{a^{n}}+frac{y^{n}}{b^{n}}=2 ) touches the line ( frac{x}{a}+frac{y}{b}=2 ) at the point
A ( .(b, a) )
в. ( (a, b) )
c. (1,1)
D ( cdotleft(frac{1}{a}, frac{1}{b}right) )
12
374The greatest value of the function ( f(x)=x e^{-x} ) in ( [0, infty), ) is
A .
B.
( c cdot-c )
D.
12
375The minimum and maximum values of ( mathbf{f}(mathbf{x})=sin (cos mathbf{x})+cos (sin mathbf{x}) forall-frac{boldsymbol{pi}}{mathbf{2}} leq )
( x leq frac{pi}{2} ) are respectively
A . ( cos 1 ) and ( 1+sin 1 )
B. ( sin 1 ) and ( 1+cos 1 )
c. ( cos 1 ) and ( cos left(frac{1}{sqrt{2}}right)+sin left(frac{1}{sqrt{2}}right) )
D. ( frac{1}{sqrt{2}} )
12
376Write the set of values of ( a ) for which
( boldsymbol{f}(boldsymbol{x})=cos boldsymbol{x}+boldsymbol{a}^{2} boldsymbol{x}+boldsymbol{b} ) is strictly
increasing on ( boldsymbol{R} )
12
377The distance, from the origin, of the
normal to the curve, ( boldsymbol{x}=mathbf{2} cos boldsymbol{t}+ )
( mathbf{2} t sin t, y=2 sin t-2 t cos t ) at ( t=frac{pi}{4}, ) is
( A cdot 2 )
B. 4
( c cdot sqrt{2} )
D. ( 2 sqrt{2} )
12
378The radius of the sphere is measured
( operatorname{as}(10 pm 0.02) c m . ) The error in the
measurement of its volume is
A ( .25 .1 c c )
B. 25.21cc
c. ( 2.51 c c )
D. ( 251.2 c c )
12
379The rate of change of the volume of a cone with respect to the radius of its
base is-
( mathbf{A} cdot pi^{2} h )
в. ( frac{4}{3} pi r h )
c. ( frac{4}{3} pi r^{2} h )
D. ( frac{2}{3} pi r h )
12
380The equation of tangent to the curve ( y=e^{-|x|} ) at the point where the curve
cuts the line ( x=1 ) is
A. ( x+y=e )
В. ( e(x+y)=1 )
c. ( y+e x=1 )
D. none of these
12
38111. Let f(x) = sinºx+2 sin? x, <x<. Find the
intervals in which a should lie in order that f(x) has exactly
one minimum and exactly one maximum. (1985 – 5 Marks)
12
382Let ( x ) be a number which exceeds its
square by the greatest quantity. Then ( x )
is equal to
A ( cdot frac{1}{2} )
B. ( frac{1}{4} )
( c cdot frac{3}{4} )
D. none of these
12
383Assertion
The maximum value of
( (sqrt{-3+4 x-x^{2}}+4)^{2}+(x-5)^{2} )
(where ( 1 leq x leq 3 ) ) is 36
Reason
The maximum distance between the
point (5,-4) and the point on the circle
( (x-2)^{2}+y^{2}=1 ) is 6
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
384If ( f(x)=x^{2} e^{2 x}(x>0,) ) then find the
local maximum value of ( boldsymbol{f}(boldsymbol{x}) )
( A cdot frac{2}{e^{2}} )
B. ( frac{-1}{e^{2}} )
( c cdot e^{2} )
D. ( frac{1}{e^{2}} )
12
385If ( boldsymbol{y}=boldsymbol{x}^{n}, ) then the ratio of relative
errors in ( y ) and ( x ) is
A . 1: 1
B . 2: 1
( c cdot 1: n )
D. ( n: 1 )
12
386ff ( y=m log x+n x^{2}+x ) has its
extreme values at ( x=2 ) and ( x=1 )
then ( 2 m+10 n ) is equal to
A . -1
B. -4
c. -2
D.
E. -3
12
387[2011]
d²x
equals :
12
388If there is an error of ( k % ) in measuring the edge of a cube, then the percent error in estimating its volume is
( A cdot k )
B. ( 3 k )
( c cdot frac{k}{3} )
D. none of these
12
389If there is an error of ( 0.01 mathrm{cm} ) in the
diameter of a sphere then percentage error in surface area when the radius ( = )
( 5 c m, ) is
A . ( 0.005 % )
B. ( 0.05 % )
( c cdot 0.1 % )
D. ( 0.2 % )
12
390The minimum value of ( 64 sec theta+ )
( 27 cos e c theta ) when ( theta operatorname{lies} operatorname{in}left(0, frac{pi}{2}right) ) is
A . 125
в. 625
c. 25
D. 1025
12
391On which of the following intervals is the function ( x^{100}+sin x-1 )
decreasing?
A ( cdotleft(0, frac{pi}{2}right) )
в. (0,1)
c. ( left(frac{pi}{2}, piright) )
D. None of the above.
12
392Find the local maxima and local
minima, if any, of the following functions. Find the sum of the local
maximum and the local minimum
values for:
( g(x)=x^{3}-3 x )
12
393The radius of a sphere is changing at the rate of ( 0.1 mathrm{cm} / ) sec. The rate of its
surface area when the radius is ( 200 mathrm{cm} )
is
A ( cdot 8 pi c m^{2} / ) sec
В . ( 12 pi c m^{2} / ) sec
c. ( 160 pi c m^{2} / ) sec
D. ( 200 pi c m^{2} / ) sec
12
394The circumference of a circle is
measured as ( 28 c m ) with an error of
( 0.01 mathrm{cm} . ) The percentage error in the
area is
A ( cdot frac{1}{14} )
в. 0.01
( c cdot frac{1}{7} )
D. none of these
12
395A particle moves in a line with velocity given by ( frac{d s}{d t}=s+1 . ) The time taken by the particle to cover a distance of 9 meter is
( mathbf{A} cdot mathbf{1} )
B. ( log 10 )
( c cdot 2 log , 10 )
D. 10
12
396The greatest value of ( f(x)=(x+1)^{frac{1}{3}}- )
( (x-1)^{frac{1}{3}} ) on [0,1] is
( mathbf{A} cdot mathbf{1} )
B. 2
( c cdot 3 )
D.
12
397A point on the parabola ( y^{2}=18 x ) at
which the ordinate increase at twice the
rate of the abscissa is
( mathbf{A} cdot(9 / 8,9 / 2) )
в. (2,-4)
c. ( (-9 / 8,9 / 2) )
D. (2,4)
12
398The rate of increase of length of the
shadow of a man 2 metres height, due
to a lamp at 10 metres height, when he is moving away from it at the rate of ( 2 m / s e c, ) is
A ( cdot frac{1}{2} mathrm{m} / mathrm{sec} )
B. ( frac{2}{5} ) m/sec
c. ( frac{1}{3} mathrm{m} / mathrm{sec} )
D. ( 5 m / s e c )
12
399The number of stationary points of ( mathbf{f}(mathbf{x})=cos mathbf{x} ) in ( [mathbf{0}, mathbf{2} boldsymbol{pi}] ) are
( mathbf{A} cdot mathbf{1} )
B. 2
( c .3 )
D.
12
400The sum of intercepts of the tangent to the curve ( sqrt{x}+sqrt{y}=sqrt{a} ) upon the coordinates axes is
( mathbf{A} cdot 2 a )
в.
c. ( 2 sqrt{2} a )
D. None of these
12
40120.
The length of a longest interval in which the function
3 sin x – 4 sinºx is increasing, is
(2002)
12
402The maximum value of ( sin left(x+frac{pi}{5}right)+cos left(x+frac{pi}{5}right), ) where
( boldsymbol{x} inleft(0, frac{pi}{2}right) ) is attained at
( ^{A} cdot frac{pi}{20} )
в. ( frac{pi}{15} )
c. ( frac{pi}{10} )
D.
12
403( frac{1}{2}-frac{1}{2.2^{2}}+frac{1}{3.2^{3}}+ldots ldots= )
A ( cdot )
[
begin{array}{l}text { B } cdot frac{1}{2} log _{2}left(frac{3}{2}right) \ text { c. } log _{e}left(frac{2}{3}right) \ text { D } cdot frac{1}{2} log _{e}left(frac{2}{3}right)end{array}
]
12
404Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a differentiable
function for all values of ( x ) and has the
peoperty that ( f(x) ) and ( f^{prime}(x) ) have opposite signs for all values of ( x ). Then
A. ( f(x) ) is an increasing function
B. ( f(x) ) is a decreasing function
C ( cdot f^{2}(x) ) is a decreasing function
D. ( |f(x)| ) is an increasing function
12
405Assertion
Equation of tangent to the curve ( boldsymbol{y}= )
( x^{2}+1 ) at the point where slope of
tangent is equal the function value of
the curve is ( y=2 x )
Reason
( boldsymbol{f}^{prime}(boldsymbol{x})=boldsymbol{f}(boldsymbol{x}) )
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
406A circular disc of radius ( 3 mathrm{cm} ) is being
heated. Due to expansion, its radius increases at the rate of ( 0.05 mathrm{cm} / mathrm{s} ). Find
the rate at which its area is increasing when radius is ( 3.2 mathrm{cms} )
12
407Prove that the function ( f(x)=log _{e} x ) is
increasing on ( (mathbf{0}, infty) )
12
408Water poured into an inverted conical vessel of which the radius of the base is
( 2 mathrm{m} ) and height ( 4 mathrm{m}, ) and the rate of 77 litres/minute. The rate at which the
water level is rising at the instant when the depth is ( 70 mathrm{cm} ) is: (use ( pi=22 / 7 ) )
A. ( 10 mathrm{cm} / mathrm{min} )
в. ( 20 mathrm{cm} / mathrm{min} )
c. ( 40 mathrm{cm} / mathrm{min} )
D. None
12
409( cos theta+frac{1}{3} cos ^{3} theta+frac{1}{5} cos ^{5} theta+ldots= )
( mathbf{A} cdot log (tan theta) )
B. ( log (cot theta) )
( ^{mathbf{c}} cdot log left(tan frac{theta}{2}right) )
D. ( log left(cot frac{theta}{2}right) )
12
41016.
(20
For all x € (0,1)
(a) exx
(b) log (1 + x)x
12
411If ( f(x)=x^{3 / 2}(3 x-10), x geq-0 ) ), then
( f(x) ) is decreasing in
B. ( (2, infty) )
c. ( (-infty,-1] cup[1, infty) )
12
412The point for the curve ( y=x e^{x} ) is
A. ( x=-1 ) is minimum
B. ( x=0 ) is minimum
c. ( x=-1 ) is maximum
D. ( x=0 ) is maximum
12
413Find the minimum value of a, such that
function ( f(x)=x^{2}+a x+5, ) is
increasing in interval [1,2]
12
414Mark the correct alternative of the
following. The sum of two non-zero numbers is 8
the minimum value of the sum of their
reciprocals is?
A ( cdot frac{1}{4} )
B. ( frac{1}{2} )
( c cdot frac{1}{8} )
D. None of these
12
415( Delta A B C ) is not right-angled and is inscribed in a fixed circle. If ( a, A, b, B ) be slightly varied keeping ( c, C ) fixed, then ( frac{d a}{cos A}+frac{d b}{cos B}=? )
A ( .2 R )
в. ( pi )
( c cdot 0 )
D. none of these
12
416The function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{a} boldsymbol{x}+boldsymbol{b} ) is strictly
increasing for all real ( boldsymbol{x} ), if
( mathbf{A} cdot a>0 )
B. ( a<0 )
( mathbf{c} cdot a=0 )
D. ( a leq 0 )
12
417The point of intersection of the tangents drawn to the curve ( x^{2} y=1-y ) at the
point where it is intersected by the
curve ( x y=1-y, ) is given by
A ( .(0,1) )
в. (1,1)
c. (0,-1)
D. none of these
12
418Find the maximum value of ( boldsymbol{y}= )
( 2 sin ^{2} x-3 sin x+1 forall x in R )
12
41936. Twenty metres of wire is available for fencing off a flower-
bed in the form of a circular sector. Then the maximum area
(in sq. m) of the flower-bed, is:
(JEEM 2017]
(a 30
(b) 12.5
(c) 10
(d) 25
12
4206.
The function defined by f(x) = (x + 2) exis (1994)
(a) decreasing for all x
(b) decreasing in (-0, -1) and increasing in (-1,..)
(c) increasing for all x
(d) decreasing in (-1,0) and increasing in (-00,-1)
12
421The value of ( b ) for which the function
( f(x)=sin x-b x+c ) is a strictly
decreasing function ( forall x in R )
A. ( b in(-1,1) )
в. ( b in(-infty, 1) )
C. ( b in(1, infty) )
D. ( b in[1, infty) )
12
42217.
Iff(x)= xetll-*), then f (x) is
(20015)
(a) increasing on [-1/2, 1] (b) decreasing on R
(c) increasing on R (d) decreasing on [-1/2,1]
12
423The rate of growth of population of a city at any time is proportional to the size of the population at that time. For a certain city, the consumer of proportionality is ( 0.04 . ) The population of the city after 25 years, if the initia population is 10,000 is ( (e=2.7182) )
( mathbf{A} cdot 27182 )
B. 27164
c. 27000
D. 27272
12
424Mark the correct alternative of the
following. ( f(x)=1+2 sin x+3 cos ^{2} x, 0 leq x leq )
( frac{2 pi}{3} ) is?
A. Minimum at ( x=pi / 2 )
B . Maximum at ( x=sin ^{-1}(1 / sqrt{3}) )
c. Minimum at ( x=pi / 6 )
D. Maximum at ( sin ^{-1}(1 / 6) )
12
42517. Let f: R → (0,0) and g: R → R be twice differentiable
functions such that f” and g” are continuous functions on
R. Suppose f'(2) = g(2)= 0, f'(2) #0 and g'(2) + 0. If
lim-81_) =1 then
. f(x)g(x)
x+2 f (x)g'(x)
(a) f has a local minimum at x=2
(b) f has a local maximum at x=2
(c) f”(2)>f(2)
(d) f(x)-f”(x)=0 for at least one x e R
12
426The equation of the normal to the curve
( boldsymbol{y}=(1+boldsymbol{x})^{y}+sin ^{-1}left(sin ^{2} boldsymbol{x}right) boldsymbol{a t} quad boldsymbol{x}=mathbf{0} )
is
A. ( x+y=1 )
B. x-y+1=0
c. ( 2 x+y=2 )
D. 2x-y+
12
427The approximate change in the volume of a cube of side ( x ) metres caused by
increasing the side by ( 3 % ) is :
B. ( 0.6 x^{3} m^{3} )
D. ( 0.9 x^{3} m^{3} )
12
428Find the slope of the tangent at (1,2) on the curve ( y=x^{2}-4 x+5 )12
4290
120,
0
The maximum value of sin x(1 + cos x) will be at the
(b) x = *
(c) x = 7
(d) x = 1
12
430If ( boldsymbol{P}=boldsymbol{x}^{3}-frac{mathbf{1}}{boldsymbol{x}^{3}} ) and ( boldsymbol{Q}=boldsymbol{x}-frac{mathbf{1}}{boldsymbol{x}}, boldsymbol{x} in )
( (1, infty) ) then minimum value of ( frac{P}{Q^{2}} ) is:
( A cdot ) is ( 2 sqrt{3} )
B. is ( 4 sqrt{3} )
c. does not exist
D. None of these
12
431Which of the following functions is always increasing?
( mathbf{A} cdot x+sin 2 x )
B. ( x-sin 2 x )
c. ( 2 x+sin 3 x )
( mathbf{D} cdot 2 x-sin x )
12
432The radius and height of a cylinder are equal. If the radius of the sphere is equal to the height of the cylinder, then the ratio of the rates of increase of the
volume of the sphere and the volume of the cylinder is
A .4: 3
в. 3: 4
( mathbf{c} cdot 4: 3 pi )
( mathbf{D} cdot 3: 4 pi )
12
43314. If y = sec(tan-x), thena r-
atx=1 is equal to :
(JEE M200
12
434Time period ( T ) of a simple pendulum of length ( l ) is given by ( T=2 pi sqrt{frac{l}{g}} . ) If the length is increased by ( 2 % ), then an approximate change in the time period
is
A . ( 2 % )
B . ( 1 % )
c. ( frac{1}{2} % )
D. None of these
12
435If the line ( a x+b y+c=0 ) is a norma
to the curve ( boldsymbol{x} boldsymbol{y}=mathbf{1}, ) then
This question has multiple correct options
A ( . a>0, b>0 )
В. ( a>0, b<0 )
c. ( a0 )
D. ( a<0, b<0 )
12
436A man ( 160 mathrm{cm} ) tall, walks away from a source of light situated at the top of a
pole ( 6 mathrm{m} ) high, at the rate of ( 1.1 mathrm{m} / mathrm{sec} )
How fast is the length of his shadow increasing when he is ( 1 mathrm{m} ) away from the pole?
12
4377.
If f(x)=-, , for every real number x, then the minimum
x +1°
value off
(1998 – 2 Marks)
(a) does not exist because f is unbounded
(b) is not attained even though f is bounded
(c) is equal to 1
(d) is equal to -1
12
438The interval a for which the local
minimum value of the function ( boldsymbol{f}(boldsymbol{x})= ) ( 2 x^{3}-21 x^{2}+60 x+a ) is positive. is
в. ( (-infty,-25) )
c. ( (25, infty) )
D. ( (-25, infty) )
12
439If ( a_{1}, a_{2}, a_{3}, dots a_{n} in R ) then ( (x- )
( left.a_{1}right)^{2}+left(x-a_{2}right)^{2}+ldotsleft(x-a_{n}right)^{2} )
assumes least value at ( boldsymbol{x}= )
A ( cdot a_{1}+a_{2}+a_{3}+ldots+a_{n} )
В . ( a_{n} )
( mathbf{c} cdot nleft(a_{1}+a_{2}+ldots+a_{n}right) )
D. ( frac{left(a_{1}+a_{2}+ldots+a_{n}right)}{n} )
12
440The value of ( a_{1}+a_{2} ) is equal to
A . 30
B. -30
c. 27
D. -27
12
44116. Ifp and q are positive real numbers such that p2 + q2 = 1,
then the maximum value of (p+q) is
[2007]
(a)
(d) 2.
12
442If ( f(x)=x^{2}+2 b x+2 c^{2} ) and ( g(x)= )
( -x^{2}-2 c x+b^{2} ) are such that min
( f(x)>max g(x), ) then the relation
between ( b ) and ( c ) is
A. no relation
в. ( 0<c<b / 2 )
C ( cdot|c||b| sqrt{2} )
12
443An object stars from rest at ( t=0 ) and
accelerates at a rate given by ( a=6 t )
What i) its velocity and ii) its
displacement at any time ( t ? )
A ( cdot t^{3}, 3 t^{2} )
B. ( 3 t^{2}, t^{3} )
( mathbf{c} cdot t^{2}, t^{3} )
D. ( 2 t^{2}, 3 t^{3} )
12
444( f(x)=x^{3}-6 x^{2}+12 x-16 ) is strictly
decreasing for
( mathbf{A} cdot x in R )
в. ( x in R-{1} )
c. ( x in R^{+} )
D. ( x in phi )
12
445f ( a, b, c ) are real number, then find the
intervals in which ( boldsymbol{f}(boldsymbol{x})= )
[
begin{array}{ccc}
boldsymbol{x}+boldsymbol{a}^{2} & boldsymbol{a} boldsymbol{b} & boldsymbol{a c} \
boldsymbol{a b} & boldsymbol{x}+boldsymbol{b}^{2} & boldsymbol{b c} \
boldsymbol{a c} & boldsymbol{b c} & boldsymbol{x}+boldsymbol{c}^{2}
end{array} mid
]
increasing or decreasing
12
446Let h(x)=f(x)-([(x))2 + (F(x))for every real number x. Then
(1998 – 2 Marks
(a) h is increasing whenever fis increasing
(b) h is increasing whenever fis decreasing
(c) h is decreasing whenever fis decreasing
(d) nothing can be said in general.
12
447The approximate value of ( sqrt[5]{33} ) correct
to 4 decimal places is
A . 2.0000
B. 2.1001
c. 2.0125
D. 2.0500
12
448Find the rate of change of the area of a
circular disc with respect to its
circumference, when the radius is 3
( mathrm{cm} .left(text { incm }^{2}right) )
12
449Maximum value of ( sin theta+cos theta ) in
( left[0, frac{pi}{2}right] ) is
A ( cdot sqrt{2} )
B. 2
c. 0
D. ( -sqrt{2} )
12
450Let tangent at a point ( mathrm{P} ) on the curve ( x^{2 m} y^{frac{n}{2}}=a^{frac{4 m+n}{2}} ) meets the x-axis
and y-axis at A and B respectively if AP:
( mathrm{PB} ) is ( frac{n}{lambda m} ) where ( mathrm{P} ) lies between ( mathrm{A} ) and ( mathrm{B} )
then find the value of ( lambda )
12
451How fast is the farther end of the
A. ( 2 k m / h r )
в. ( 4 k m / h r )
( mathrm{c} .6 mathrm{km} / mathrm{hr} )
D. ( 8 k m / h r )
12
452Suppose that ( f(0)=-3 ) and ( f^{prime}(x) leq 5 )
for all values of ( x ). Then the largest value
which ( f(2) ) can attain is
A. 7
B. -7
c. 13
D.
12
453If ( a>b ) maximum value of ( a sin ^{2} x+ )
( b cos ^{2} x )
A . a
B.
( c cdot a+b )
D. ( sqrt{a^{2}+b^{2}} )
12
454The greatest value of the function
( f(x)=sin ^{2} x-20 cos x+1 ) is
A . 20
B. 1
c. 21
D.
12
455The maximum value of ( left(frac{1}{x}right)^{x} ) is
A ( cdot(1 / e)^{e} )
B . ( e^{1 / c} )
( c )
D. none of these
12
456Function ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}^{2}-mathbf{2}}{sqrt{mathbf{1}+boldsymbol{x}^{2}}} )
A. is always increasing
B. is always decreasing
C. has exactly one point of minima
D. has exactly one point of maxima
12
457The absolute minimum & maximum
values of ( f(x)=frac{x^{2}-3 x+4}{x^{2}+3 x+4} ) are
respectively
A ( cdot frac{1}{7} ) and 7
B. ( 5 & 7 )
c. ( frac{1}{7} & 1 )
D. None of these
12
458The radius of a circle is increasing
uniformly at the rate of ( 3 mathrm{cm} / mathrm{s} ). Find the rate at which the area of the circle is
increasing when the radius is ( 10 mathrm{cm} )
12
4599.
The function f(x)= t (e – 1) (t – 1) (t 2)} (t – 3) dt has
-1
a local minimum atx=
(1999 – 3 Marks)
(a) o (6) 1 (c) 2
(d)3
1
.1
.21.
10.
191 –
1 Alan
12
460The area of a triangle is computed using the formula ( S=frac{1}{2} b c sin A . ) If the relative errors made in measuring b, ( c ) and calculating ( mathrm{S} ) are respectively 0.02
0.01 and 0.13 the approximate error in ( A ) when ( A=pi / 6 ) is
c. 0.05 degree
D. 0.01 degree
12
461The local maximum value of ( x(1-x)^{2}, 0 leq x leq 2 ) is
( A cdot 2 )
в. ( frac{4}{27} )
( c .5 )
D. ( 2, frac{4}{27} )
12
462Let a function ( boldsymbol{f}:[mathbf{0}, mathbf{5}] rightarrow boldsymbol{R} ) be
continuous, ( boldsymbol{f}(1)=boldsymbol{3} ) and ( boldsymbol{F} ) be defined
as:
( boldsymbol{F}(boldsymbol{x})=int_{1}^{x} boldsymbol{t}^{2} boldsymbol{g}(boldsymbol{t}) boldsymbol{d} t, ) where ( boldsymbol{g}(boldsymbol{t})= )
( int_{1}^{t} boldsymbol{f}(boldsymbol{u}) boldsymbol{d} boldsymbol{u} )
Then for the function ( F ) the point ( x=1 )
is
A. a point of local minima
B. a point of local maxima
c. not a critical point
D. a point of inflection
12
463A particle moves along the curve ( y= )
( x^{3 / 2} ) in the first quadrant in such a way
that its distance from the origin increases at the rate of 11 units per
second. The value of when ( x=3 ) is
A . 4
B. ( frac{9}{2} )
c. ( frac{3 sqrt{3}}{2} )
D. none of these
12
464( operatorname{Let} g(x)=2 fleft(frac{x}{2}right)+f(x-2) ) and
( boldsymbol{f}^{prime prime}(boldsymbol{x})<mathbf{0} forall boldsymbol{x} in(mathbf{0}, mathbf{2}), ) then ( boldsymbol{g}(boldsymbol{x}) )
increases in
A ( cdotleft(frac{1}{2}, 2right) )
в. ( left(frac{4}{3}, 2right) )
c. (0,2)
(年. ( 0,2,2) )
D. ( left(0, frac{4}{3}right) )
12
465A stone is dropped into a quiet lake and
waves move in a circle at a speed of
( 3.5 c m / )sec. At the instant when the
radius of the circular wave is ( 7.5 mathrm{cm} )
The enclosed area increases as fastly
as.
A ( cdot 52.5 pi mathrm{cm}^{2} / mathrm{sec} )
в. ( 50.5 pi mathrm{cm}^{2} / mathrm{sec} )
( mathbf{c} cdot 57.5 pi mathrm{cm}^{2} / mathrm{sec} )
D. ( 62.5 pi mathrm{cm}^{2} / mathrm{sec} )
12
466In the following increasing function is
A ( cdot e^{x^{2}}^{2} )
B cdot ( e^{x^{3}} )
( c cdot e^{0} )
D. all the above
12
467The volume of metal in a hollow sphere
is constant.If the inner radius is
increasing at the rate of ( 1 mathrm{cm} / mathrm{sec} ), then the rate of increase of the outer radius
when the radii are ( 4 c m ) and ( 8 c m )
respectively is.
A. ( 0.75 mathrm{cm} / mathrm{sec} )
B. ( 0.25 mathrm{cm} / mathrm{sec} )
( mathrm{c} cdot 1 mathrm{cm} / mathrm{sec} )
D. ( 0.50 mathrm{cm} / mathrm{sec} )
12
468The function ( f(x)=sqrt{3} cos x+sin x )
has an amplitude of
A . 1.37
в. 1.73
( c cdot 2 )
D. 2.73
E . 3.46
12
469Using differentials, find the approximate value of ( left(frac{17}{81}right)^{frac{1}{4}} )12
470The greatest value of ( boldsymbol{f}(boldsymbol{x})=(boldsymbol{x}+ )
1) ( ^{1 / 3}-(x-1)^{1 / 3} ) on [0,1] is
( mathbf{A} cdot mathbf{1} )
B.
( c cdot 3 )
D. ( 1 / 3 )
12
471Let ( boldsymbol{f}(boldsymbol{x})=(boldsymbol{x}-boldsymbol{p})^{2}+(boldsymbol{x}-boldsymbol{q})^{2}+(boldsymbol{x}- )
( boldsymbol{r})^{2} . ) Then ( boldsymbol{f}(boldsymbol{x}) ) has a minimum at ( boldsymbol{x}=boldsymbol{lambda} )
where ( lambda ) is equal to
A. ( frac{p+q+r}{3} )
B. ( sqrt[3]{p q r} )
c. ( frac{3}{frac{1}{p}+frac{1}{q}+frac{1}{r}} )
D. none of these
12
472Illustration 2.37 The particle’s position as a function of
time is given as x = 5t2 – 9t + 3. Find out the maximum value
of position co-ordinate? Also, plot the graph.
12
473Normal to the curve ( x^{2}=4 y ) which
passes through the point ( (mathbf{1}, mathbf{2}) )
A. ( x+y=3 )
в. ( x-y=3 )
c. ( 2 x+y=4 )
D. ( x+2 y=5 )
12
474Find all points of local maxima and local minima of the function f given by
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-boldsymbol{3} boldsymbol{x}+boldsymbol{3} )
12
475A tangent to the hyperbola ( boldsymbol{y}=frac{boldsymbol{x}+mathbf{9}}{boldsymbol{x}+mathbf{5}} )
passing though the origin is
A . ( x+25 y=0 )
в. ( 5 x+y=0 )
c. ( 5 x-y=0 )
D. ( x-25 y=0 )
12
476The maximum value of ( f(x)=100- ) ( |45-x| ) is
A . 100
в. 145
c. 55
D. 45
12
477Tangent to parabola ( boldsymbol{y}^{2}=mathbf{4} boldsymbol{x}+mathbf{5} ) which
is parallel to ( boldsymbol{y}=2 boldsymbol{x}+mathbf{7} )
A. ( y-2 x-3=0 )
В. ( y=x+3 )
c. ( y-2 x+1=0 )
D. ( y=x+1 )
12
478A stone is dropped into a quiet lake and
waves move in circles at the speed of 5 ( mathrm{cm} / mathrm{s} ) At the instant when the radius of
the circular wave is ( 8 mathrm{cm} ) how fast is the
enclosed area increasing?
( mathbf{A} cdot 84 pi mathrm{cm}^{2} / mathrm{s} )
B. ( 80 pi mathrm{cm}^{2} / mathrm{s} )
( mathbf{c} cdot 90 pi mathrm{cm}^{2} / mathrm{s} )
D. ( 96 pi mathrm{cm}^{2} / mathrm{s} )
12
479Write the equation of tangent at (1,1) on the curve ( 2 x^{2}+3 y^{2}=5 )12
480Consider ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+boldsymbol{c} )
Parameters ( a, b, c ) are chosen,
respectively, by throwing a die three times. Then the probability that ( f(x) ) is
an increasing function is
A ( .5 / 12 )
в. ( 2 / 9 )
c. ( 4 / 9 )
D. ( 1 / 3 )
12
481If the points of local extremum of
( f(x)=x^{3}-3 a x^{2}+3left(a^{2}-1right) x+1 )
lies between ( -2 & 4, ) then ( ^{prime} a^{prime} ) belongs to
A. (-2,2)
B ( cdot(-infty,-1) cup(3, infty) )
c. (-1,3)
( D cdot(3, infty) )
12
482If a point is moving in a line so that its velocity at time ( t ) is proportional to the square of the distance covered, then its
acceleration at time t varies as
A. cube of the distance
B. the distance
c. square of the distance
D. none of these
12
483Find the equation of the tangent and the normal to the following curves at the indicated points.
( boldsymbol{y}=boldsymbol{x}^{4}-boldsymbol{6} boldsymbol{x}^{3}+1 mathbf{3} boldsymbol{x}^{2}-mathbf{1 0} boldsymbol{x}+mathbf{5} ) at ( boldsymbol{x}= )
( mathbf{1} )
12
484Find the equation of the tangent line to
the curve ( y=x^{2}-2 x+7 ) which
is perpendicular to the line ( 5 y-15 x= )
( mathbf{1 3} )
12
485A spherical raindrop evaporates at a rate proportional to its surface area. If originally its radius be ( 3 m m ) and
1 hour later it reduces to 2 mm, find an
expression for the radius of the rain
drop at any time ( t )
12
486Which of the following function has extreme point?
( mathbf{A} cdot 2^{x} )
B . ( log _{10} x )
c. ( x-[x] )
D. all of these
12
487I et P(x)= a, + a,x2+aart + …… +a x2n be a polynomial in
a real variable x with
Oxao < a <a2 <….. < an.. The function P(x) has
(a) neither a maximum nor a minimum (1986-2 Marks)
(b) only one maximum
(c) only one minimum
(d) only one maximum and only one minimum
(e) none of these.
12
488If ( f(x)=frac{x}{1+x tan x}, x inleft(0, frac{pi}{2}right) ) then
This question has multiple correct options
A. ( f(x) ) has exactly one point of minima
B. ( f(x) ) has exactly one point of maxima
C ( f(x) ) is many one in ( left(0, frac{pi}{2}right) )
D. ( f(x) ) has maximum at ( x_{0} ) where ( x_{0}=cos x_{0} )
12
489A ladder ( 10 mathrm{m} ) long rests against a vertical wall with the lower end on the
horizontal ground. The lower end of the ladder is pulled along the ground away
from the wall at the rate of ( 3 c m / s ) The
height of the upper end while it is descending at the rate of ( 4 mathrm{cm} / mathrm{s} ), is
A. ( 4 sqrt{3} m )
В. ( 5 sqrt{3} m )
c. ( 5 sqrt{2} m )
D. ( 6 m )
12
490Find Stationary points of ( f(x)=sin x )
where ( 0<x<2 pi )
12
491Two pipes running together can fill a cistern in ( 3 frac{1}{13} ) minutes. If one pipe take
3 minutes more than other to fill it,
find the time in which each pipe can fill
the tank.
12
49228.
A line is drawn through the point (1,2) to meet the coordinate
axes at Pand Q such that it forms a triangle OPQ, where O is
the origin. If the area of the triangle OPQ is least, then the
slope of the line PQ is :
[2012]
(a)
·
(b) – 4
(C) -2
(d)
12
493Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{a} boldsymbol{x}^{2}-boldsymbol{b}|boldsymbol{x}|, ) where ( boldsymbol{a} ) and ( boldsymbol{b} ) are
constants. Then at ( x=0, f(x) ) has
This question has multiple correct options
A. A maxima whenever ( a>0, b>0 )
B. A maxima whenever ( a>0, b0, b0, b<0 )
12
49417. A window of perimeter P (including the base of the arch) is
in the form of a rectangle surmounded by a semi circle. The
semi-circular portion is fitted with coloured glass while the
rectangular part is fitted with clear glass transmits three
times as much light per square meter as the coloured glass
does.
What is the ratio for the sides of the rectangle so that the
window transmits the maximum light? (1991 – 4 Marks)
12
495Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} sqrt{mathbf{4} boldsymbol{a} boldsymbol{x}-boldsymbol{x}^{2}},(boldsymbol{a}>mathbf{0}) . . ) Then
( f(x) ) is decreasing in:
This question has multiple correct options
A. ( (5 a, infty) )
)
в. ( (-infty, 0) U(4 a, infty) )
c. Always increasing
D. None of the above
12
496The radius of a circular plate is
increased at ( 0.01 mathrm{cm} / ) sec. If the area is
increased at the rate of ( frac{pi}{10} . ) Then its radius is
( A cdot 5 mathrm{cm} )
B. ( 10 mathrm{cm} )
c. ( 15 mathrm{cm} )
D. ( 20 mathrm{cm} )
12
49714. The perimeter of a sector is p. The area of the sector is
(b) =
12
498π
π
Let the function g:(-0,00) → be given by
g(u) = 2 tan-(e”) . Then, g is
(2008)
(a) even and is strictly increasing in (0,0)
n odd and is strictly decreasing in (-0,0)
odd and is strictly increasing in (-00,00)
(d) neither even nor odd, but is strictly increasing in
(-00,00)
12
499If ( mathbf{f}(mathbf{x}) ) is minimum at ( mathbf{x}=mathbf{a} ) then
A. There exists ( delta>0 ) such that ( a-delta<x<a Rightarrow f(x)0 ) such that ( a<xf(a) )
C. There exists ( delta>0 ) such that ( a-delta<x
0 ) such that ( a-delta<x<a+delta Rightarrow )
( f(x) leq f(a) )
12
500If ( 2 x-7-5 x^{2} ) has maximum value at
( x=a, ) then ( a=dots dots )
A. ( -1 / 5 )
в. ( 1 / 5 )
( c cdot 34 / 5 )
D. ( -34 / 5 )
12
50123.
Tangent is drawn to ellipse
* + y2 = 1 at (3-13 cos 0, sin e) (where 0 € (0,7/2)).
27
Then the value of such that sum of intercepts on axes
made by this tangent is minimum, is
(2003)
(a) T3 (6) Tu6 (C) T8 (d) T4
Teco 32
12
502Let ( boldsymbol{f}(boldsymbol{x})=(boldsymbol{x}-mathbf{1})^{boldsymbol{m}}(boldsymbol{2}-boldsymbol{x})^{n} ; boldsymbol{m}, boldsymbol{n} boldsymbol{epsilon} boldsymbol{N} )
and ( boldsymbol{m}, boldsymbol{n}>2 )
12
503The equation of the common normal at the point of contact of the curves ( x^{2}=y )
and ( x^{2}+y^{2}-8 y=0 )
A ( . x=y )
B. ( x=0 )
( mathbf{c} cdot y=0 )
D. ( x+y=0 )
12
504Using differentials, find the approximate value of ( (3.968)^{frac{3}{2}} )12
505Mark the correct alternative of the
following.
The least and greatest values of ( f(x)= ) ( x^{3}-6 x^{2}+9 x ) in ( [0,6], ) are?
A .3,4
в. 0,6
c. 0,3
D. 3,6
12
506The function ( boldsymbol{f}(boldsymbol{x})=frac{mathbf{1}}{mathbf{1}+boldsymbol{x}^{2}} ) is
decreasing in the interval
( mathbf{A} cdot(-infty,-1] )
в. ( (-infty, 0] )
c. ( [1, infty) )
(i)
D. ( (0, infty) )
12
507Observe the following lists Let ( f(x) ) be
any function
List – I
A) ( f^{prime}(mathrm{a})=0 ) and
I) ( f(x) ) is increasing at ( x=a )
[
f^{prime prime}(mathrm{a})0 text { then } quad text { at } x=a
]
3) ( f(x) ) has
C) ( f^{prime}(a) neq 0 ) then neither maximum nor minimum
D) ( f^{prime}(a)>0 )
4) ( f(x) ) has minimum value at ( x=a )
5) ( f(x) ) is decreasing at ( x=a )
( A cdot A-4, B-2, C-3, D-5 )
B. A -2,B -4, C -3, D -1
C. ( A-2, B-4, C-3, D-5 )
D. A -2,B -4, C -5, D-1
12
508If two curves ( boldsymbol{y}=boldsymbol{a}^{x} ) and ( boldsymbol{y}=boldsymbol{b}^{x} )
intersect at an angle ( alpha ) then find the
value of tana
A ( cdotleft|frac{ln a-ln b}{1+ln a ln b}right| )
в. ( left|frac{ln a-ln b}{1-ln a ln b}right| )
c. ( mid frac{ln a+ln b}{1+ln a ln b} )
D. ( mid frac{ln a+ln b}{1-ln a ln b} )
12
509The function ( boldsymbol{f}(boldsymbol{x})=log _{e}left[boldsymbol{x}^{3}+sqrt{boldsymbol{x}^{6}+mathbf{1}}right] )
is an
This question has multiple correct options
A. even function
B. odd function
c. increasing function
D. decreasing function
12
510If the area of circle increases at a
uniform rate, then prove that the perimeter varies inversely as the radius.
12
511The maximum value of the function
( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-boldsymbol{4}| ) exists at
( mathbf{A} cdot x=0 )
B. ( x=2 )
c. ( x=4 )
D. ( x=-4 )
12
512Find the stationary point of ( boldsymbol{y}=boldsymbol{x}^{2}+ )
( 5 x-6 )
12
513Investigate the behaviour of the
function ( boldsymbol{y}=left(boldsymbol{x}^{3}+boldsymbol{4}right)(boldsymbol{x}+mathbf{1})^{3} ) and
construct its graph. How many
solutions does the equation ( left(x^{3}+right. )
4)( (x+1)^{3}=c ) possess?
12
514Find the points at which the function ( boldsymbol{f} ) ( operatorname{given} operatorname{by} f(x)=(x-2)^{4}(x+1)^{3} )
has local maxima
12
515The function ( f(x)=sin x-k x-c )
where ( k ) and ( c ) are constants, decreases
always when
( mathbf{A} cdot k>1 )
в. ( k geq 1 )
c. ( k<1 )
D. ( k leq 1 )
E . ( k<-1 )
12
516Assertion
If ( f(x)=(x-2)^{3} ) then ( f(x) ) has neither
maximum nor minimum at ( boldsymbol{x}=mathbf{2} )
Reason
( boldsymbol{f}^{prime}(boldsymbol{x})=mathbf{0}=boldsymbol{f}^{prime prime}(boldsymbol{x}) ) when ( boldsymbol{x}=mathbf{2} )
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
517Illustration 3.24 Find the maximum value of 13 sin x +
cos x and x for which a maximum value occurs.
12
518If metallic circular plate of radius ( 50 mathrm{cm} )
is heated so that its radius increases at
the rate of ( 1 mathrm{mm} ) per hour, then the rate
at which the area of the plate increases ( left(operatorname{in} c m^{2} / h rright) ) is
A . ( 5 pi )
в. ( 10 pi )
c. ( 100 pi )
D. ( 50 pi )
12
519The two curves ( x=y^{2}, x y=a^{3} ) cut
orthogonally at a point. Then ( a^{2} ) is equal
to
A ( cdot frac{1}{3} )
B. 3
( c cdot 2 )
D.
12
520If an edge of a cube measure ( 2 mathrm{m} ) with a
possible error of ( 0.5 mathrm{cm} . ) Find the corresponding error in the calculated volume of the cube.
( mathbf{A} cdot 0.6 m^{3} )
В. ( 0.06 m^{3} )
c. ( 0.006 m^{3} )
D. ( 0.0006 mathrm{m}^{3} )
12
521Find the least value of ( k ) for which the
function ( x^{2}+k x+1 ) is an increasing
function in the interval ( 1<x<2 )
A . 1
B. –
( c cdot 2 )
D. –
12
522Let for a function ( boldsymbol{f}(boldsymbol{x}), boldsymbol{h}(boldsymbol{x})= )
( (f(x))^{2}+(f(x))^{3} ) for every real
number ( boldsymbol{x} ). Then

This question has multiple correct options
A. ( h ) is increasing whenever ( f ) is increasing
B. ( h ) is increasing whenever ( f ) is decreasing
C. ( h ) is decreasing whenever ( f ) is decreasing
D. nothing can be said in general

12
523Find the interval of increase and
decrease of the following functions. ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}+ln (1-boldsymbol{4} boldsymbol{x}) )
12
524In a culture, the bacteria count is ( 1,00,000 . ) The number is increased by ( 10 % ) in 2 hours. In how many hours will
the count reach ( 2,00,000, ) if the rate of growth of bacteria is proportional to the
number present?
12
52522. Letf:R
→ R be defined by
f(x) = k-2x, if xs-1
J(x) = 2x+3, if x >-1
Iff has a local minimum at x=-1, then a possible value of
k is
[2010]
(a) O
(b)
(C)
-1 (d) 1
12
526If ( x>0, ) then find greatest value of the
expression ( frac{boldsymbol{x}^{100}}{1+boldsymbol{x}+boldsymbol{x}^{2}+boldsymbol{x}^{3}+ldots .+boldsymbol{x}^{200}} )
12
527Find the derivative of ( f(x)=3 x ) at ( x= )
2
12
528If the radius of a sphere is measured as ( 9 mathrm{cm} ) with an error of ( 0.02 mathrm{cm}, ) then find
the approximate error in calculating its
volume
12
529The length of the longest interval in
which the function ( 3 sin x-4 sin ^{3} x ) is
increasing is
A.
в.
c. ( frac{3 pi}{2} )
D.
12
530Find the values of ( a ) for which the
function ( f(x)=sin x-a x+4 ) is
increasing function on ( boldsymbol{R} )
12
531The absolute maximum of ( boldsymbol{y}=boldsymbol{x}^{3}- )
( 3 x+2 ) in ( 0 leq x leq 2 ) is:
A . 4
B. 6
( c cdot 3 )
D.
12
532On the interval [0,1] the function
( f(x)=x^{1005}(1-x)^{1002} ) assumes
maximum value equal to.
A ( cdot frac{(1005)^{1002}}{(2007)^{2007}} )
в. ( frac{(2007)^{2007}}{(1005)^{12055}(1002)^{1002}} )
c. ( frac{(2007)^{2007}}{(1005)^{1202}(1002)^{1005}} )
D. ( frac{(1005)^{1005}(1002)^{1002}}{(2007)^{2007}} )
12
533Assertion
Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a function such that
( f(x)=x^{3}+x^{2}+3 x+sin x . ) Then, ( f ) is
one-one.
Reason
( f(x) ) is decreasing function
A. Assertion and Reason are correct and the reason is the correct explanation for the assertion
B. Assertion and Reason are correct and the reason is Not the correct explanation for the assertion.
c. Assertion is correct while the Reason is incorrect
D. Assertion is incorrect while the Reason is correct
12
534The greatest value of ( boldsymbol{f}(boldsymbol{x})= )
( (x+1)^{1 / 3}-(x-1)^{1 / 3} ) on [0,1] is
( A )
B. 2
( c cdot 3 )
D. ( 1 / 3 )
12
535The function which has neither
maximum nor minimum at ( x=0=0 ) is
A ( cdot f(x)=x^{2} )
B. ( f(x)=cos x )
( c cdot f(x)=x^{3}-8 )
D. ( f(x)=cos h x )
12
536A particle’s velocity ( v ) at time ( t ) is given by ( v=2 e^{2 t} cos frac{pi t}{3} . ) The least value of ( t ) at
which the acceleration becomes zero is
( mathbf{A} cdot mathbf{0} )
в. ( frac{3}{2} )
( ^{mathrm{c}} cdot frac{3}{pi} tan ^{-1}left(frac{6}{pi}right) )
D. ( frac{3}{pi} cot ^{-1}left(frac{6}{pi}right) )
12
537()
1.JWs
3. If by dropping a stone in a quiet lake a wave moves in
circle at a speed of 3.5 cm/sec, then the rate of increase of
the enclosed circular region when the radius of the circular
22
wave is 10 cm, is
(a) 220 sq. cm/sec
(c) 35 sq. cm/sec
(b) 110 sq. cm/sec
(d) 350 sq. cm/sec
12
538If ( f(x)=frac{80}{3 x^{4}+8 x^{3}-18 x^{2}+60}, ) then
the points of local maxima for the
function ( boldsymbol{f}(boldsymbol{x}) ) are
A . 1,3
в. -3,1
c. -1,3
D. -1,-3
12
539Show that the function ( f ) given by
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-boldsymbol{3} boldsymbol{x}^{2}+boldsymbol{4} boldsymbol{x}, boldsymbol{x} in boldsymbol{R} ) is strictly
increasing.
12
540If 1 degree ( =0.017 ) radians, then the
approximate value of ( sin 46 ) degrees is
A .0 .7194
B. ( frac{0.017}{sqrt{2}} )
c. ( frac{1.017}{sqrt{2}} )
D. none of these
12
54140.
The maximum volume (in cu.m) of the right circular como
having slant height 3 m is: [JEE M 2019-9 Jan (M)
(a) 61
(b) 3131
(d) 213 T
12
542Function ( boldsymbol{f}(boldsymbol{x})=(boldsymbol{x}+mathbf{2}) boldsymbol{e}^{-boldsymbol{x}} ) is
A. decreasing
B. decreasing in ( (-infty,-1) ) and increasing in ( (-1, infty) )
c. increasing
D. decreasing in ( (-1, infty) ) and increasing in ( (-infty,-1) )
12
543A balloon is pumped at the rate of a cm ( ^{3} ) /minute. The rate of increase of its
surface area when the radius is ( b mathrm{cm} ), is
A ( cdot frac{2 a^{2}}{b^{4}} mathrm{cm}^{2} / mathrm{min} )
B. ( frac{a}{2 b} mathrm{cm}^{2} / mathrm{min} )
c. ( frac{2 a}{b} mathrm{cm}^{2} / mathrm{min} )
D. none of these
12
544A stone is dropped into a quiet lake. If
the waves moves in circle at the rate of
( 30 mathrm{cm} / mathrm{sec} ) when the radius is ( 50 mathrm{m} ), the
rate of increase of enclosed area is
A ( .30 pi m^{2} / ) sec
B. ( 30 m^{2} / )sec
c. ( 3 pi m^{2} / )sec
D. none of these
12
545Find the sum of minimum and
maximum values of ( y ) in ( 4 x^{2}+12 x y+ )
( mathbf{1 0 y}^{2}-mathbf{4 y}+mathbf{3}=mathbf{0} )
12
54610. f(x) is cubic polynomial with f(2)= 18 and f(1) = -1. Also
f(x) has local maxima at x=-1 and f'(x) has local minima at
x=0, then
(2006 – 5M, -1)
(a) the distance between (-1,2) and (a f(a)), where x=a is
the point of local minima is 275
(b) f(x) is increasing for x = [1,2 V5]
(c) f(x) has local minima at x=1
(d) the value of f(0) = 15
12
547At what time ( t ) will the volume of the
sphere be 27 times its volume at ( t=0 )
12
548The position vector of a particle at time ( t^{prime} ) is given by ( vec{r}=t^{2} hat{t}+t^{3} hat{j} ). The velocity
makes an angle ( theta ) with positive ( x ) -axis. Find the of ( frac{boldsymbol{d} boldsymbol{theta}}{boldsymbol{d} boldsymbol{t}} ) at ( boldsymbol{t}=frac{sqrt{mathbf{2}}}{mathbf{3}} )
12
549For what value of ‘a’ the function
( f(x)=x+cos x-a ) increases
( A cdot 0 )
B.
( c .-1 )
D. Any value
12
550Let ( boldsymbol{f}(boldsymbol{x})=(boldsymbol{x}-mathbf{1})^{4} cdot(boldsymbol{x}-boldsymbol{2})^{n}, boldsymbol{n} boldsymbol{epsilon} boldsymbol{N} )
Then ( f(x) ) has
This question has multiple correct options
A. a maximum at ( x=1 ) if ( n ) is odd
B. a maximum at ( x=1 ) if ( n ) is even
c. a minimum at ( x=2 ) if ( n ) is even
D. a maximum at ( x=2 ) if ( n ) is odd
12
551The rate of change of surface area of a
increasing at the rate of ( 2 mathrm{cm} / ) sec is
proportional to
A ( cdot frac{1}{r^{2}} )
B. ( frac{1}{r} )
( c cdot r^{2} )
D.
12
552The total number of local maxima and
local minima of the function ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{cc}(2+x)^{3}, & -3<x leq-1 \ x^{2 / 3}, & -1<x<2end{array}right. )
( A cdot O )
B.
( c cdot 2 )
( D )
12
5532. A ladder 5 m in length is resting against vertical wall. The
bottom of the ladder is pulled along the ground away from
the wall at the rate of 1.5 m/sec. The length of the highest
point of the ladder when the foot of the ladder 4.0 m away
from the wall decreases at the rate of
(a) 2 m/sec
(b) 3 m/sec
(c) 2.5 m/sec
d) 1.5 m/sec
12
554If ( A, B, C ) are angles of a triangle, then the minimum value of ( tan ^{2} frac{A}{2}+ ) ( tan ^{2} frac{B}{2}+tan ^{2} frac{C}{2} ) is12
555Consider the function ( boldsymbol{f}:(-infty, infty) rightarrow )
( (-infty, infty) ) defined by ( boldsymbol{f}(boldsymbol{x})= )
( frac{x^{2}-a x+1}{x^{2}+a x+1}, 0<a<2 )
Which of the following is true?
A. ( f(x) ) is decreasing on (1,1) and has a local minimum at ( x=1 )
B. ( f(x) ) is increasing on (1,1) and has a local maximum at ( x=1 )
c. ( f(x) ) is increasing on (1,1) but has neither a local maximum nor a local minimum at ( x=1 )
maximum nor a local minimum at ( x=1 )
12
556A point is moving on ( y=4-2 x^{2} ). The ( x )
coordinate of the point is decreasing at the rate of 5 units per second. Then the rate at which y-coordinate of the point
is changing when the point is at (1,2)
is.
A . 5 units/sec
B. 10 units/sec
c. 15 units/sec
D. 20 units/sec
12
557Set up an equation of a tangent to the graph of the following function. ( y=4 x-x^{2} ) at the points of its
intersections with the ( 0 x ) axis.
12
558Number of critical points of the function ( f(x)=(x-2)^{frac{2}{3}}(2 x+1) ) is equal to12
559Let, ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{B} ) be an invertible function
If ( f(x)=2 x^{3}+3 x^{2}+x-1, ) then
( f^{prime-1}(5)= )
A ( cdot frac{1}{13} )
B.
( c cdot 6 )
D. can not be determined
12
560Using differentials, find the sum of digits approximate value of the
following up to 3 places of decimal. ( (0.999)^{frac{1}{10}} )
12
5613×2+12x-1,
-1<x<2,
5.
{ 37-x
2<x<3 then: (1993 – 2 Marks)
(a) f(x) is increasing on [-1,2]
(b) f(x) is continues on [-1,3]
(c) f'(2) does not exist
d) f(x) has the maximum value at x=2
12
562The minimum value of ( x log x ) is equal
to.
( A )
B. ( frac{1}{e} )
( c cdot-frac{1}{e} )
( D cdot 2 )
12
563What is minimum value of ( sec ^{2} theta+ )
( cos ^{2} theta ? )
12
564Find the approximate value of
( (0.009)^{1 / 3} )
( mathbf{A} cdot 0.208 )
B. 0.108
c. 0.205
D. 0.204
12
565Find the intervals in which the function
( f(x) ) is
(i) increasing,
(ii) decreasing
( f(x)=2 x^{3}-9 x^{2}+12 x+15 )
12
566The equation of the normal to the curve
( x^{4}=4 y ) through the point (2,4) is
A. ( x+8 y=34 )
B. ( x-8 y+30=0 )
c. ( 8 x-2 y=0 )
D. ( 8 x+y=20 )
12
56720. Find the equation of the normal to the curve
y = (1+x)” + sin (sin? x) at x =0 (1993 – 3 Marks)
12
56833. Find a point on the curve x2 + 2y2 = 6 whose distance from
the line x+y=7, is minimum.
(2003 – 2 Marks)
12
569The length of the longest interval, in
which ( f(x)=3 sin x-4 sin ^{2} x ) is
increasing, is
A.
в.
( c cdot frac{3 pi}{2} )
D.
12
570Write the set of values of ( a ) for which
( boldsymbol{f}(boldsymbol{x})=cos boldsymbol{x}+boldsymbol{a}^{2} boldsymbol{x}+boldsymbol{b} ) is strictly
increasing on ( boldsymbol{R} )
12
571Show that the function ( x^{2}-x+1 ) is
neither increasing nor decreasing on (0,1)
12
572Consider the following statements:
1. The function ( f(x)=x^{2}+2 cos x ) is
increasing in the interval ( (0, pi) )
2. The function ( f(x)=ln (sqrt{1+x^{2}}-x) )
is decreasing in the interval ( (-infty, infty) ) Which of the above statements is lare
correct?
A. 1 only
B. 2 only
c. Both 1 and 2
D. Neither 1 nor 2
12
573Assertion
The equation ( boldsymbol{f}(boldsymbol{x})left(boldsymbol{f}^{prime prime}(boldsymbol{x})right)^{2}+ )
( boldsymbol{f}(boldsymbol{x}) boldsymbol{f}^{prime}(boldsymbol{x}) boldsymbol{f}^{prime prime prime}(boldsymbol{x})+left(boldsymbol{f}^{prime}(boldsymbol{x})right)^{2} boldsymbol{f}^{prime prime}(boldsymbol{x})= )
0 has atleast 5 real roots
Reason
The equation ( f(x)=0 ) has atleast 3
real distinct roots
& if ( f(x)=0 ) has ( k )
real distinct roots, then ( f^{prime}(x)=0 ) has
atleast k-1 distinct roots.
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
574Find the slope of the tangent and the
normal to the curve ( x^{2}-2 y-y^{2}=1 ) at
(-1,-2)
12
575The equation ( x+e^{x}=0 ) has
A. only one real root
B. only two real roots
c. no real root
D. none of these
12
576If the rate of change of area of a square plate is equal to that of the rate of change of its perimeter, then length of
the side is?
A . 1 unit
B. 2 unit
c. 3 unit
D. 4 unit
12
577The greatest value of ( f(x)=x-2 ln x )
in ( [1, e] ) is attained at ( x= )
( mathbf{A} cdot mathbf{1} )
B. ( sqrt{e} )
( c cdot 2 )
D.
12
578Find the slope of tangent of the curve
( boldsymbol{x}=boldsymbol{a} sin ^{3} boldsymbol{t}, boldsymbol{y}=boldsymbol{b} cos ^{3} boldsymbol{t} ) at ( boldsymbol{t}=frac{pi}{2} )
A . cott
B. ( – ) tant
c. ( – ) cott
D. not defined at ( frac{pi}{2} )
12
579The maximum slope of the curve ( mathbf{y}= ) ( -x^{3}+3 x^{2}+9 x-27 ) is
( A )
B. 12
( c cdot 6 )
D.
12
58024.
Iff(x)= x3 + bx2 + cx + d and 0<b2 <c, then in (-0,00)
(a) f(x) is a strictly increasing function (2004)
(b) f(x) has a local maxima
(c) f(x) is a strictly decreasing function
(d) f (x) is bounded
12
581The normal to the curve ( y(x-2)(x- )
( mathbf{3})=boldsymbol{x}+boldsymbol{6} ) at the point where the curve
intersects the y-axis passes through the point.
( ^{A} cdotleft(-frac{1}{2},-frac{1}{2}right) )
в. ( left(frac{1}{2}, frac{1}{2}right) )
( ^{mathrm{c}} cdotleft(frac{1}{2},-frac{1}{3}right) )
D ( cdotleft(frac{1}{2}, frac{1}{3}right) )
12
5825. The sum of two numbers is fixed. Then its multiplication
is maximum, when
(a) Each number is half of the sum
13
(b) Each number is
espectively of the sum
(c) Each number is
respectively of the sum
(d) None of these
12
583The values of ( a ) and ( b ) for which the
function ( boldsymbol{y}=boldsymbol{a} log _{e} boldsymbol{x}+boldsymbol{b} boldsymbol{x}^{2}+boldsymbol{x}, ) has
extremum at the points ( x_{1}=1 ) and
( boldsymbol{x}_{2}=2 ) are
( ^{A} cdot_{a}=frac{2}{3}, b=-frac{1}{6} )
B. ( quad a=-frac{2}{3}, b=-frac{1}{6} )
c. ( _{a=-frac{2}{3}, b=frac{1}{6}} )
D. ( _{a=-frac{1}{3}, b=-frac{1}{6}} )
12
584Which one of the following is correct in respect of the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}(boldsymbol{x}-mathbf{1})(boldsymbol{x}+mathbf{1}) ? )
A. The local maximum value is larger than local minimum value
B. The local maximum value is smaller than local minimum value
c. The function has no local maximum
D. The function has no local minimum
12
585The slope of the tangent to the curve
( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}} cos boldsymbol{x} ) is minimum at ( boldsymbol{x}=boldsymbol{alpha}, boldsymbol{0} leq )
( a leq 2 pi, ) then the value of ( alpha ) is
A . 0
в. ( pi )
c. ( 2 pi )
D. ( frac{3 pi}{2} )
12
586If ( f(x)=sin x+a^{2} x+b ) is an
increasing function for all values of ( x )
then
A ( . a epsilon(-infty,-1) )
B. ( a in R )
( mathbf{c} cdot a epsilon(-1,1) )
D. none of these
12
5872. The adjacent sides of a rectangle with given perimeter as
100 cm and enclosing maximum area are
(a) 10 cm and 40 cm (b) 20 cm and 30 cm
(c) 25 cm and 25 cm (d) 15 cm and 35 cm
1
.11
1
.
12
588The angle made by the tangent at any
point on the curve ( boldsymbol{x}=boldsymbol{a}(boldsymbol{t}+ )
( sin t cos t), y=a(1+sin t)^{2} ) with ( x ) -axis
is
A ( cdot frac{pi}{2} )
B. ( frac{pi}{4} )
c. ( _{pi+frac{t}{2}} )
D. ( frac{pi}{4}+frac{t}{2} )
12
589Function ( f(x)=a^{x} ) is increasing on ( R )
if
( mathbf{A} cdot a>0 )
B. ( a<0 )
c. ( 0<a1 )
12
590Let ( f ) be function defined on ( [a, b] ) such
that ( f^{prime}(x)>0, ) for all ( x in(a, b) . ) Then
prove that ( f ) is an increasing function on ( (a, b) )
12
591The equation of the tangent to the curve ( y=sqrt{9-2 x^{2}} ) at the point where the
ordinate and abscises are equal is?
A. ( 2 x+y-3 sqrt{3}=0 )
В. ( 2 x+y+3 sqrt{3}=0 )
c. ( 2 x-y-3 sqrt{3}=0 )
D. ( 2 x-y+3 sqrt{3}=0 )
12
592The values of ( x ) at which ( f(x)=sin x ) is stationary are given by
A. ( mathrm{n} pi, forall n in Z )
в. ( (2 mathrm{n}+1) frac{pi}{2}, forall n in Z )
c. ( frac{mathrm{n} pi}{4}, forall n in Z )
D. ( frac{mathrm{n} pi}{2}, forall n in Z )
12
593The critical points of the function ( f(x)=(x-2)^{2 / 3}(2 x+1) ) are
A. 1 and 2
B. 1 and ( -frac{1}{2} )
c. -1 and 2
( D )
12
594Sand is pouring from a pipe at the rate of ( 12 mathrm{cm}^{3} / mathrm{s} ). The falling sand forms a cone on the ground in such a way that
the height of the cone is always onesixth of the radius of the base. How fast
is the height of the sand cone increasing when the height is ( 4 mathrm{cm} )
12
595The nearest point on the line ( 3 x-4 y= )
25 from the origin is :
A ( cdot(-4,5) )
В ( cdot(3,-4) )
( mathbf{c} cdot(3,4) )
D ( cdot(3,5) )
12
596The total revenue received from the sale
of ( x ) units is given by ( R(x)=10 x^{2}+ )
( 20 x+1500 . ) The marginal revenue
when ( x=2015, ) is
A. 4032
B. 40320
( c .403 )
D. 40300
12
5971.
The function f(x)=sin4 x + cos4 x increases if
(1999 – 2 Marks)
(a) 0<x< 1/8
(b) /4<x<31/8
(c) 37/8<x< 571/8
(d) 51/8<x<31/4
12
598If ( boldsymbol{x}, boldsymbol{y}, boldsymbol{z}>mathbf{0} ) and ( boldsymbol{x} boldsymbol{y} boldsymbol{z}=mathbf{1} )
Then
This question has multiple correct options
( mathbf{A} cdot x^{2}+y^{2}+z^{2} leq x^{3}+y^{3}+z^{3} )
B . ( x^{2}+y^{2}+z^{2} geq x^{3}+y^{3}+z^{3} )
( mathbf{c} cdotleft(x^{4}+y^{4}+z^{4}right) geqleft(x^{3}+y^{3}+z^{3}right) )
D. ( left(x^{4}+y^{4}+z^{4}right) leqleft(x^{3}+y^{3}+z^{3}right) )
12
599A rectangular tank is ( 80 m ) long and ( 25 m ) broad. Water flows into it through
a pipe whose cross-section is ( 25 mathrm{cm}^{2}, ) at
the rate of ( 16 k m ) per hour. How much
the level of the water rises in the tank in
45 minutes?
12
600( mathbf{f}(mathbf{x})=frac{mathbf{a x}+mathbf{b}}{mathbf{c x}+mathbf{d}}(boldsymbol{a d}-boldsymbol{b c} neq mathbf{0}) )
A. Has a maximum
B. Has a minimum
c. Neither max nor min is true
D. Both max and min are true
12
601For all ( a, b in R ) the function ( f(x)= )
( 3 x^{4}-4 x^{3}+6 x^{2}+a x+b ) has
A. no extremumm
B. exactly one extremum
c. exactly two extremum
D. three extremum
12
602Two sides of a triangle are given. If the area of the triangle is maximum then the angle between the given sides is
A ( cdot 45^{circ} )
B. ( 30^{circ} )
( c cdot 60^{circ} )
D. ( 90^{circ} )
12
603Let ( boldsymbol{E}=boldsymbol{x}^{3}left(boldsymbol{x}^{3}+mathbf{1}right)left(boldsymbol{x}^{mathbf{3}}+mathbf{2}right)left(boldsymbol{x}^{mathbf{3}}+right. )
( mathbf{3}) ; boldsymbol{x} epsilon boldsymbol{R} ). Then minimum value of ( boldsymbol{E} ) be
( A )
B. – –
( c cdot-1 )
D. Minimum value is not attained
12
60409
19.

cos(2x) cos(2x) sin(2x)
If f(x) = -COS X COS X -sin x , then
sin x sin x cOS X
(a f'(x)=0 at exactly three points in (-T, T)
(b) f'(x) = 0 at more than three points in (-1, T)
(C f(x) attains its maximum at x=0
(d) f(x) attains its minimum at x=0
12
605If ( f(x)=x^{2} e^{-x^{2} / a^{2}} ) is an increasing
function then (for ( a>0 ) ), x lies in the
interval
( mathbf{A} cdot[a, 2 a] )
в. ( (-infty,-a) cup(0, a) )
c ( .(-a, 0) )
D. None of these
12
6062.
Let x and y be two real variables such that x>0 and xy=1.
Find the minimum value of x+y. (1981 – 2 Marks)
1
.
:
.
:
.
C
…..
.
….
..
12
607A particle moves on a line according to
the law ( s=a t^{2}+b t+c . ) If the
displacement after 1 sec is ( 16 mathrm{cm}, ) the velocity after 2 sec is ( 24 mathrm{cm} / mathrm{sec} ) and acceleration is ( 8 mathrm{cm} / mathrm{sec}^{2}, ) then
A. ( a=4, b=8, c=4 )
В. ( a=4, b=4, c=8 )
c. ( a=8, b=4, c=4 )
D. none of these
12
60813. Investigate for maxima and minima the function
(1988 – 5 Marks)
f(x)= [2(t – 1)(t – 2)2 + 3(t – 1)(t – 2)?]dt
12
609Using differentials, find the sum of digits approximate value of the
following up to 3 places of decimal. ( (0.0037)^{frac{1}{2}} )
12
61018. A cubic f (x) vanishes at x = 2 and has relative minimum/
14.
maximum at x = -1 and x = = if
.
Ix =
, find the
cubic f(x).
(1992 – 4 Marks)
12
611The function ( f(x)=4-3 x+3 x^{2}-x^{3} )
is
A. decreasing on ( R )
B. increasing on ( R )
c. strictly decreasing on ( R )
D. strictly increasing on ( R )
12
612Find the set of values of ( b ) for which
( boldsymbol{f}(boldsymbol{x})=boldsymbol{b}(boldsymbol{x}+cos boldsymbol{x})+boldsymbol{4} ) is decreasing
on ( boldsymbol{R} )
12
613For ( mathbf{0} leq boldsymbol{x} leq mathbf{1}, ) the function ( boldsymbol{f}(boldsymbol{x})= )
( |boldsymbol{x}|+|boldsymbol{x}-mathbf{1}| ) is
A. Monotonically increasing
B. Monotonically decreasing
c. constant function
D. Identity function
12
61426.
A spherical balloon is filled with 45001 cubic meters of helium
gas. If a leak in the balloon causes the gas to escape at the
rate of 72īt cubic meters per minute, then the rate (in meters
per minute) at which the radius of the balloon decreases 49
minutes after the leakage began is:
[2012]
(a)
=
(b)
12
615Show that the function ( x^{x} ) is minimum
at ( boldsymbol{x}=frac{mathbf{1}}{boldsymbol{e}} )
12
616Which one of the following statements is correct?
A ( cdot e^{x} ) is a increasing function
B. ( e^{x} ) is a decreasing function.
( mathrm{c} cdot e^{x} ) is neither increasing nor decreasing function
D. ( e^{x} ) is a constant function
12
617If ( f(x)=frac{x^{2}-1}{x^{2}+1}, ) for every real ( x, ) then
the minimum value of ( boldsymbol{f} ) is
A. does not exist because fis unbounded
B. is not attained even through f is bounded
c. is equal to 1
D. is equal to -1
12
618A spherical balloon is being inflated so that its volume increases uniformly at
the rate of ( 40 mathrm{cm}^{3} / ) min. When ( r=8 )
then the increase in radius in the next
( 1 / 2 m i n ) is
A ( .0 .025 mathrm{cm} )
B . ( 0.050 mathrm{cm} )
c. ( 0.075 mathrm{cm} )
D. ( 0.01 mathrm{cm} )
12
61913. Let
interval
(a) (-00,-2)
(c) (1,2)
– 1)(x – 2)dx . Then f decreases in the
(2000)
(b) (-2,-1)
(d) (2,700)
12
620To find the equation of tangent and normal to the circle ( x^{2}+y^{2}-3 x+ )
( 4 y-31=0 ) at the point (2,3)
12
621Construct the graphs of the following functions and carry out a complete investigation
( y=sin ^{4} x+cos ^{4} x )
12
622Maximum value of ( f(x)=frac{x^{2}-x+1}{x^{2}+x+1} )
is
A.
B. 3
( c cdot frac{3}{7} )
D.
12
623If ( x=a(cos 2 t+2 t sin 2 t) ) and ( y= )
( a(sin 2 t-2 t cos 2 t) ) then find ( frac{d^{2} y}{d x^{2}} )
12
624The equation of one of the tangents to the curve ( boldsymbol{y}=cos (boldsymbol{x}+boldsymbol{y}),-2 boldsymbol{pi} leq boldsymbol{x} leq )
( 2 pi ; ) that is parallel to the line ( x+2 y= )
( 0, ) is
A. ( x+2 y=1 )
B. ( x+2 y=frac{pi}{2} )
c. ( x+2 y=frac{pi}{4} )
D. None of these
12
625The function ( frac{sin (x+alpha)}{sin (x+beta)} ) has no
maximum or minimum if (k an integer)
A. ( beta-alpha=k pi )
B. ( beta-alpha neq k pi )
c. ( beta-alpha=2 k pi )
D. none of the above
12
626Find a point of inflation for the curve ( y=frac{x+1}{x^{2}+1} )12
627if ( f^{prime}(x)<0 forall x in R ) and ( g(x)=fleft(x^{2}-right. )
2) ( +boldsymbol{f}left(boldsymbol{6}-boldsymbol{x}^{2}right) ) then
( A cdot g(x) ) is an increasing in ( [2, infty) )
B. ( g(x) ) is an increasing in [-2,0]
c. ( g(x) ) has a local minima ( a t x=-2 )
D. ( g(x) ) has a local maxima at ( x=2 )
12
628An edge of a variable cube is increasing at the rate of ( 5 mathrm{cm} / mathrm{s} ). How fast is the
volume of the cube increasing when the
edge is ( 10 mathrm{cm} ) long?
12
629Find the slopes of the tangent and the normal to the following curves at the
indicated points.
( boldsymbol{x}^{2}+mathbf{3} boldsymbol{y}+boldsymbol{y}^{2}=mathbf{5} ) at ( (mathbf{1}, mathbf{1}) )
12
630Find the slope of a line. Which bisects the first quadrant angle.12
631syualt
* 1933 y = 10 then the making point of us is
tu Nile ul uitst
4. If x + y = 10, then the maximum value of xy is
(a) 5
(b) 20
(c) 25
(d) None of these
12
632If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) is a differentiable function
such that ( f^{prime}(x)>2 f(x) ) for all ( x in R )
and ( f(0)=1, ) then
This question has multiple correct options
A ( cdot f(x) ) is decreasing in ( (0, infty) )
B ( cdot f^{prime}(x)e^{2 x} ) in ( (0, infty) )
12
63325. For xe ( 0.574), define \$(w) = vi sint dt. Thenſ has
[2011]
(a) local minimum at n and 21
(b) local minimum at n and local maximum at 21
(c) local maximum at n and local minimum at 21
(d) local maximum at n and 21
12
634A particle starts moving from rest from a fixed point in a fixed direction. The
distance ( s ) from the fixed point at a time
( t ) is given by ( s=t^{2}+a t-b+17, ) where
( a, b ) are real numbers. If the particle
comes to rest after 5 sec at a distance
of ( s=25 ) units from the fixed point
then values of ( a ) and ( b ) are respectively.
A .10,-33
B. -10,-33
c. -8,33
D. -10,33
12
635total number of local maxima and local minima of the
29. The total numbe
is
(2008)
function f(x) = (2+x)}, -3,
[(2+x), -3<xs-1
7×2/3,-1<x<2
(6) 1 (c) 2
(+2/3
(a) 0
.
(d)
3
12
636Let ( f ) be a decreasing function in ( (a, b] )
then which of the following must be true?
A. ( f ) is continuous at ( b )
B ( cdot f^{prime}(b)<0 )
c. ( lim _{x rightarrow b} f(x) leq f(b) )
D. ( lim _{x rightarrow b} f(x) geq f(b) )
12
637The function ( boldsymbol{f}(boldsymbol{x})=int_{0}^{x} boldsymbol{e}^{-boldsymbol{x}^{2} / 2}left(boldsymbol{x}^{2}-right. )
( 3 x+2) d x ) is maximum at ( x= )
( A )
B. 2
( c cdot 3 )
D. 4
12
638If ( mathbf{f}(boldsymbol{theta})=sin ^{99} boldsymbol{theta} cos ^{94} boldsymbol{theta} ; boldsymbol{theta} inleft(-frac{boldsymbol{pi}}{2}, frac{boldsymbol{pi}}{2}right) )
attains a maximum at ( boldsymbol{theta} ) equals
A ( cdot tan ^{-1} sqrt{frac{94}{99}} )
B. ( tan ^{-1} sqrt{frac{99}{94}} )
c.
D.
12
639II (X)= and e(r) * where 0<x s 1, the
tan x
this interval
(1997 – 2 Marks)
(a) both f(x) and g(x) are increasing functions
(b) both f(x) and g(x) are decreasing functions
(C) f(x) is an increasing function
(d) g(x) is an increasing function.
TLC
12
640Find the co-ordinates of the points on
the ellipse ( x^{2}+2 y^{2}=9 ) at which tangent has slope ( frac{1}{4} . ) Also find the equation of normal.
12
641While measuring the side of an
equilateral triangle an error of ( k % ) is marked, the percentage error in its area is
A. ( k % )
в. ( 2 k % )
( c cdot frac{k}{2} % )
D. ( 3 k % )
12
642Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be defined by
( boldsymbol{f}(boldsymbol{x})=left{begin{array}{lll}boldsymbol{k}-boldsymbol{2} boldsymbol{x}, & boldsymbol{i} boldsymbol{f} & boldsymbol{x} leq-1 \ 2 boldsymbol{x}+boldsymbol{3}, & boldsymbol{i f} & boldsymbol{x}>-1end{array}right. )
If ( f ) has a local minimum at ( x=-1 )
then the possible value of ( k ) is
A. ( -1 / 2 )
B. –
( c cdot 1 )
( D )
12
643If the ratio of base radius and height of a cone is 1: 2 and percentage error in radius is ( lambda %, ) then the error in its volume is
A . ( lambda % )
B . 2lambda%
c. ( 3 lambda % )
D. none of these
12
644If ( boldsymbol{f}(boldsymbol{x})=sum_{boldsymbol{r}=mathbf{0}}^{boldsymbol{n}} boldsymbol{a}_{boldsymbol{r}} boldsymbol{x}^{boldsymbol{r}} )
for
( boldsymbol{a}_{boldsymbol{r}} boldsymbol{epsilon} boldsymbol{R} ; boldsymbol{r} boldsymbol{epsilon} boldsymbol{N} ; boldsymbol{n} geq mathbf{3} )
If ( boldsymbol{f}(boldsymbol{x}) neq mathbf{0} ) for ( boldsymbol{x} boldsymbol{epsilon}(boldsymbol{alpha}, boldsymbol{beta}) )
Then, if ( (boldsymbol{alpha}<boldsymbol{t}<boldsymbol{beta}) )
This question has multiple correct options
A. ( f(x) ) is continuous and Differentiable over ( (alpha, beta) ) atleast
B. ( (x-alpha)(x-beta) f(x) ) is continuous and Differentiable over ( (alpha, beta) ) atleast
C ( .(x-alpha)(x-beta) f(x) ) is continuous, but Not
differentiable over ( (alpha, beta) )
( frac{f^{prime}(t)}{f(t)}=frac{1}{alpha-t}+frac{1}{beta-t}, ) for atleast one ( ^{prime} t^{prime} ) in ( (alpha, beta) )
12
645a, b>c, x>-cis (Va-c+Vb-c)2.
(1979)
Tet x and y be two real variables such that x > 0 and xy = 1.
Find the minimum value of x+y.
(1981 – 2 Marks)
12
64613. A triangular park is enclosed on two side
park is enclosed on two sides by a fence and on
the third side by a straight river bank. The two sid
fence are of same length x. The maximum area enclosed
the park is
[2006]
(a)
3
(c) 1 x²
(d) rx²
12
647Example 2.2 From point A located on a highway as shown
in Fig. 2.41, one has to get by car as soon as possible to point B
located in the field at a distance I from the highway. It is known
that the car moves in the field time slower on the highway. At
what distance from point D one must turn off the highway?
X
Fig. 2.41
12
648The length ( x ) of an rectangle is
decreasing at the rate ( 2 mathrm{cm} / ) sec and width y is increasing at the rate of
( 2 c m / )sec. When ( x=12 c m ) and ( y= )
( 12 c m, ) the rate of change of the area of
the rectangle is ( k c m^{2} / s e c, ) then ( k-9 ) is
12
6493.
The normal to the curve x = a (cos + O sin o),
y = a (sin 0-0 cos O) at any point ‘O’ is such that
(1983 – 1 Mark)
(a) it makes a constant angle with the x-axis
(b) it passes through the origin
(c) it is at a constant distance from the origin
(d) none of these
12
650Find the values of ( x, ) for which the
function ( f(x)=x^{3}+12 x^{2}+36 x+6 ) is
increasing.
12
651sin tax
12. Let f(x) = 2 ,x>0
Let x, <x, <x,<…….. <x<……. be all the points of local
maximum off and y, <y, <y,< ……. <y, 2
(b) x, el 21,2n + „ }for every n
(©) |xn – yul>1 for every n
(d) x,<y,
12
652f ( boldsymbol{y}=cos ^{2}left(45^{circ}+xright)(sin x-cos x)^{2} )
then the maximum value of y is:
12
653The slope of the curve ( y=sin x+cos ^{2} x )
is zero at the point where –
A ( cdot x=frac{pi}{4} )
B. ( x=frac{pi}{2} )
c. ( x=pi )
D. No where
12
654( operatorname{Let} frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}=ln sqrt{frac{1+x}{1-x}}, ) then find ( x )12
655where Il 15 UISLUMUUUU
13. Let h(x)=min {x, r2), for every real number of x,
(1998 – 2 Marks)
(a) h is continuous for all x
(b) h is differentiable for all x
(c) h'(x)=1, for all x >1
(d) h is not differentiable at two values of x.
12
65638. Let f(x) = x² + + and g(x) = x
, XER-{-1,0,1}. If
x
h(x)=-***, then the local minimum value of h(x) is :
g
(a) -3
(b) -2/2
(c) 2/2
[JEE M 2018]
(d) 3
12
657Prove that the function given by ( boldsymbol{f}(boldsymbol{x})= ) ( cos x ) is strictly increasing in ( (pi, 2 pi) )12
658If the percentage error in the edge of a cube is ( 1, ) then error in its volume is
A. ( 1 % )
B. 2%
( c .3 % )
D. none of these
12
659A cylindrical tank of radius ( 10 m ) is being filled with wheat at the rate of 314
cubic metre per hour. then the depth of the wheat is increasing at the rate of
A ( cdot 1 m^{3} / h )
B. ( 0.1 m^{3} / h )
( mathbf{c} cdot 1.1 m^{3} / h )
D. ( 0.5 m^{3} / h )
E. None of these
12
66022. For every pair of continuous functions f, g: [0, 1] → R such
that max f(x): x €[0,1]} = max {g(x): x €[0,11), the
correct statement(s) is (are):
(a) f(c))2 +3f (c)= (g(c))2 + 3g(c) for some ce [0, 1]
(b) (c)2 +f(c)=(g(c))2 +3g(c) for some c € [0, 1]
(c) (c))2 + 3f (c)= (g(c))2+g(c) for some c e [0, 1]
(d) (C))2 = (g(c))2 for some c e [0, 1]
12
661The approximate value of ( f(x)=x^{3}+ )
( 5 x^{2}-7 x+9=0 ) at ( x=1.1 ) is
A . 8.6
B. 8.5
( c .8 .4 )
D. 8.3
12
662Let ( f(x)=(x-2)^{33}(x-3)^{44} ) then
which of the following is true
A. ( x=2 ) is point of inflexion
B. ( x=3 ) is point of minima
c. ( _{x}=frac{17}{7} ) is point of maxima
D. All of these
12
6634.
Use the function f(x) = x1/x , x>0. to determine the bigger
of the two numbers er and me
(1981 – 4 Marks)
12
664The function ( f(x)=x^{2}+frac{lambda}{x} ) then:
A. minimum at ( x=2 ) if ( lambda=16 )
B. maximum at ( x=2 ) if ( lambda=16 )
c. maximum for no real value of ( lambda )
D. point of inflection at ( x=1 ) if ( lambda=-1 )
12
66521.
The equation of the tangent to the curve y=x+
2 , that
is parallel to the x-axis, is
(a) y=1 (b) y=2
[2010]
(d) y=0
(c) y=3
12
66614. Find all maxima and minima of the function
y = x(x – 1)2,0 < x < 2
(1989 – 5 Marks)
Also determine the area bounded by the curvey=x (x – 1)2,
the y-axis and the line y=2.
12
667Let the absolute maxima/minima value
of ( boldsymbol{f}(boldsymbol{x})=mathbf{3} boldsymbol{x}^{4}-boldsymbol{8} boldsymbol{x}^{3}+mathbf{1} boldsymbol{2} boldsymbol{x}^{2}-boldsymbol{4} boldsymbol{8} boldsymbol{x}+ )
( mathbf{2 5} ; boldsymbol{x} in[mathbf{0}, mathbf{3}] ) be max, min. Find
( 2(max )+min ? )
12
668Find the greatest and the least values of the following functions. ( f(x)=left(2^{x}+2^{-x}right) / I n 2 ) on the
interval [-1,2]
12
669Minimum value of the polynomial ( boldsymbol{p}(boldsymbol{x})=mathbf{4} boldsymbol{x}^{2}-boldsymbol{6} boldsymbol{x}+mathbf{1} )
A. ( -frac{3}{4} )
B. ( -frac{5}{4} )
( c cdot-frac{5}{16} )
D. – œ
12
670A sphere of radius ( 100 mathrm{mm} ) shrinks to radius ( 98 mathrm{mm} ), then the approximate decrease in its volume is
A. ( 12000 pi m m^{3} )
В. ( 800 pi m m^{3} )
c. ( 80000 pi m m^{3} )
D. ( 120 pi m m^{3} )
12
671(
6 то п
опало –
(1995)
12. The function flx) = max (1-x)(1+x), 2}, xe(-0, 0) is
(a) continuous at all points
(6) differentiable at all points
(© differentiable at all points except at x=landx=-1
(d) continuous at all points except at x = 1 and x = -1.
where it is discontinuous
Loof, Then
7
12
672Assertion ( (A): ) If ( f(x)=|x|, ) then ( f ) has
minimum value at ( x=0 )
Reason (R): A function f(x) has
minimum value at ( x=a ) if ( f^{prime}(a)=0 ) and
( mathbf{f}^{prime prime}(mathbf{a})>mathbf{0} )
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true and R is not the correct explanation of A
C. A is true and R is false
D. A is false and R is true
12
673The function ( x^{5}-5 x^{4}+5 x^{3}-10 ) has a
maximum, when ( boldsymbol{x}= )
( A cdot 3 )
B . 2
( c .1 )
D.
12
674Find limits of the error when ( frac{mathbf{9 1 6}}{mathbf{1 9 1}} ) is taken for ( sqrt{mathbf{2 3}} )12
675Find intervals in which ( f(x)=frac{4 x^{2}+1}{x} )
is increasing and decreasing.
12
676The equation of normal to the curve ( boldsymbol{y}=tan boldsymbol{x} ) at the point ( (boldsymbol{0}, boldsymbol{0}) ) is
A. ( x+y=0 )
В. ( x-y=0 )
c. ( x+2 y=0 )
D. None of these
12
677A particle is moving in a straight line such that its distance at any time ( t ) is given by ( s=frac{t^{4}}{4}-2 t^{3}+4 t^{2}-7 . ) The
acceleration of the particle is minimum
when ( t= )
( A cdot 1 )
B. 2
( c .3 )
( D )
12
67823. Let f:
R R be a function defined by
f(x)=min (x +1,|x+1),Then which of the following is true ?
(a) f(x) is differentiable everywhere
[2007]
(6) f() is not differentiable at x=0
© f(x) > 1 for all X ER
(d) f(x) is not differentiable at x=1
12
679Find the rate of change of the area of a
circle with respect to its radius ( r ) when
(i) ( r=3 mathrm{cm} )
(ii) ( r=4 mathrm{cm} )
( mathbf{A} cdot 6 pi, 8 pi )
B . ( 5 pi, 8 pi )
( mathbf{c} .4 pi, 10 pi )
D . ( 2 pi, 8 pi )
12
680Assertion(A): If the tangent at any point
( P ) on the curve ( x y=a^{2} ) meets the axes
at ( A ) and ( B ) then ( A P: P B=1: 1 )
Reason(R): The tangent at ( P(x, y) ) on
the curve ( boldsymbol{X}^{boldsymbol{m}} cdot boldsymbol{Y}^{boldsymbol{n}}=boldsymbol{a}^{boldsymbol{m}+boldsymbol{n}} ) meets the
axes at ( A ) and ( B ). Then the ratio of ( P )
divides ( overline{A B} ) is ( n: m )
A. Both A and R are true R is the correct explanation of
B. Both A and R are true but R is not correct explanation of A
c. A is true but R is false
D. A is false but R is true
12
681The point on the curve ( y^{2}=8 x ) for
which the abscissa and ordinate
change at the same rate is.
A ( cdot(4,2) )
в. (-4,2)
( c cdot(2,4) )
D. (-2,-4)
12
682Find the critical points of the function ( f(x)=(x-2)^{2 / 3}(2 x+1) )
( A .-1 ) and 2
B.
c. 1 and -2
D. 1 and 2
12
683the radius of a sphere increases at a rate of ( 2 mathrm{cm} / mathrm{sec} . ) Find the rate at which
its volume and area increases when
radius is ( 4 mathrm{cm} )
12
684Find the values of ( x ) for which ( y= )
( [x(x-2)]^{2} ) is an increasing function.
12
685Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+boldsymbol{5} sin ^{2} boldsymbol{x} ) be
an increasing function on the set ( R )
Then, ( a ) and ( b ) satisfy
A ( cdot a^{2}-3 b-15>0 )
B . ( a^{2}-3 b+15>0 )
c. ( a^{2}-3 b+150 ) and ( b>0 )
0
12
686ff ( f(x)=x^{2}-4 x+5 ) on [0,3] then the
absolute maximum value is:
( A cdot 2 )
B. 3
( c cdot 4 )
D. 5
12
687A point on the parabola ( y^{2}=18 x ) at
which the ordinate increases at twice
the rate of the abscissa is
A. (2,6)
(年) (2,6)
в. (2,-6)
( ^{mathrm{c}} cdotleft(frac{9}{8},-frac{9}{2}right) )
D ( cdotleft(frac{9}{8}, frac{9}{2}right) )
12
688If ( f ) is an increasing and ( g ) is a decreasing function and fog is defined,
then fog will be
A. increasing function
B. decreasing funtion
c. neither increasing nor decreasing
D. None of these
12
689The side of a square sheet is increasing at the rate of ( 4 mathrm{cm} ) per minute. The rate by which the area increasing when the side is ( 8 mathrm{cm} ) long is.
( mathbf{A} cdot 60 mathrm{cm}^{2} / mathrm{minute} )
B. ( 66 mathrm{cm}^{2} / ) minute
c. ( 62 mathrm{cm}^{2} / mathrm{minute} )
D. ( 64 mathrm{cm}^{2} / mathrm{minute} )
12
690If the rate of change in the circumference of a circle of ( 0.3 mathrm{cm} / mathrm{s} ) then the rate of change in the area of the circle when the radius is ( 5 mathrm{cm}, ) is:
A. ( 1.5 mathrm{sq} mathrm{cm} / mathrm{s} )
B. ( 0.5 mathrm{sq} mathrm{cm} / mathrm{s} )
c. ( 5 mathrm{sq} mathrm{cm} / mathrm{s} )
D. 3 sq cm/s
12
691(2006 – 5M, -1)
15. Iff(x)=min {1, x2, x3, then
(a) f(x) is continuous VXER
(b) f(x) is continuous and differentiable everywhere.
© f(x) is not differentiable at two points
(d) f(x) is not differentiable at one point
12
69219. Let f(x)=(1 + b2)x2 + 2bx + 1 and let m(b) be the minimum
value of f(x). As b varies, the range of m(b) is (20015)
(a) [0,1] (b) (0,1/2] (c) [1/2,1] (d) (0,1]
12
693( boldsymbol{f}(boldsymbol{x})left{begin{array}{l}=mathbf{2} boldsymbol{x}^{2}+frac{2}{x^{2}} text { for }-mathbf{2} leq boldsymbol{x}<mathbf{0} boldsymbol{a} \ =mathbf{1} quad text { for } boldsymbol{x}=mathbf{0}end{array}right. )
Determine the greatest and least values. What is the minimum value of
the function?
A. 0
B. 1
( c cdot 4 )
( D )
12
694The maximum value of ( frac{ln x}{x} ) is
A ( . e )
B. ( frac{1}{e} )
( c cdot frac{2}{e} )
D.
12
6954.
Let f and g be increasing and decreasing functions,
respectively from [0, 0 ) to [0, 0). Let h(x)=f(g(x)). If
h(0)=0, then h(x)-h(1) is
(1987 – 2 Marks)
(a) always zero
b) always negative
(©) always positive d) strictly increasing
(e) None of these.
12
696The family of curves represented by ( frac{d y_{1}}{d x}=frac{x^{2}+x+1}{y^{2}+y+1} ) and the family
represented by ( frac{boldsymbol{d} boldsymbol{y}_{2}}{boldsymbol{d} boldsymbol{x}}+frac{boldsymbol{y}^{2}+boldsymbol{y}+mathbf{1}}{boldsymbol{x}^{2}+boldsymbol{x}+mathbf{1}}=mathbf{0} )
A. touch each other
B. orthogonal to each other
c. identical
D. intersect at an angle of ( frac{pi}{4} )
12
697The values of ( a ) for which ( f(x)= ) ( frac{a^{2} x^{3}}{3}+frac{3 a x^{2}}{2}+2 x+1 ) is strictly
decreasing at ( boldsymbol{x}=mathbf{1} )
A. ( a in(-2,-1) )
в. ( a in(-1,0) )
c. ( a in(1,2) )
D. ( a in(-2,1) )
12
6986. If sum of two numbers is 3, then maximum value of the
product of first and the square of second is
(a) 4 (6) 3 (c) 2 (d) 1
Tecn
ie footh 24
.1.
12
699If the percentage error in measuring the
surface area of a sphere is ( alpha % ), then the error in its volume is
A ( cdot frac{3}{2} alpha % )
в. ( frac{2}{3} alpha % )
( c .3 alpha % )
D. none of these
12
700The normal to the curve ( sqrt{x}+sqrt{y}=sqrt{a} ) is perpendicular to ( x ) axis at the point
в. ( (a, 0) )
c. ( left(frac{a}{4}, frac{a}{4}right) )
D. No where
12
70138. Ify=(sinx + cosec x)2 + (cosx + sec x)?, then the minimum
value of y, Vxe R, is
b. 3 o levo !
c. 9
d. 0
a.
7
000
12
702If the distance ‘s’ metres transversed by
a particle in ( t ) seconds is given by ( s= )
( t^{3}-3 t^{2}, ) then the velocity of the particle
when the acceleration is zero, in metre/sec is
( A cdot 3 )
B. – –
( c cdot-3 )
D.
12
703ff ( x y(y-x)=2 a^{3}, ) at what point does ( y )
have a minimum value
A . a, 2a
B. a,-a
c. २а,२а
D. None of these
12
704( f(x)=x+sin x ) is decreasing when ( x )
lies in the interval
A ( cdot[-1,1] )
B. [2,3]
c. [3,4]
D. No value of ( x )
12
705The function ( f(x)=frac{lambda sin x+2 cos x}{sin x+cos x} ) is
increasing, if
( A cdot lambda1 )
c. ( lambda2 )
12
706The function ( f(x)=2-3 x+3 x^{2}- )
( x^{3}, x varepsilon R ) is
A. neither increasing nor decreasing
B. increasing
c. decreasing
D. none of these
12
707The distance between the origin and the tangent to the curve ( y=e^{2 x}+x^{2} ) drawn
at the point ( x=0 ) is
A ( cdot frac{1}{sqrt{5}} )
в. ( frac{2}{sqrt{5}} )
c. ( frac{-1}{sqrt{5}} )
D. ( frac{2}{sqrt{3}} )
12
708A point particle moves along a straight line such that ( x=sqrt{t}, ) where ( t ) is time.
Then, ratio of acceleration to cube of the
velocity is
A . -1
в. -0.5
c. -3
D. –
12
709The slope of the normal to the curve
( boldsymbol{x}=boldsymbol{a}(boldsymbol{theta}-sin theta), boldsymbol{y}=boldsymbol{a}(1-cos boldsymbol{theta}) ) at
point ( boldsymbol{theta}=frac{boldsymbol{pi}}{2} ) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot-1 )
D. ( frac{1}{sqrt{2}} )
12
71012. For the function
x x
x
f (x) = x COS-, X21,
(2009)
(a) for at least onex in the interval [1,00 ), f(x + 2)-f(x)00
(c) for all x in the interval [1,0), f (x + 2)-f(x)>2
(d) f'(x) is strictly decreasing in the interval [1,0 )
12
7111. A man 2 metre high walks at a uniform speed 5 metre/hour
away from a lamp post 6 metre high. The rate at which the
length of his shadow increases is
(a) 5 m/h (b) m/h (0
m/h (a) m/h
12
712Area of the greatest rectangle that can be inscribed in the
ellipse
+
=lis
[20051
2
64
(a) zab
(b) ab
(c)
Jab
(d) 1 2
12
713In a bank, principal increases
continuously at the rate of ( 5 % ) per year. An amount of ( R s .1000 ) is deposited
with this bank, how much will it worth
after 10 years( left(e^{0.5}=1.648right) )
12
7141 ui
10.
Define the collections {E, E., Ez, ……} of ellipses and
{R1, R2, Rz, …..} of rectangles as follows:
x2 y = 1;
E: 74
R : rectangle of largest area, with sides parallel to the axes,
inscribed in Ej;
12
E: ellipse – + 5=1 of largest area inscribed in R .
a b-
n>1;
R, : rectangle oflargest area, with sides parallel to the axes,
inscribed in E n >1.
Then which of the following options is/are correct?
(a) The eccentricities of E, and E, are NOT equal
(b) The length of latus rectum of E, is –
(C)
(area of R.)< 24, for each positive integer N
n=1
(d)
The distance of a focus from the centre in E, is
را در
12
715Let ( f(x)=x^{3}+3 x^{2}-9 x+2 . ) Then
A. ( f(x) ) has a maximum at ( x=1 )
B. ( f(x) ) has neither a minimum nor a maximum at ( x= ) -3
c. ( f(x) ) has a minimum at ( x=1 )
D. none of these
12
716If ( f^{prime}(x)=g(x)(x-a)^{2}, ) where ( g(a) neq 0 )
and ( g ) is continuous at ( x=a ) then
This question has multiple correct options
A. ( f ) is increasing near a if ( g(a)>0 )
B. ( f ) is increasing near a if ( g(a)0 )
D. ( f ) is decreasing near a if ( g(a)<0 )
12
717The set of values of ( p ) for which the
points of extremum of the function,
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-boldsymbol{p} boldsymbol{x}^{2}+boldsymbol{3}left(boldsymbol{p}^{2}-mathbf{1}right) boldsymbol{x}+mathbf{1} ) lie
in the interval (-2,4) is
A. (-3,5)
в. (-3,3)
c. (-1,3)
D. (-1,5)
12
718Consider the function ( f(x)=frac{x^{2}-1}{x^{2}+1} )
where ( boldsymbol{x} boldsymbol{epsilon} boldsymbol{R} )
At what value of ( x operatorname{does} f(x) ) attain minimum value?
A . -1
B.
c. 1
D.
12
719The distance (in metre) travelled by a
vehicle in time ( t ) (in seconds) is given by the equation ( s=t^{3}+2 t^{2}+t+1 . ) The
difference in the acceleration between
( boldsymbol{t}=boldsymbol{2} ) and ( boldsymbol{t}=boldsymbol{4} ) is
A ( cdot 12 m / s^{2} )
B . ( 18 m / s^{2} )
c. ( 16 m / s^{2} )
D. ( 14 m / s^{2} )
12
720The focal length of a mirror is given by ( frac{1}{v}-frac{1}{u}=frac{2}{f} . ) If equal errors ( (alpha) ) are
made in measuring ( u ) and ( v ), then the
relative error in ( boldsymbol{f} ) is
A ( cdot frac{2}{alpha} )
B ( cdot alphaleft(frac{1}{u}+frac{1}{v}right) )
( ^{c} cdot alphaleft(frac{1}{u}-frac{1}{v}right) )
D. none of these
12
721Find the derivative of ( f(x)=3 x ) at ( x= )
2
12
72218.
Iff:RRis a differentiable function such that f'(x) >2f(x)
for all x e R, and f(0) = 1, then
(a) f(x) is increasing in (0,00)
(b) f(x) is decreasing in (0,0)
(c) f(x)>e2x in (0,00)
(d) f'(x) <e2x in (0,0)
12
723The points on the curve ( 12 y=x^{3} ) whose
ordinate and abscissa change at the
same rate, are
A ( cdot(-2,-2 / 3),(2,2 / 3) )
в. ( (-2,2 / 3),(2 / 3,2) )
c. ( (-2,-2 / 3) ) only
D. ( (2 / 3,2) ) only
12
724If the volume of spherical ball is increasing at the rate of ( 4 pi c c / ) sec then
the rate of change of its surface area
when the volume is ( 288 pi c c ) is
A ( cdot frac{4}{3} pi c m^{2} / ) sec
в. ( frac{2}{3} pi c m^{2} / ) sec
( mathrm{c} cdot 4 pi mathrm{cm}^{2} / mathrm{sec} )
D. ( 2 pi c m^{2} / ) sec
12
725( f(x)=x+2 cos x ) is increasing in
A ( cdotleft(0, frac{pi}{2}right) )
В ( cdotleft(frac{-pi}{2}, frac{pi}{6}right) )
c. ( left(frac{pi}{2}, piright) )
D. ( left(frac{-pi}{2} frac{pi}{2}right) )
12
726Using differentials, find the approximate value of each of the
following up to 3 places of decimal.
(i) ( sqrt{25.3} )
(ii) ( sqrt{49.5} )
(iii) ( sqrt{mathbf{0 . 6}} )
( (i v)(0.009)^{frac{1}{3}} )
( (v)(0.999)^{frac{1}{10}} )
( (v i)(15)^{frac{1}{4}} )
(vii) ( (26)^{frac{1}{3}} )
( (text { viii })(255)^{frac{1}{4}} )
( (i x)(82)^{frac{1}{4}} )
( (x)(401)^{frac{1}{2}} )
( (x i)(0.0037)^{frac{1}{2}} )
12
727From a variable point of an ellipse ( frac{x^{2}}{d^{2}}+ ) ( frac{boldsymbol{y}^{2}}{boldsymbol{b}^{2}}=1 ) normal is drawn to the ellipse.
Find the maximum distance of the
normal from the centre of the ellipse.
( mathbf{A} cdot a+b )
B. ( a-b )
c. ( a^{2}-b^{2} )
D. ( -a+b )
12
728A particle moving on a curve has the
position given by ( boldsymbol{x}=boldsymbol{f}^{prime}(boldsymbol{t}) sin boldsymbol{t}+ )
( f^{prime prime}(t) cos t, y=f^{prime}(t) cos t-f^{prime prime}(t) sin t ) at
time ( t ) where ( f ) is a thrice-differentiable function.Then the velocity of the particle
at time ( t ) is
( mathbf{A} cdot f^{prime prime prime}(t) )
B . ( f^{prime}(t)+f^{prime prime prime}(t) )
c. ( f^{prime}(t)+f^{prime prime}(t) )
D. ( f^{prime}(t)-f^{prime prime prime}(t) )
12
729A particular point moves on the parabola ( y^{2}=4 a x ) in such a way that
its projection on ( y ) -axis has a constant
velocity. Then its projection on ( x ) -axis
moves with
This question has multiple correct options
A. constant velocity
B. constant acceleration
c. variable velocity
D. variable acceleration
12
7309. x and y be two variables such that x > 0 and xy = 1. Then
the minimum value of x + y is
(a) 2 (6) 3 (c) 4 (d) o
12
731If ( frac{x^{2}}{f(4 a)}=frac{y^{2}}{fleft(a^{2}-5right)} ) respresents and
ellipse with major axis as y-axis and ( boldsymbol{f} ) is a decreasing function, then
A ( . a in(-infty, 1) )
B . ( a in(5, infty) )
c. ( a in(1,4) )
D. ( a in(-1,5) )
12
732What is the value of ( x, ) when ( f(x)= )
( 6+(x-2)^{2} ) is at its minimum?
A . -6
B. –
c. 0
D. 2
E . 5
12
733For the function ( f(x)=x cos frac{1}{x}, x geq 1 )
which of the following is/are true??
This question has multiple correct options
A. There is at least one ( x ) in the interval ( [1, infty) ) for which ( f(x+2)-f(x)2 ) for all ( x ) in the interval ( [1, infty) )
D. ( f^{prime}(x) ) is strictly decreasing in the interval ( [1, infty) )
12
734If displacement ( s ) at time ( t ) is ( s=t^{3}- ) ( 3 t^{2}-15 t+12, ) then acceleration at
time ( t=1 ) sec is
A ( cdot ) 6units ( / ) sec ( ^{2} )
B. – 6units/sec ( ^{2} )
( c cdot 0 )
D. 4units ( / )sec( ^{2} )
12
735Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+boldsymbol{5} boldsymbol{s} boldsymbol{i} boldsymbol{n}^{2} boldsymbol{x} ) be
an increasing function on the set ( R ) Then, a and b satisfy
A ( cdot a^{2}-3 b-15>0 )
B . ( a^{2}-3 b+15>0 )
c. ( a^{2}-3 b+150 ) and ( b>0 )
12
736A balloon which always remains spherical, is being inflated by pumping in 900 cubic centimetres of gas per second. The rate at which the radius of
the balloon is increasing when the radius is ( 15 mathrm{cm} ) is.
A ( cdot frac{1}{pi} c m / s e c )
B ( cdot frac{2}{pi} c m / ) sec
c. ( pi c m / s e c )
D. ( frac{pi}{2} mathrm{cm} / mathrm{sec} )
12
73712.
Consider the following statments in S and R (2000S)
S: Both sin x and cos x are decreasing functions in the
interval
,
R: If a differentiable function decreases in an interval
(a, b), then its derivative also decreases in (a, b).
Which of the following is true ?
(a) Both S and R are wrong
(b) Both S and R are correct, but R is not the correct
explanation of S
Sis correct and Ris the correct explanation for S
(d) Sis correct and R is wrong
12
738The surface area of a spherical balloon
is increasing at the rate of ( 2 mathrm{cm}^{2} / mathrm{sec} )
At what rate is the volume of the balloon
is increasing when the radius of the
balloon is ( 6 mathrm{cm} ? )
12
739Let ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}^{2}+mathbf{2}}{[boldsymbol{x}]}, mathbf{1} leq boldsymbol{x} leq mathbf{3}, ) where [
represents greatest integer function,
then
A ( . f(x) ) is increasing in [1,3]
B. Least value of ( f(x) ) is 3
c. Greatest value of ( f(x) ) is ( frac{11}{2} )
D. ( f(x) ) has no greatest value
12
740Find the point of local maxima & local
minima of the function
( boldsymbol{f}(boldsymbol{x})=sin ^{4} boldsymbol{x}+cos ^{4} boldsymbol{x} boldsymbol{i n}[mathbf{0}, boldsymbol{pi}] )
12
741Find the set of values of ( a ) for which
( f(x)=x+cos x+a x+b ) is
increasing on ( boldsymbol{R} )
12
742Prove that the following functions are
increasing on ( boldsymbol{R} )
(i) ( f(x)=3 x^{5}+40 x^{3}+240 x )
(ii) ( f(x)=4 x^{3}-18 x^{2}+27 x-27 )
12
743A particle moves along the ( x ) -axis obeying the equation ( boldsymbol{x}=boldsymbol{t}(boldsymbol{t}-mathbf{1})(boldsymbol{t}- )
2), where ( x ) is in meter and ( t ) is in
second
Find the acceleration of the particle when its velocity is zero.
12
744For all x in [0, 1], let the second derivative f” (x) of a function
f(x) exist and satisfy f” (x) < 1. Iff(0)=f(1), then show that
f(x)<1 for all x in [0, 1]. (1981 – 4 Marks)
12
745Write the set of values of ( k ) for which
( f(x)=k x-sin x ) is increasing on ( R )
12
746A stone is dropped into a quiet lake and
waves move in a circle at a speed of 3.5 ( mathrm{cm} / mathrm{sec} . ) At the instant when the radius
of the circular wave is ( 7.5 mathrm{cm}, ) how fast is the enclosed area increasing?
12
747Find the greatest and the least values of the following function:
( f(x)=cos 3 x-15 cos x+8 ) where
( boldsymbol{x} epsilonleft[frac{boldsymbol{pi}}{mathbf{3}}, frac{boldsymbol{3} boldsymbol{pi}}{boldsymbol{2}}right] )
12
74827. Let a, b e R be such that the function f given byf(x)= In x+
bx2 + ax, x = 0 has extreme values at x=-1 and x=2
Statement-1: f has local maximum at x = -1 and at
x=2.
Statement-2 : a = 1 and b= -1
[2012]
. 4
(a) Statement-1 is false, Statement-2 is true.
(b) Statement-1 is true, statement-2 is true; statement-2 is
a correct explanation for Statement-1.
Statement-1 is true, statement-2 is true; statement-2 is
not a correct explanation for Statement-1.
(d) Statement-1 is true, statement-2 is false.
12
749Illustration 2.36 Find the minimum and maximum values
of the function y = x – 3x + 6. Also find the values of x at
which these occur.
12
750The displacement of a body varies with
the time as ( S=t^{3}+3 t^{2}+2 t-1 . ) If the
velocity at ( t=4 sec ) is ( 2+12 K m / s )
then find ( k )
( A cdot 6 )
B. 3
( c cdot 2 )
D.
12
751Find the maximum value of ( f(x)=left(frac{1}{x}right)^{x} )12
752The function ( boldsymbol{y}=boldsymbol{a} log |boldsymbol{x}|+boldsymbol{b x}^{2}+boldsymbol{x} )
has its extremum values at ( x=-1 ) and
( x=2 ) then
A ( a=2, b=-1 )
– ( 1, a=2, b=2=2 )
В. ( a=2, b=-1 / 2 )
c. ( a=-2, b=1 / 2 )
D. None of these
12
753Arrange ( A, B, C, D ) in ascending order
A) Maximum value of ( sin 5 x )
B) Minimum value of ( x+frac{1}{x}(x>0) )
C) Minimum value of ( 4 times 2^{left(x^{2}-3right)^{3}+27} )
D) Minimum value of ( 5 sin ^{2} x+3 cos ^{2} x )
( A cdot A, B, D, C )
B. A, C, B, D
c. ( A, D, B, C )
D. B, D, A, C
12
754Coffee is coming out from a conical filter, with height and diameter both ( 25 mathrm{cm} ) into a cylindrical coffee pot with
diameter ( 15 mathrm{cm} . ) The constant rate at
which coffee comes out from the filter
into the pot is ( 100 mathrm{cm}^{3} / ) min. The rate in
cm / min at which the level in the pot is rising at the instance when the coffee in the pot is ( 10 mathrm{cm} ), is
A ( cdot frac{9}{16 pi} )
В ( cdot frac{25}{9 pi} )
c. ( frac{5}{3 pi} )
D. ( frac{16}{9 pi} )
12
75512. The function f(x) = * +2 has a local minimum at [2006]
(a) x=2
(C) x=0
(b) x= -2
(d) x=1
12
756Find the local maxima and local
minima for the given function and also find the local maximum and local
minimum values ( f(x)=sin x- )
( cos x, 0<x<2 pi )
12
757The function ( y=sqrt{2 x-x^{2}} )
A. increases in (0,1) but decreases in (1,2)
B. decreases in (0,2)
C. increases in (1,2) but decreases in (0,1)
D. increases in (0,2)
12
758( y=[x(x-3)]^{2} ) is increasing when
A ( cdot 0<x<frac{3}{2} )
B. ( 0<x<infty )
c. ( -infty<x<0 )
D. ( 1<x<3 )
12
759If ( x ) and ( y ) are sides of two squares such
that ( y=x-x^{2} ). Find the rate of change
of area of second square (side ( y ) ) with respect to area of the first square (side
( x) ) when ( x=1 mathrm{cm} )
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. 3
12
760The function has ( boldsymbol{f}(boldsymbol{x})= )
( (log (x-1))^{2}(x-1)^{2} ) has
A. local extremum at ( x=1 )
B. point of inflection at ( x=1 )
c. local extremum at ( x=2 )
D. point of inflection at ( x=2 )
12
761Find the values of ( a ) and ( b ), if the slope of
the tangent to the curve ( boldsymbol{x} boldsymbol{y}+boldsymbol{a} boldsymbol{x}+ )
( b y=2 ) at (1,1) is 2
12
762Bacteria multiply at a rate proportional to the number present. If the original number ( N ) double in 3 hours, the
number of the bacteria will be ( 4 N ) is (in
hours)
( mathbf{A} cdot mathbf{6} )
B. 4
( c .5 )
D.
12
763A computer solved several problems in succession. The time it took the
computer to solve each successive
problem was the same number of times smaller than the time it took it to solve
the preceding problem. How many problems were suggested to the
computer if it spent ( 63.5 mathrm{mm} ) to solve al the problems except for the first, 127 ( mathrm{mm} ) to solve all the problems except for the last one, and ( 31.5 mathrm{mm} ) to solve all the
problems except for the first two?
12
764Mark the correct alternative of the
following. For the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}+frac{mathbf{1}}{boldsymbol{x}} )
A. ( x=1 ) is a point of maximum
B. ( x=-1 ) is a point of minimum
c. Maximum value> minimum value
D. Maximum value < minimum value
12
765Consider the function ( f(x)=0.75 x^{4}- )
( x^{3}-9 x^{2}+7 )
Consider the following statements:
1. The function attains local minima at
( x=-2 ) and ( x=3 )
2. The function increases in the interval
(-2,0)
Which of the above statements is/are
correct?
A. 1 only
B. 2 only
c. Both 1 and 2
D. Neither 1 nor
12
766The radius of a spherical soap bubble is increasing at the rate of ( 0.2 mathrm{cm} / mathrm{sec} ) Find the rate of increase of its surface
area, when the radius is ( 7 mathrm{cm} )
12
767The function ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}}{1+boldsymbol{x} tan boldsymbol{x}} ) has
A. one point of minimum in the interval ( (0, pi / 2) )
B. one point of maximum in the interval ( (0, pi / 2) )
C. no point of maximum,no point of minumum in the interval ( (0, pi / 2) )
D. two points of maximum in the interval ( (0, pi / 2) )
12
768Let the function ( f(x)=sin x+cos x )
be defined in ( [0,2 pi], ) then ( f(x) )
A ( cdot ) increases in ( left(frac{pi}{4}, frac{pi}{2}right) )
B. decreases in ( left[frac{pi}{4}, frac{5 pi}{4}right] )
C . increases in ( left[0, frac{pi}{4}right] cup[pi, 2 pi] )
D. decreases in ( left[0, frac{pi}{4}right) cupleft(frac{pi}{2}, 2 piright] )
12
769Let the function ( f(x) ) be defined as
follows: ( f(x)= )
( left{begin{array}{cc}x^{3}+x^{2}-10 x & -1 leq x<0 \ cos x & 0 leq x<frac{pi}{2} \ 1+sin x & frac{pi}{2} leq x leq piend{array}, ) then right.
which of the following statement(s) is/are correct
This question has multiple correct options
A. Local maximum at ( x=0 )
B . Local maximum at ( x=frac{pi}{2} )
c. Absolute maxima at ( x=-1 )
D. Absolute minima at ( x=pi )
12
770Find the derivative of ( f(x)=tan x ) at
( boldsymbol{x}=mathbf{0} )
12
771Two roads ( O A ) and ( O B ) intersect at an
angle fo ( 60^{circ} . ) A car driver approaches 0
from ( A, ) where ( A O=800 ) metres, at a
uniform speed of 20 metres per second.
Simultaneously a runner starts running
from ( boldsymbol{O} ) towards ( boldsymbol{B} ) at uniform speed of ( mathbf{5} )
metres per second. Find the time when the car and the runner are closest.
A ( cdot frac{240}{7} ) sec
B. ( frac{260}{7} ) sec
c. ( frac{250}{7} mathrm{sec} )
D. ( frac{210}{7} mathrm{sec} )
12
772The radius of the balloon is variable.
Find the rate of change of its volume, when the radius is ( 5 mathrm{cm} ) ?
12
773The intercept on x-axis made by tangent to the curve, ( y=int_{0}^{x}|t| d t, x in R, ) which
are parallel to the line ( y=2 x, ) are equal
to
A . ±1
B. ±2
( c .pm 3 )
( mathrm{D} cdot pm 4 )
12
774( operatorname{Let} y=left(x+frac{4}{x^{2}}right) ) and ( x in+R )
The minimum value of ( y ) is
A . -4
B. 3
c. 8
D. 7
12
775( f(x)=x^{3}+a x^{2}+b x+c ) has a max.
at ( x=-1 ) and ( min . ) at ( x=3 . ) Determine
the constants ( a, b, c )
( mathbf{A} ldots a=-3, b=-3, c=0 )
В ( ldots a=-3, b=-9, c=3 )
( mathbf{c} ldots a=-3, b=9, c=9 )
D ( ldots a=3, b=9, c=-3 )
12
776The real number ( x ) when added to its
inverse gives the minimum value of the
( operatorname{sum} operatorname{at} x= )
( (boldsymbol{x} in+boldsymbol{R}) )
( mathbf{A} cdot mathbf{1} )
B. 3
( c cdot 2 )
D.
12
777( f(x) ) is cubic polynomial which has local maximum at ( mathbf{x}=-1 . ) If ( mathbf{f}(mathbf{2})= )
( mathbf{1 8}, mathbf{f}(mathbf{1})=-mathbf{1} ) and ( mathbf{f}(mathbf{x}) ) has local
minima at ( mathbf{x}=mathbf{0}, ) then
This question has multiple correct options
A. the distance between (-1,2) and ( (a, f(a)), ) where ( x=a ) is the point of local minima is ( 2 sqrt{5} )
B. f(x) is increasing for ( x in[1,2 sqrt{5} )
( mathrm{c} . mathrm{f}(mathrm{x}) ) has local minima at ( mathrm{x}=1 )
D. the value of ( f(0)=5 )
12
778Given ( boldsymbol{f}(boldsymbol{x})=cos ^{2} boldsymbol{x}, ) find whether ( boldsymbol{f}(boldsymbol{x}) )
is increasing or decreasing in the range ( left[0, frac{pi}{2}right] )
12
779The sides of an equilateral triangle are increasing at the rate of ( 2 mathrm{cm} / mathrm{s} ). The rate at which the area increases when
the side is ( 10 mathrm{cm} ), is
( mathbf{A} cdot sqrt{3} c m^{2} / s )
B. ( 10 mathrm{cm}^{2} / mathrm{s} )
c. ( 10 sqrt{3} mathrm{cm}^{2} / mathrm{s} )
D. ( frac{10}{sqrt{3}} c m^{2} / s )
12
780A particle moves along a straight line
according to the law ( s=16-2 t+3 t^{3} )
where ( s ) metres is the distance of the
particle from a fixed point at the end of ( t ) second. The acceleration of the particle
at the end of 2 s is
A ( cdot 3.6 m / s^{2} )
B. ( 36 m / s^{2} )
c. ( 36 k m / s^{2} )
D. ( 360 m / s^{2} )
12
781The maximum value of ( left(frac{log x}{x}right) ) is
A ( cdotleft(frac{1}{e}right. )
B. ( frac{2}{e} )
( c cdot e )
D.
12
782If there is an error of ( 2 % ) in measuring the length of a simple pendulum, then percentage error in its period is
A . ( 1 % )
B. 2%
( c .3 % )
D. ( 4 % )
12
783: The function ( f(x)=2 x^{3}-3 x^{2}- )
( 12 x+8 ) attains minimum value at ( x= )
2
II: The function of ( f(x)=x^{4}-6 x^{2}+ )
( 8 x+11 ) attains minimum value at ( x= )
2 which of the above statements are
true
A. onlyı
B. only II
c. both I and II
D. neither I nor II
12
784***
v
y
u
taurus
7. The sides of an equilateral triangle are increasing at the
rate of 2 cm/sec. The rate at which the area increases, when
the side is 10 cm is
(a) V3 sq. unit/sec (b) 10 sq. unit/sec
10
(c) 103 sq. unit/sec
(d) to sq. unit/sec
12
785The total revenue in Rupees received
from the sale of ( x ) units of a product
is given by ( boldsymbol{R}(boldsymbol{x})=mathbf{3} boldsymbol{x}^{2}+mathbf{3 6} boldsymbol{x}+mathbf{5} . ) The
marginal revenue, when ( boldsymbol{x}=mathbf{1 5} ) is.
A . 116
B. 96
( c cdot 90 )
D. 126
12
786Consider the following statements:
Statement I
( x>sin x ) for all ( x>0 )
Statement II:
( f(x)=x-sin x ) is an increasing
function for all ( x>0 )
Which one of the following is correct in respect of the above statements?
A. Both Statements I and II are true and Statement II is the correct explanation of statement
B. Both Statements I and II are true and Statement II is the not correct explanation of Statement
c. statement lis true but Statement II is false
D. Statement I is true but Statement II is true
12
787Find the minimum distance of any point on the curve ( x^{2}+y^{2}+2 x y=8 ) from
the origin.
12
788Find intervals in which the function ( operatorname{given} operatorname{by} f(x)=sin 3 x, x inleft[0, frac{pi}{2}right] ) is
decreasing.
12
789If ( y=7 x-x^{3} ) and ( x ) increases at the
rate of 4 units per second, how fast is
the slope of the curve changing when
( x=2 ? )
12
790If ( y=6 x-x^{3} ) and ( x ) increases at the
rate of 5 units per second, the rate of
change of slope when ( x=3 ) is
A. -90 units/sec
B. 90 units/ sec
c. 180 units/sec
D. -180 units/sec
12
791Find the value of ( theta ) for attaining a max
value of ( sin ^{p} theta cos ^{q} theta ) is
A ( cdot tan ^{-1} frac{p}{q} )
B. ( tan ^{-1} frac{sqrt{p}}{sqrt{q}} )
c. ( tan ^{-1}(p+q) )
( mathbf{D} cdot tan ^{-1} q )
12
792cosine of the angle of intersection of curves ( y=3^{x-1} ln x ) and ( y=x^{x}-1 ) is
( mathbf{A} cdot mathbf{1} )
B. ( 1 / 2 )
( c cdot 0 )
D. ( 1 / 3 )
12
793A stone dropped into a pond of still water sends out concentric circular
waves from the point of disturbance of water at the rate of ( 4 mathrm{cm} / ) sec. Find the rate of change of disturbed area at the instant when the radius of wave ring is
( mathbf{1 5} c m )
12
7949. A spherical balloon is being inflated at the rate of
35 cc/min. The rate of increase of the surface area of the
balloon when its diameter is 14 cm is
(a) 7 sq. cm/min (b) 10 sq. cm/min
(c) 17.5 sq. cm/min (d) 28 sq. cm/min
.
12
795The rate of change of the area of a circle with respect to its radius ( r ) at ( r=6 c m )
is.
( mathbf{A} cdot 10 pi )
в. ( 12 pi )
( c cdot 8 pi )
D. ( 11 pi )
12
796The equation normal to the curve ( boldsymbol{x}^{2 / 3}+boldsymbol{y}^{2 / 3}=boldsymbol{a}^{2 / 3} ) at the point ( (boldsymbol{a}, boldsymbol{0}) ) is
( mathbf{A} cdot x=a )
B. ( x=-a )
( mathbf{c} cdot y=a )
D. ( y=-a )
12
797The largest term of the sequence ( a_{n}= ) ( frac{n}{left(n^{2}+10right)} ) is
A ( cdot frac{3}{19} )
B. ( frac{2}{13} )
c. 1
D.
12
798If the line ( a x+b y+c=0 ) is a normal
to the rectangular hyperbola ( boldsymbol{x} boldsymbol{y}=mathbf{1} )
then
This question has multiple correct options
A ( . a>0, b>0 )
в. ( a>0, b<0 )
c. ( a0 )
D. ( a<0, b<0 )
12
799Example 2.4 Two bodies start moving in the same straight
line at the same instant of time from the same origin. The first
body moves with a constant velocity of 40 ms, and the second
starts from rest with a constant acceleration of 4 ms. Find the
time that elapses before the second catches the first body. Find
also the greatest distance between them prior to it and time
at which this occurs.
12
800A point on the parabola ( y^{2}=18 x ) at
which the ordinate increases at twice
the rate of the abscissa is
в. (2,-4)
( ^{mathrm{c}} cdotleft(-frac{9}{8}, frac{9}{2}right) )
D ( cdotleft(frac{9}{8}, frac{9}{2}right) )
12
801If the length of the diagonal of a square is increasing at the rate of ( 0.2 mathrm{cm} / mathrm{sec} ) then rate of increase of its area when its
side is ( 30 / sqrt{2} mathrm{cm}, ) is
( mathbf{A} cdot 3 mathrm{cm}^{2} / mathrm{sec} )
в. ( frac{6}{sqrt{2}} mathrm{cm}^{2} / mathrm{sec} )
( c cdot sqrt[3]{2} mathrm{cm}^{2} / mathrm{sec} )
D. ( 6 mathrm{cm}^{2} / mathrm{sec} )
12
802The function ( f(x)=frac{x}{1+x tan x} ) has
a point of minimum in the interval ( left(0, frac{pi}{2}right) )
Bne point of maximum in the interval ( left(0, frac{pi}{2}right) )
C. No point of maximum, no point of minimum in the interval ( left(0, frac{pi}{2}right) )
Two points of maxima in the interval ( left(0, frac{pi}{2}right) )
12
803The maximum value of ( frac{log x}{x} ) in ( (2, infty) )
is
A. 5
B. ( frac{5}{e} )
( mathbf{c} cdot e^{e} )
D.
12
804A particle is moving in a straight line such that its distance at any time ( t ) is given by ( s=frac{1}{4} t^{4}-2 t^{3}+4 t^{2}-7 . ) Find
twhen its velocity is maximum( ( t_{v} ) ) and
acceleration minimum(t ( _{a} ) ).
A ( cdot t_{v}=2-2 / sqrt{3}, t_{a}=4 )
В ( cdot t_{v}=2, t_{a}=1 )
C ( cdot t_{v}=2-2 / sqrt{3}, t_{a}=2 )
D. ( t_{v}=-2 / sqrt{3}, t_{a}=2 )
12
805Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. ( (26)^{frac{1}{3}} )12
806Solve: ( boldsymbol{x}^{4}-boldsymbol{x}^{mathbf{3}}+boldsymbol{x}^{2}-boldsymbol{x}+mathbf{1}=mathbf{0} )12
807( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}| ) has
A. minimum at ( x=0 )
B. maximum at ( x=0 )
c. neither a maximum nor a minimum at ( x=0 )
D. none of these
12
808Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) if ( boldsymbol{y}=[boldsymbol{x}+sqrt{boldsymbol{x}+} sqrt{boldsymbol{x}}]^{1 / 2}, ) at ( boldsymbol{x}= )
( mathbf{1} )
A. ( frac{3+4 sqrt{2}}{8 sqrt{2}(sqrt{1+sqrt{2}})} )
B. Not defined
( c cdot 0 )
D.
12
809A spherical balloon is being inflated so that its volume increase uniformly at
the rate of ( 40 mathrm{cm}^{3} / ) minute. The rate of
increase in its surface area when the
radius is ( 8 mathrm{cm} ) is
( mathbf{A} cdot 10 mathrm{cm}^{2} / mathrm{minute} )
B . ( 20 mathrm{cm}^{2} / ) minute
( mathbf{c} cdot 40 mathrm{cm}^{2} / mathrm{minute} )
D. none of these
12
810If the sum of the lengths of the hypothesis and another side of a right angle is given, show that the area of the triangle is maximum when the angle
between these sides is ( frac{pi}{3} )
12
811If ( f ) and ( g ) are two decreasing function such that ( f o g ) is defined, then fog will
be
A. increasing function
B. decreasing function
c. neither increasing nor decreasing
D. None of these
12
812In a ( Delta A B C ) the sides b and ( c ) are given.
If there is an error ( Delta A ) in measuring
angle ( A, ) then the error ( Delta a ) in side a is
given by
( ^{text {A }} cdot frac{S}{2 a} Delta A )
в. ( frac{2 S}{a} Delta A )
( c cdot b c sin A Delta A )
D. none of these
12
813Using differential, find the approximate value of the following:
( sqrt{401} )
12
814Find the absolute maximum and
minimum values of a function ( f ) given
by ( f(x)=2 x^{3}-15 x^{2}+36 x+1 ) on the
interval ( [mathbf{1}, mathbf{5}] )
12
815The acceleration of a particle starting from rest moving in a straight line with uniform acceleration is ( 8 m / s^{2} . ) The
time taken by the particle to move the second metre is
A ( cdot frac{sqrt{2}-1}{2} s )
B. ( frac{sqrt{2}+1}{2} s )
( mathbf{c} cdot(1+sqrt{2}) s )
D. ( (sqrt{2}-1) s )
12
816defined by ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{l}k-2 x, text { if } x leq 1 \ 2 x+3, text { if } x>-1end{array}right} ), if has a loca
minimum at ( x=-1, ) then a pair
12
817The displacement ( x ) of a particle moving in one dimension under the action of a
constant force is related to time ( t ) by the
equation ( t=sqrt{x}+3, ) where ( x ) is in
meter and ( t ) is in second. Find the
displacement of the particle when its velocity is zero.
12
81821. The point(s) on the curve y3 + 3×2 = 12y where the tange
is vertical, is (are)
(2002)
(c) (0,0)
12
81927.
The tangent to the
le tangent to the curve y = et drawn at the point (c, e)
intersects the line joining the points (C – 1, e ) and
(c +1, ec+1)
(2007 -3 marks)
(a) on the left of x = c . (b) on the right of x = 0
(C) at no point
(d) at all points
12
820Find the point on the parabola ( y^{2}=18 x )
at which the ordinate increases at twice
the rate of the abscissa.
12
821A man ( 1.5 mathrm{m} ) tall walks away from a
lamp post ( 4.5 mathrm{m} ) high at the rate of 4 km/hr. How fast is the farther end of
A. ( 4 mathrm{km} / mathrm{hr} )
B. ( 2 mathrm{km} / mathrm{hr} )
c. ( 6 mathrm{km} / mathrm{hr} )
D. None of these
12
822In which one of the following intervals
is the function ( f(x)=x^{2}-5 x+6 )
decreasing?
A ( cdot x<frac{5}{2} )
в. ( xfrac{2}{5}} )
D. none
12
823Let ( f ) be a function defined on ( R ) (the set
of all real numbers) such that ( mathbf{f}^{prime}(mathbf{x})= ) ( 2010(x-2009)(x-2010)^{2}(x- )
2011)( ^{3}(x-2012)^{4}, ) for all ( x in R ). If ( g ) is a
function defined on ( mathbf{R} ) with values in the
interval ( (0, infty) ) such that ( f(x)= ) ( ln (mathrm{g}(mathrm{x})), ) for all ( mathrm{x} in mathrm{R}, ) then the number
of points in ( mathbf{R} ) at which ( mathbf{g} ) has a local maximum is
A .
B.
( c cdot 2 )
D. 3
12
824The minimum value of ( 4 cos ^{2} x+ )
( 5 sin ^{2} x ) is
12
825Assertion
( f(x) ) is increasing with concavity
upwards, then concavity of ( boldsymbol{f}^{-1}(boldsymbol{x}) ) is
also upwards.
Reason
If ( boldsymbol{f}(boldsymbol{x}) ) is decreasing function with
concavity upwards, then concavity of
( f^{-1}(x) ) is also upwards
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 and Reason are correct
12
826(a)
1
(0
)
2
The shortest distance between the line y – x = 1 and the
curve x=y2 is:
[2009]
(a
12
827Assertion
Let ( f ) and ( g ) be increasing and
decreasing functions respectively from ( [0, infty] ) to ( [0, infty] . operatorname{Let} h(x)=f(g(x)) . ) If
( h(0)=0, ) then ( h(x) ) is always zero
Reason
( h(x) ) is an increasing function of ( x )
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
828Find the approximate error in the volume of a cube with edge ( x mathrm{cm}, ) when the edge is increased by ( 2 % )
A . ( 4 % )
в. ( 2 % )
( c .6 % )
D. ( 8 % )
12
829Function ( x-sin x ) has
A. a maxima
B. a minima
c. a maxima and a minima
D. no maxima and no minima
12
830Assertion
The largest term in the sequence ( a_{n}= ) ( frac{n^{2}}{n^{3}+200}, n in N ) is the 7 th term.
Reason
The function ( f(x)=frac{x^{2}}{x^{3}+200} ) attains
local maxima at ( x=7 )
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
831Mark the correct alternative of the
following. The maximum value of ( x^{1 / x}, x>0 ) is?
A ( cdot e^{1 /} )
( ^{mathrm{B}}left(frac{1}{e}right) )
c. 1
D. None of these
12
832For ( boldsymbol{a} in[boldsymbol{pi}, boldsymbol{2} boldsymbol{pi}] ) and ( boldsymbol{n} in boldsymbol{Z}, ) the critical
points of ( boldsymbol{f}(boldsymbol{x})=frac{1}{mathbf{3}} sin boldsymbol{a} tan ^{3} boldsymbol{x}+ )
( (sin a-1) tan x+sqrt{frac{a-2}{8-a}} ) are
A . ( x=n pi )
B. ( x=2 n pi )
c. ( x=(2 n+1) pi )
D. None of these
12
833The top of a ladder 6 meters long is resting against a vertical wall.Suddenly the ladder begins to slide outwards. At the instant when the foot of the ladder is
4 meters from the wall, it is sliding away at the rate of ( 0.5 mathrm{m} / ) sec. How fast
is the top sliding downward at this
moment?
12
83443.
If the tangent to the curve, y=x3 + ax – b at the point
(1,-5) is perpendicular to the line, -x+y+4=0,
then which one of the following points lies on the curve?
JJEEM 2019-9 April (M)
(a) (-2,1)
(b) (-2,2)
(c) (2,-1)
(d) (2,-2)
12
835Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal.
1
( (255)^{overline{4}} )
12
836If the radius of a circle is diminished by
( 10 %, ) then its area is diminished by:
A . ( 10 % )
B. ( 19 % )
( c cdot 20 % )
D. ( 36 % )
12
837If ( e^{d y / d x}=x+1 ) given that when ( x= )
( mathbf{0}, boldsymbol{y}=mathbf{3} ) then minimum (local) value of
( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) ) is
12
838The approximate value of ( (1.0002)^{3000} ) is
A . 1.2
B. 1.4
c. 1.6
D. 1.8
12
839The two tangents to the curve ( a x^{2}+ )
( 2 h x y+b y^{2}=1, a>0 ) at the points
where it crosses the X-axis, are
A. parallel
B. perpendicular
C . inclined at an angle ( frac{pi}{4} )
D. None of these
12
840If an error of ( 1^{circ} ) is made in measuring
the angle of a sector of radius ( 30 mathrm{cm} ) then the approximate error in its area is
A ( cdot 450 mathrm{cm}^{2} )
В ( cdot 25 pi c m^{2} )
( mathbf{c} cdot 2.5 pi c m^{2} )
D. none of these
12
841Find the derivative of the following function at the indicated points. ( 2 cos x ) at ( x=frac{pi}{2} )12
842If a function ( f(x) ) has ( f^{prime}(a)=0 ) and
( f^{prime prime}(a)=0, ) then
A. ( x=a ) is a maximum for ( f(x) )
B. ( x=a ) is minimum for ( f(x) ),
c. It is difficult to say ( (a) ) and ( (b) )
D. ( f(x) ) is necessarily a constant function.
12
843I. If the curve ( y=x^{2}+b x+c ) touches
the straight line ( y=x ) at the point
(1,1) then ( b ) and ( c ) are given by 1,1
Il. If the line ( boldsymbol{P x}+boldsymbol{m} boldsymbol{y}+boldsymbol{n}=boldsymbol{0} ) is a
normal to the curve ( x y=1, ) then ( P> )
( mathbf{0}, boldsymbol{m}<mathbf{0} )
Which of the above statements is
correct
A. onlyı
B. only II
c. both I and II
D. Neither I nor II
12
844Consider the function.
( f(x)=3 x^{4}-20 x^{3}-12 x^{2}+288 x+1 )
In which one of the following intervals is the function decreasing?
( mathbf{A} cdot(-1,0) )
в. (0,2)
c. ( (2, infty) )
D. none of these
12
845The function ( f(x)=2 x^{3}-15 x^{2}+ )
( 36 x+6 ) is strictly decreasing in the
interval
A. (2,3)
(年) (2,3,3)
в. ( (-infty, 2) )
c. (3,4)
(年. ( (3,4)) )
D. ( (-infty, 3) cup(4, infty) )
E ( .(-infty, 2) cup(3, infty) )
12
84613.
If
2)(t-3)dt for all x € (0,00), then
(2012)
(a) f has a local maximum at x=2
(b) f is decreasing on (2,3)
(c) there exists somece (0,0), such that f'(c)=0
(d) f has a local minimum at x=3
12
847The value of ( x ) at which tangent to the
curve ( boldsymbol{y}=boldsymbol{x}^{3}-boldsymbol{6} boldsymbol{x}^{2}+boldsymbol{9} boldsymbol{x}+boldsymbol{4}, boldsymbol{0} leq boldsymbol{x} leq )
5 has maximum slope is
( mathbf{A} cdot mathbf{0} )
B. 2
( c cdot frac{5}{2} )
( D )
12
848The slope of the tangent to the curve ( boldsymbol{x}=mathbf{3} boldsymbol{t}^{2}+mathbf{1}, boldsymbol{y}=boldsymbol{t}^{3}-mathbf{1} ) at ( boldsymbol{x}=mathbf{1} ) is
A ( cdot frac{1}{2} )
B.
( c cdot-2 )
( D cdot infty )
12
849For which region is ( f(x)=3 x^{2}-2 x+ )
1 strictly increasing?
This question has multiple correct options
( mathbf{A} cdot(2,5) )
B. ( left(frac{1}{3}, inftyright) )
c. ( left(-1, frac{1}{3}right] )
D. ( left(-infty, frac{1}{3}right) )
12
850The difference between the greatest and the least value of ( boldsymbol{f}(boldsymbol{x})= ) ( cos ^{2} frac{x}{2} sin x, x in[0, pi] ) is
( A cdot frac{3 sqrt{3}}{8} )
в. ( frac{sqrt{3}}{8} )
( c cdot frac{3}{8} )
D. ( frac{1}{2 sqrt{2}} )
12
851A particle starts with some initial velocity with an acceleration along the direction of motion. Draw a graph
depicting the variation of velocity
( (v) )
along y-axis with the variation of displacement
( (s) ) along ( x ) -axis.
12
852The minimum value of ( boldsymbol{f}(boldsymbol{x})= ) ( max {x, 1+x, 2-x} ) is
A ( cdot frac{1}{2} )
B. ( frac{3}{2} )
c.
D. 0
E . 2
12
85310. Find all the tangents to the curve
y = cos(x + y), – 21 Sxs 21, that are parallel to the line
x+2y=0.
(1985 – 5 Marks)
12
854ff ( boldsymbol{y}=mathbf{8} boldsymbol{x}^{3}-boldsymbol{6} mathbf{0} boldsymbol{x}^{2}+mathbf{1 4 4} boldsymbol{x}+mathbf{2 7} ) is a
strictly decreasing function in the interval
A. (-5,6)
В ( cdot(-infty, 2) )
c. (5,6)
(年. ( 5,6,6) )
D. ( (3, infty) )
E ( .(2,3) )
12
855In the interval ( boldsymbol{x} epsilonleft(boldsymbol{e}^{2 r boldsymbol{pi}+boldsymbol{pi} / 4}, boldsymbol{e}^{2 boldsymbol{r} boldsymbol{pi}+boldsymbol{5} boldsymbol{pi} / boldsymbol{4}}right) )
in which the function ( f(x)= )
( sin left(log _{e} xright)+cos left(log _{e} xright) ) is
A. decreasing
B. increasing
c. const.
D. no monotonicity
12
856The maximum value of the function
( y=frac{1}{4 x^{2}+2 x+1} ) is
A ( cdot frac{4}{3} )
в. ( frac{5}{2} )
c. ( frac{13}{4} )
D. None of these
12
857A particle moves along the curve ( y= )
( (2 / 3) x^{3}+1 . ) Find the points on the
curve at which the y-coordinate is changing twice as fast as the ( x ) coordinate.
12
85850 COLUuus
The function f(x)=max {(1-x), (1+x), 2), xe(-0, 0) is
(a) continuous at all points
(1995)
(6) differentiable at all points
c) differentiable at all points except at x = 1 and x=-1
(d) continuous at all points except at x = 1 and x = -1,
where it is discontinuous
12
859The equation of tangent to the curve ( y=b e^{-x / a} ) where it cuts the ( y ) -axis is
A ( cdot frac{x}{a}+frac{y}{b}=1 )
B. ( frac{x}{a}+frac{y}{b}=-1 )
c. ( frac{x}{a}-frac{y}{b}=1 )
D. none of these
12
860Assertion
( f(x)=frac{1}{x-7} ) is decreasing ( forall x epsilon R )
{7}
Reason
( boldsymbol{f}^{prime}(boldsymbol{x})<mathbf{0} forall boldsymbol{x} neq mathbf{7} )
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
861The minimum value of ( 2^{left(x^{2}-3right)^{3}+27} ) is
( A cdot 2^{27} )
B.
( c cdot 2 )
D. None of the above
12
862For what values of a does the minimum
value of the function ( y=x^{2}-4 a x- )
( a^{4} ) assume the greatest value?
12
863If the distance of the point on ( y=x^{4}+ )
( 3 x^{2}+2 x ) which is nearest to the line
( boldsymbol{y}=2 boldsymbol{x}-mathbf{1} ) is ( mathbf{p}, ) Find ( mathbf{5} boldsymbol{p}^{2} )
12
864A swimming pool is to be drained for cleaning. If L represents the number of liters of water in the pool t seconds after
the pool has been plugged off to drain and ( L=200(10-t)^{2} . ) What is the
average rate at which the water flows out during the first 5 seconds ( left(frac{text {litres}}{text {sec}}right) )
7
12
865It is desired to construct a cylindrical
vessel of capacity 500 cubic metres open at the top.What should be the dimensions of the vessel so that the
material need is minimum, given that the thickness of the material used is 2
( mathrm{cm} )
( ^{mathbf{A}} cdot quad r=left(frac{100}{pi}right)^{1 / 3}, h=5left(frac{100}{pi}right)^{1 / 3} )
B. ( _{r}=left(frac{500}{pi}right)^{1 / 3}=h )
( r=left(frac{1000}{pi}right)^{1 / 3}, h=left(frac{125}{pi}right)^{1 / 3} )
( r=left(frac{20}{pi}right)^{1 / 3}, h=left(frac{100}{pi}right)^{1 / 3} )
12
866Find the maximum and the minimum
values, if any, without using derivatives
of the following function:
( f(x)=4 x^{2}-4 x+4 ) on ( R )
12
867A window is in the shape of a rectangle surmounted by a semi circle. If the perimeter of the window is of fixed
length I then the maximum area of the window is
A ( cdot frac{I^{2}}{2 pi+4} )
B. ( frac{I^{2}}{pi+8} )
c. ( frac{I^{2}}{2 pi+8} )
D. ( frac{I^{2}}{8 pi+4} )
12
868( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}+frac{1}{x}, boldsymbol{x} neq 0, ) then
A. ( f(x) ) has no point of local maxima
B. ( f(x) ) has no point local minima
C ( . f(x) ) has exactly has one point of local minima
D. ( f(x) ) has exactly has two point of local minima
12
86910. A population p(t) of 1000 bacteria introduced into nutrient
medium grows according to the relation p(t) = 1000 +
1000t
“. The maximum size of this bacterial population
100+ 2
is
(a) 1100
(c) 1050
(b) 1250
(d) 5250
12
870Calculate the rate of flow of glycerin of
density ( 1.25 times 10^{3} k g / m^{3} ) through the
conical section of a pipe, if the radii of its ends are ( 0.1 m ) and 0.04 m and the
pressure drop across its length is
( mathbf{1 0} N / boldsymbol{m}^{2} )
A ( cdot 6.43 times 10^{-4} m^{3} / s )
B. ( 5.43 times 10^{-4} mathrm{m}^{3} / mathrm{s} )
c. ( 5.44 times 10^{-3} mathrm{m}^{3} / mathrm{s} )
D. ( 6.43 times 10^{-3} mathrm{m}^{3} / mathrm{s} )
12
87131. Let f(x) be a polynomial of degree four having extreme values
[
f(x)
at x= 1 and x=2. If lim
1+-
then f(2) is equal to :
x>0L
(a) o
(6) 4
(6 -8
[JEEM 2015]
(d) -4
12
872A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases
when the radius is ( 15 mathrm{cm} )
12
873( operatorname{Let} h(x)=f(x)-(f(x))^{2}+(f(x))^{3} ) for
every real number ( x, ) then
A. ( h ) is increasing whenever ( f ) is increasing and decreasing whenever ( f ) is decreasing
B. ( h ) is increasing whenever ( f ) is decreasing
c. ( h ) is decreasing whenever ( f ) is increasing
D. Nothing can be said in general
12
874The maximum value of ( x^{1 / x} ) is
A ( cdot frac{1}{e^{e}} )
B.
( mathbf{c} cdot e^{1 / e} )
D.
12
875Find an angle ( theta, 0<theta<frac{pi}{2}, ) which
increases twice as fast as its sine.
12
87615.
Let f(x)= 11
|xl, for 0 < x < 2 1
then at x=0, f has
for x = 0
(a) a local maximum
(c) a local minimum
(2000)
(b) no local maximum
(d) no extremum
12
877Which one of the following curves cut the parabola
y2 = 4ax at right angles?
(1994)
(a) x2 + y2 = a?
(b) y=e-x/2a
(c) y = ax
(d) x2 = 4ay
12
878The real value of ( k ) for which ( f(x)= ) ( x^{2}+k x+1 ) is increasing on ( (1,2), ) is
A . -2
B. – 1
( c .1 )
D. 2
12
879Illustration 3.42 If a + B = 90°, find the maximum value
of sin a sin ß.
12
880The function ( f(x)=x^{2}+2 x-5 ) is
strictly increasing in the interval
B ( cdot(-infty,-1] )
( c cdot[-1, infty) )
D. ( (-1, infty) )
12
8819
10. Divide 20 into two parts such that the product of one part
and the cube of the other is maximum. The two parts are
(a) (10, 10)
(b) (5, 15)
(c) (13,7)
(d) None of these
12
882The function ( f(x)=x^{3}-3 x^{2}+6 ) is an
increasing function for:
A. ( 0<x<2 )
B. ( x2 ) or ( x<0 )
D. all ( x )
12
88335. The normal to the curve y(x – 2)(x-3)=x+6 at the point
where the curve intersects the y-axis passes through the
JEE M 2017]
point:
12
884A small hole is made at the button of a
symmetrical jar as shown in figure.
liquid is filled into the jar upto a certain
height. The rate of descension of liquid
is independent of the level of liquid in
the jar. Then the surface of jar is a
surface of revolution of the curve
( mathbf{A} cdot y=k x^{4} )
( mathbf{B} cdot y=k x^{2} )
( mathbf{c} cdot y=k x^{3} )
D. ( y=k x^{5} )
12
885A stone is dropped into a quiet lake and waves move in circles at a speed of 4 ( mathrm{cm} / mathrm{sec} . ) At the instant when the radius
of the circular wave is ( 10 mathrm{cm}, ) how fast is the enclosed area increasing?
12
886Given that carbon ( 14left(C_{14}right) ) decays at a
constant rate in such a way that it
reduces to ( 50 % ) in 5568 years. Find the age of an old wooden piece in the carbon
is only ( 12 frac{1}{2} % ) of the original.
12
887Show that ( f(x)=sin x ) is an increasing
function on ( (-pi / 2, pi / 2) ? )
12
888( f(x)=sin x ) -ax is decreasing in ( R ) if
( mathbf{A} cdot mathbf{a}>1 )
B. ( afrac{1}{2}} )
D. ( a<frac{1}{2} )
12
889If ( x ) changes from 4 to 4.01 then find the
approximate change in ( log x )
12
890OLLOIOL 21001
20. Given P(x)=x4 +ar3 + bx2 + cx + d such that x=0 is the only
real root of P'(x) = 0. If P(-1)< P(1), then in the interval
[-1,1]:
[2009]
(a) P(-1) is not minimum but P(1) is the maximum of P
(b) P(-1is the minimum but P(1) is not the maximum of P
(C) Neither P(-1) is the minimum nor P(1) is the maximum
of P
(d) P(-1) is the minimum and P(1) is the maximum ofP
12
891The value of ( K ) in order that ( f(x)= )
( sin x-cos x-K x+5 ) decreases for all
positive real values of ( x ) is given by
A ( . Ksqrt{2} )
D. ( K<sqrt{2} )
12
892The coordinates of a point ( boldsymbol{P} ) on the line
( 2 x-y+5=0 ) such that ( |P A-P B| ) is
maximum where ( A ) is (4,-2) and ( B ) is
(2,-4) will be
A ( .(11,27) )
B. (-11,-17)
c. (-11,17)
D. (0.5)
12
893If the global maximum value of ( boldsymbol{f}(boldsymbol{x})= )
( -x^{2}+a x-a^{2}-2 a-2 ) for ( x epsilon[0,1] ) be
-5′, then ‘a’ can take the value
This question has multiple correct options
( ^{mathrm{A}} cdot frac{-4+2 sqrt{13}}{3} )
B. ( frac{-4-2 sqrt{13}}{3} )
c. 1
D. – 3
12
894N characters of information are held on
magnetic tape, in batches of ( x ) characters each,the batch processing
time is ( alpha+beta x^{2} ) seconds, ( alpha ) and ( beta ) are
constants. The optical value of ( x ) for fast
processing is,
( mathbf{A} cdot alpha / beta )
в. ( beta / alpha )
c. ( sqrt{alpha / beta} )
D. ( sqrt{beta / alpha} )
12
895If the radius of a sphere is measured as ( 9 m ) with an error of ( 0.03 m, ) then find
the approximate error in calculating in surface area.
12
896( ln a Delta A B C ) if sides a and b remain
constant such that ( alpha ) is the error in ( C )
then relative error in its area is
A. ( alpha cot C )
B. ( alpha sin C )
( c cdot alpha tan C )
D. ( alpha cos C )
12
89732.
Consider :
X
1.
A normal to y=f(x) at x =
also passes through the point :
[JEE M 2016]
(a) (5.0) (0) (69) (C) (0,0)(a) (637
12
898The side of a square of metal is increasing at the rate of ( 5 mathrm{cm} / ) minute. At what are its area is increasing when
the side is ( 20 mathrm{cm} ) long?
12
899The pressure ( boldsymbol{P} ) and the volume ( boldsymbol{v} ) of ( mathbf{a} )
gas are connected by the relation ( p v^{1.4}= ) const. Find the percentage error in ( p ) corresponding to a decrease of ( frac{1}{2} % )
in ( boldsymbol{v} )
12
900A spherical balloon is filled with ( 4500 pi )
cubic meters of helium gas. If a leak in
the balloon causes the gas to escape at the rate of ( 72 pi ) cubic meters per
minute, then the rate (in meters per minute) at which the radius of the
balloon decreases 49 minutes after the
leakage began is:
A ( cdot 6 / 7 )
B. ( 4 / 9 )
c. ( 2 / 9 )
D. None of these
12
901The velocity of any particle at maximum
height is equal to
12
902Let ( f(x) ) and ( g(x) ) are two functions
which are defined and differentiable for
all ( boldsymbol{x} geq boldsymbol{x}_{0} . ) If ( boldsymbol{f}left(boldsymbol{x}_{0}right)=boldsymbol{g}left(boldsymbol{x}_{0}right) ) and ( boldsymbol{f}^{prime}(boldsymbol{x})> )
( g^{prime}(x) ) for all ( x>x_{0} ) then
A ( cdot f(x)x_{0} )
B. ( f(x)=g(x) ) for some ( x>x_{0} )
C. ( f(x)>g(x) ) only for some ( x>x_{0} )
D. ( f(x)>g(x) ) for all ( x>x_{0} )
12
903A spherical balloon is being inflated so that its volume increases uniformly at
the rate of ( 40 mathrm{cm}^{3} / mathrm{min.} ) At ( r=8 mathrm{cm}, ) its
surface area increases at the rate
( mathbf{A} cdot 8 mathrm{cm}^{2} / mathrm{min} )
B . ( 10 mathrm{cm}^{2} / mathrm{min} )
c. ( 20 mathrm{cm}^{2} / mathrm{min} )
D. none of these
12
904The smallest value of ( x^{2}-3 x+3 ) in the interval ( left(-3, frac{3}{2}right) ) is
A ( cdot frac{3}{4} )
B. 5
( c .-15 )
D . -20
12
905Show that the function ( boldsymbol{f}(boldsymbol{x})= )
( cot ^{-1}(sin x+cos x) ) is decreasing on
( left(0, frac{pi}{4}right) ) and increasing on ( left(frac{pi}{4}, frac{pi}{2}right) )
12
90625. If f(x)=x2 + 2br + 2.2 and g(x) = -x2 – 2cx+bsuch
that min f(x) > max g(x), then the relation between b and c,
is
(2003)
(a) no real value of b&c (b) 0<c<bv2
(c) lcl|b|v2
11
12
907For what values of ( x ) is the rate of
increase of ( x^{3}-5 x^{2}+5 x+8 ) is twice
the rate of increase of ( x ? )
A ( cdot-3,-frac{1}{3} )
в. ( _{-3, frac{1}{3}} )
c. ( _{3,-frac{1}{3}} )
D. ( _{3, frac{1}{3}} )
12
90831. The least value of a E R for which 4ax2 +-21, for allx>0, is
12
909The rate of change of area of a circle with respect to its radius at ( r=2 mathrm{cm} ) is
A .4
в. ( 2 pi )
( c cdot 2 )
D. ( 4 pi )
12
910Find the greatest and the least values of the following functions. Fin the extrema of the function ( boldsymbol{f}(boldsymbol{x})= ) ( 2 x sin 2 x+cos 2 x-sqrt{3} ) on the
interval ( [-boldsymbol{pi} / 2, mathbf{3} boldsymbol{pi} / mathbf{8}] )
12
911Find the point on the curve ( y^{2}=8 x ) for
which the abscissa and ordinate
change at the same rate.
12
912Find the maximum and minimum
values, if any of the following function given by
¡) ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}+mathbf{2}|-mathbf{1} )
ii) ( boldsymbol{g}(boldsymbol{x})=-|boldsymbol{x}+mathbf{1}|+mathbf{3} )
iii) ( boldsymbol{f}(boldsymbol{x})=|sin 4 boldsymbol{x}+boldsymbol{3}| )
12
913Find the maximum and minimum value
of the function:
( f(x)=2 x^{3}-21 x^{2}+36 x-20 )
12
91411. A ladder 10 m long rests against a vertical wall with the
lower end on the horizontal ground. The lower end of the
ladder is pulled along the ground away from the wall at
the rate of 3 cm/sec. The height of the upper end while it
is descending at the rate of 4 cm/sec is
(a) 413m
(b) 6 m
(c) 572m
(d) 8 m
12
915The two curves ( x^{3}-3 x y^{2}+2=0 ) and
( mathbf{3} boldsymbol{x}^{2} boldsymbol{y}-boldsymbol{y}^{3}-boldsymbol{2}=mathbf{0} )
A. cut at right angles
B. touch each other
c. cut at an angle ( frac{pi}{3} )
D. cut at an angle ( frac{pi}{4} )
12
916A ( 13 f t . ) ladder is leaning against a wall when its base starts to slide away At the instant when the base is 12 ft. away from the wall, the base is moving away from the wall at the rate of 5 ft/sec. The
rate at which the angle ( theta ) between the ladder and the ground is changing is
A ( cdot-frac{12}{13} r a d / sec )
B. – 1 rad( / )sec.
c. ( -frac{13}{12} r a d / s e c )
12
917At present a firm manufacturing 2000 items it is estimated that the rate of
change of production (P) with respect to additional number of workers ( (x) ) is
given by ( frac{d P}{d x}=100-12 sqrt{x} . ) If the firm employes
25 more workers then the new
production is?
A . 2500
в. 3000
c. 3500
D. 4500
12
918If ( boldsymbol{x}>mathbf{0} ) and ( boldsymbol{x} boldsymbol{y}=mathbf{1}, ) the minimum
value of ( (x+y) ) is
A . -2
B.
( c cdot 2 )
D. none of these
12
919The value of ( (127)^{1 / 3} ) to four decimal places is
A .5 .0267
B. 5.4267
c. 5.5267
D. 5.001
12
920A variable triangle is inscribed in a circle of radius ( mathrm{R} ). If the rate of change of
side is ( R ) times the rate of change of the opposite angle, then that angle is
( mathbf{A} cdot pi / 6 )
в. ( pi / 4 )
c. ( pi / 3 )
D. ( pi / 2 )
12
921Let ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{4}}{mathbf{3}} boldsymbol{x}^{3}-mathbf{4} boldsymbol{x}, mathbf{0} leq boldsymbol{x} leq mathbf{2} . ) Then
the global minimum value of the function is
( mathbf{A} cdot mathbf{0} )
B. ( -8 / 3 )
c. -4
D. none of these
12
922Let ( boldsymbol{f}: boldsymbol{I} boldsymbol{R} rightarrow boldsymbol{I} boldsymbol{R} quad ) be defined as
( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}|+left|boldsymbol{x}^{2}-1right| . ) The total number
of points at which ( f ) attains either a
local maximum or local minimum is
12
923A point on the parabola ( y^{2}=18 x ) at
which the ordinate increases at twice
the rate of the absicca is
( ^{A} cdotleft(frac{9}{8}, frac{9}{2}right) )
в. (2,-4)
( ^{mathbf{c}} cdotleft(frac{-9}{8}, frac{9}{2}right) )
D. (2,4)
12
924The slope of the tangent to the locus ( y=cos ^{-1}(cos x) ) at ( x=frac{pi}{4} ) is
( mathbf{A} cdot mathbf{1} )
B.
( c cdot 2 )
D. –
12
925The focal length of a mirror is given by ( frac{2}{f}=frac{1}{v}-frac{1}{u} . ) In finding the values of
and ( v, ) the errors are equal and equal to
‘p’. The the relative error in ( f ) is
A ( cdot frac{p}{2}left(frac{1}{u}+frac{1}{v}right) )
в. ( pleft(frac{1}{u}+frac{1}{v}right) )
c. ( frac{p}{2}left(frac{1}{u}-frac{1}{v}right) )
D. ( _{p}left(frac{1}{u}-frac{1}{v}right) )
12
926The straight line which is parallel to ( x ) axis and crosses the curve ( y=sqrt{x} ) at an
angle of ( 45^{0} ) is
A ( cdot x=frac{1}{4} )
в. ( y=frac{1}{4} )
c. ( y=frac{1}{2} )
D. ( x=frac{1}{2} )
12
927The radius of a right circular cylinder increases at a constant rate. Its altitude
is a linear function of the radius and
increases three times as fast as radius.
When the radius is ( 1 mathrm{cm} ) the altitude is
( 6 mathrm{cm} . ) When the radius is ( 6 mathrm{cm} ), then
volume is increasing at the rate of ( 1 mathrm{Cu} ) ( c m / )sec. When the radius is ( 36 mathrm{cm}, ) the
volume is increasing at a rate of ( n ) cu.
( mathrm{cm} / ) sec. The value of ‘ ( n^{prime} ) is equal to:
A .12
в. 22
( c .30 )
D. 33
12
928Find the min. value of ( frac{boldsymbol{a}^{2}}{cos ^{2} boldsymbol{x}}+frac{boldsymbol{b}^{2}}{sin ^{2} boldsymbol{x}} )
A. ( (a+b)^{2} )
2) ( a+b+b^{-2} b+b^{-2} )
B . ( (a-b)^{2} )
( mathbf{c} cdot a^{2}+b^{2} )
D. ( a^{2}-b^{2} )
12
929You are given a rod of length ( L ). The
linear mass density is ( lambda ) such that ( lambda= ) ( a+b x . ) Here ( a ) and ( b ) are constant and
the mass of the rod increases as ( x )
decreases. Find the mass of the rod.
12
930The number of Points of Inflexion in ( boldsymbol{y}= )
( x^{3}-3 x^{2}+3 x ) are:
( A cdot 0 )
B.
( c cdot 2 )
D. 3
12
931The function ( f(x)=x^{3}-27 x+8 ) is
increasing when
A . ( |x|3 )
c. ( -3<x<3 )
D. none of these
12
932If ( f(x)=x^{3}-10 x^{2}+200 x-10, ) then
( (x) ) is
A. decreasing ( (-infty, 10] ) and increasing in ( (10, infty) )
B. increasing ( (-infty, 10] ) and decreasing in ( (10, infty) )
c. increasing for every value of ( x )
D. decreasing for every value of ( x )
12
933A flower bed is made in the shape of sector of a circle. ( 20 m ) of wire is
available to make a fence for the flower
bed. Find the radius of the circle so that
area of the flower bed is maximum.
12
934Show that the function ( f ) given by
( f(x)=10^{x} ) is increasing for all ( x )
12
9355.
A function y = f(x) has a second order derivative
f”(x) = 6(x-1). If its graph passes through the point (2,1)
and at that point the tangent to the graph is y=3x-5, then
the function is
[2004]
(a) (x+1)2 (b) (x – 1)3() (x+1)3 (d) (x-1)2
12
936Find the points of the maxima or local minima of the following function, using the first derivative test. Also, find the
local maximum or local minimum
values, as the case may be. ( f(x)=x^{3}(x-1)^{2} )
12
937The number of values of ( x ) where the
function ( boldsymbol{f}(boldsymbol{x})=cos boldsymbol{x}+cos (sqrt{mathbf{2}} boldsymbol{x}) )
attains its maximum is
A .
в.
( c cdot 2 )
D. infinite
12
938Let ( f ) and ( g ) be two functions defined on
an interval I such that ( f(x) geq 0 ) and
( g(x) leq 0 ) for all ( x epsilon I ) and ( f ) is strictly decreasing on ( I ) while ( g ) is strictly
increasing on ( I ) then

This question has multiple correct options
A. The product function ( f g ) is strictly increasing on ( I )
B. The product function ( f g ) is strictly decreasing on ( I )
c. ( operatorname{Fog}(x) ) is monotonically increasing on ( I )
D. ( operatorname{Fog}(x) ) is monotonically decreasing on ( I )

12
939The slope of the normal to the curve
( y^{3}-x y-8=0 ) at the point (0,2) is
equal to:
A . -3
в. -6
( c cdot 3 )
D. 6
E . 8
12
94039. If the curves y2 = 6x, 9x² + by2 = 16 intersect each other
at right angles, then the value of bis: [JEEM 2018]
(a) ₃
(1) 4
(c) 7
(d) 6
12
94139. If]f(x,)-f(x))}<(x, – x,)”, for all x,, x, € R. Find the equation
of tangent to the cuve y=f(x) at the point (1, 2).
(2005 – 2 Marks)
12
942
40. Ifp(x) be a polynomial of degree 3 satisfying p(-1)= 10,p(1)
=-6 and p(x) has maxima at x=-1 and p'(x) has minima atx
= 1. Find the distance between the local maxima and local
minima of the curve.
(2005 – 4 Marks)
12
943Find an angle ( theta, 0<theta<frac{pi}{2}, ) which
increases twice as fast as its sine.
12
944The side of a square sheet is increasing at the rate of ( 4 mathrm{cm} ) per minute. The rate by which the area increasing when the side is ( 8 mathrm{cm} ) long is-
A ( cdot 60 mathrm{cm}^{2} / mathrm{sec} )
B. ( 66 mathrm{cm}^{2} / mathrm{sec} )
( mathrm{c} cdot 62 mathrm{cm}^{2} / mathrm{sec} )
D. ( 64 mathrm{cm}^{2} / mathrm{sec} )
12
945The minimum value of ( mathbf{f}(mathbf{x})= )
( 4 sec ^{2} x+9 operatorname{cosec}^{2} x ) is
A . 5
B . 25
c. 13
( D cdot 13^{2} )
12
946The ( boldsymbol{f}(boldsymbol{x})=(3-boldsymbol{x}) e^{2 x}-4 boldsymbol{x} e^{x}-boldsymbol{x} ) has
( mathbf{A} cdot ) a maximum at ( x=0 )
B. a minimum at ( x=0 )
C. neither of two at ( x=0 )
D. ( f(x) ) is not differentiable at ( x=0 )
12
947Show that the equation of normal at any point on the curve ( x=3 cos t-cos ^{3} t )
( boldsymbol{y}=mathbf{3} sin t-sin ^{3} boldsymbol{t} ) is
( 4left(y cos ^{3} t-x sin ^{3} tright)=3 sin 4 t )
12
948The minimum value of the polynomial.
( p(x)=3 x^{2}-5 x+2 )
A ( cdot-frac{1}{6} )
B.
( c cdot frac{1}{12} )
D. ( -frac{1}{12} )
12
949Find the maximum and minimum
values, if any, without using derivatives of the following function. ( f(x)=16 x^{2}-16 x+28 ) on ( R )
12
950The least value of ( boldsymbol{f}(boldsymbol{x})=left(boldsymbol{e}^{boldsymbol{x}}+boldsymbol{e}^{-boldsymbol{x}}right) ) is
A . -2
B.
( c cdot 2 )
D. none of these
12
951Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. ( sqrt{0.6} )12
952The normal to the curve ( 2 x^{2}+y^{2}=12 )
at the point (2,2) cuts the curve again
at
( ^{mathbf{A}} cdotleft(-frac{22}{9},-frac{2}{9}right) )
в. ( left(frac{22}{9}, frac{2}{9}right) )
c. (-2,-2)
D. none of these
12
953What normal to the curve ( y=x^{2} ) form
the shortest chord?
( mathbf{A} cdot x+sqrt{2} y=sqrt{2} ) or ( x-sqrt{2} y=-sqrt{2} )
B ( . x+sqrt{2} y=sqrt{2} ) or ( x-sqrt{2} y=sqrt{2} )
c. ( x+sqrt{2} y=-sqrt{2} ) or ( x-sqrt{2} y=-sqrt{2} )
D. ( x+sqrt{2} y=-sqrt{2} ) or ( x-sqrt{2} y=sqrt{2} )
12
9546.
The normal to the curve x = a(1+cos), y = a sino at ‘o’
always passes through the fixed point
[2004]
(a) (a, a) (b) (0, a) (c) (0,0) (d) (a,0)
12
9552.
AB is a diameter of a circle and C is any point on the
circumference of the circle. Then
(1983 – 1 Mark)
(a) the area of A ABC is maximum when it is isosceles
(b) the area of A ABC is minimum when it is isosceles
(c) the perimeter of A ABC is minimum when it is isosceles
(d) none of these
12
956The side of an equilateral triangle is ‘a’
units and is increasing at the rate of ( lambda ) units /sec. The rate of increase of its
area is
A ( cdot frac{2}{sqrt{3}} lambda a )
в. ( sqrt{3} lambda a )
c. ( frac{sqrt{3}}{2} lambda a )
D. none of these
12
957Find intervals in which the function
( operatorname{given} operatorname{by} f(x)=sin 3 x, x inleft[0, frac{pi}{2}right] )
is increasing function
12
958If an error of ( k % ) is made in measuring
the radius of a sphere, then percentage error in its volume is
A. ( k % )
B. ( 3 k % )
c. ( 2 k % )
D. ( frac{2}{3} k % )
12
959Find the coordinates of the point on ( y )
axis which is nearest to the point (-2,5)
12
960A particle moves along a curve so that
its coordinates at time ( t ) are ( boldsymbol{x}=boldsymbol{t}, boldsymbol{y}= ) ( frac{1}{2} t^{2}, z=frac{1}{3} t^{3} ) acceleration at ( t=1 ) is
( mathbf{A} cdot j+2 k )
B. ( j+k )
( c cdot 2 j+k )
D. none of these
12
961The side of a square is equal to the diameter of a circle. If the side and
radius change at the same rate, then the ratio of the change of their areas is
A . ( 2: pi )
в. ( pi: 1 )
( c cdot 4: pi )
D. 1: 2
12
96226. Determine the points of maxima and minima of the function
In x-bx + x², x > 0, where b > O is a constant.
12
963If ( I^{2}+mathbf{m}^{2}=1, ) then the max values of
( boldsymbol{I}+mathbf{m} ) is
( A cdot 1 )
B. ( sqrt{2} )
c. ( frac{1}{sqrt{2}} )
D. 2
12
964If ( f^{prime}(x) ) exists for all ( x in R ) and ( g(x)= )
( boldsymbol{f}(boldsymbol{x})-(boldsymbol{f}(boldsymbol{x}))^{2}+(boldsymbol{f}(boldsymbol{x}))^{3} ) for all ( boldsymbol{x} in boldsymbol{R} )
then
A. ( g(x) ) is increasing whenever ( f ) is increasing
B. ( g(x) ) is increasing whenever ( f ) is decreasing
( mathrm{c} . g(x) ) is decreasing whenever ( f ) is increasing
D. none of these
12

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