Determinants Questions

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

List of determinants Questions

Question No Questions Class
1 Solve:
( left|begin{array}{ccc}mathbf{0} & boldsymbol{a} & -boldsymbol{b} \ -boldsymbol{a} & boldsymbol{0} & -boldsymbol{c} \ boldsymbol{b} & boldsymbol{c} & boldsymbol{0}end{array}right|=0 )
12
2 The value of ( (operatorname{adj} A) ) is equal to
A . ( 2 A )
в. ( 4 A )
( c .8 A )
D. ( 16 A )
12
3 Find the values of ( x, ) if ( left|begin{array}{ll}mathbf{3} & boldsymbol{x} \ boldsymbol{x} & mathbf{1}end{array}right|=left|begin{array}{ll}mathbf{3} & mathbf{2} \ mathbf{4} & mathbf{1}end{array}right| ) 12
4 ( fleft(begin{array}{ccc}2 & -1 & 3 \ -5 & 3 & 1 \ -3 & 2 & 3end{array}right], ) then
( A .(A d j A)= )
A ( cdotleft(A d j cdot A^{T}right) )
B. ( (text {Adj.A}) . A )
c. ( |A| . A )
D. None of these
12
5 set of all values of 2 for which the system of linear
31. The set of all value
equations :
2×1 – 2×2 + x3 = 2×1
2x – 3×2 + 2×3 = 2×2
-x + 2×2=hxz
has a non-trivial solution
[JEEM 2015]
(a) contains two elements
(b) contains more than two elements
(C) is an empty set
(d) is a singleton
12
6 1
2
3 and (adj M) =
-5
where
3
-1
To 1 al
-1 1 -17
15. Let M= 1
3 and (adi M = 8 -6 2
| 3 6 1
a and b are real numbers. Which of the following options
is/are correct ?
(JEE Adv. 2019)
(a) a+b=3
(6) det (adj M2)=81
© (adjM)-1 + adjM =-M
N
(d)
IfM
, then a-B+y = 3
12
7 ( f(x)=left|begin{array}{ccc}cos x & x & 1 \ 2 sin x & x^{2} & 2 x \ tan x & x & 1end{array}right| . ) The value of
( lim _{x rightarrow 0} frac{f(x)}{x} ) is equal to
A . 1
B. –
( c cdot 0 )
D. None of these
12
8 Using properties of determinants, prove
that:
[
left|begin{array}{ccc}
1+a & 1 & 1 \
1 & 1+b & 1 \
1 & 1 & 1+c
end{array}right|=a b c+b c+
]
( c a+a b )
12
9 Find the value of the determinant:
[
left|begin{array}{cc}
mathbf{5} & -mathbf{2} \
-mathbf{3} & mathbf{1}
end{array}right|
]
12
10 A. square matrix ( boldsymbol{A} ) of order ( mathbf{3}, ) has ( |boldsymbol{A}|= )
( 5, ) find ( mid A ) adjal
12
11 valuate: ( left|begin{array}{cccc}1 & a & a^{2} & a^{3}+b c d \ 1 & b & b^{2} & b^{3}+c d a \ 1 & c & c^{2} & c^{3}+a b d \ 1 & d & d^{2} & d^{3}+a b cend{array}right| ) 12
12 ( left|begin{array}{cc}cos 15^{circ} & sin 15^{circ} \ sin 15^{circ} & cos 15^{circ}end{array}right|=? )
A . 1
B.
c. ( frac{sqrt{3}}{2} )
D. none of these
12
13 Show that the points
( boldsymbol{A}(mathbf{1}, mathbf{2}, mathbf{7}), boldsymbol{B}(mathbf{2}, mathbf{6}, mathbf{3}) ) and ( boldsymbol{C}(mathbf{3}, mathbf{1 0},-mathbf{1}) )
are collinear.
12
14 ( boldsymbol{a} neq boldsymbol{p}, boldsymbol{b} neq boldsymbol{q}, boldsymbol{c} neq boldsymbol{r} ) and ( left|begin{array}{lll}boldsymbol{p} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{a} & boldsymbol{q} & boldsymbol{c} \ boldsymbol{a} & boldsymbol{b} & boldsymbol{r}end{array}right|=0 )
then the value of ( frac{boldsymbol{p}}{boldsymbol{p}-boldsymbol{a}}+frac{boldsymbol{q}}{boldsymbol{q}-boldsymbol{b}}+ )
( frac{r}{r-c} ) is equal to
( A )
B. 2
( c cdot 3 )
D.
12
15 Using properties of determinant solve 🙁 left|begin{array}{ccc}1 & a & a^{2}-b c \ 1 & b & b^{2}-a c \ 1 & c & c^{2}-a bend{array}right|= ) 12
16 Evaluate.
( mid begin{array}{ccc}frac{1}{z} & frac{1}{z} & -frac{(x+y)}{z^{2}} \ -frac{(y+z)}{x^{2}} & frac{1}{x} & frac{1}{x} \ -frac{y(y+z)}{x^{2} z} & frac{x+2 y+z}{x z} & -frac{y(x+y)}{x z^{2}}end{array} )
12
17 Expand:
( left|begin{array}{lll}boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{c} & boldsymbol{a} & boldsymbol{b} \ boldsymbol{b} & boldsymbol{c} & boldsymbol{a}end{array}right| )
12
18 ( boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & boldsymbol{omega} & boldsymbol{omega}^{2} \ boldsymbol{omega} & boldsymbol{omega}^{2} & mathbf{1} \ boldsymbol{omega}^{2} & mathbf{1} & boldsymbol{omega}end{array}right] ) Find Det ( mathbf{A} ) 12
19 If ( boldsymbol{A}=left[begin{array}{ccc}2 & 52 & 152 \ 4 & 106 & 358 \ 6 & 162 & 620end{array}right], ) then the
determinant of the matrix ( a d j(2 A) ) is equal to:
12
20 ( fleft|begin{array}{ccc}mathbf{6} i & -mathbf{3} i & mathbf{1} \ mathbf{4} & mathbf{3} i & -mathbf{1} \ mathbf{2 0} & mathbf{3} & boldsymbol{i}end{array}right|=boldsymbol{x}+boldsymbol{i} boldsymbol{y}, ) then
A ( . x=3, y=1 )
B. ( x=1, y=3 )
c. ( x=0, y=3 )
D. ( x=0, y=0 )
12
21 Let ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma}, boldsymbol{a}, boldsymbol{b}, boldsymbol{c}, boldsymbol{x} boldsymbol{epsilon} boldsymbol{R} ) and let
( Delta= )
( left|begin{array}{ccc}sin (x+alpha) & cos (x+alpha) & a+x sin alpha \ sin (x+beta) & cos (x+beta) & b+x sin beta \ sin (x+gamma) & cos (x+gamma) & c+x sin gammaend{array}right| )
then ( Delta ) is
This question has multiple correct options
A. independent of ( x )
B. dependent of ( alpha, beta, ),
c. dependent on ( a, b, c, c )
D. ( a ) constant
12
22 If
[] denotes the greatest integer less than or equal to the real number under
consideration, and ( -1 leq x<0, quad 0 leq )
( y<1,1 leq z<2, ) the value of the determinant ( left|begin{array}{ccc}{[boldsymbol{x}]+mathbf{1}} & {[boldsymbol{y}]} & {[boldsymbol{z}]} \ {[boldsymbol{x}]} & {[boldsymbol{y}]+mathbf{1}} & {[boldsymbol{z}]} \ {[boldsymbol{x}]} & {[boldsymbol{y}]} & {[boldsymbol{z}]+mathbf{1}}end{array}right| )
is
( A cdot[x] )
в. ( [y] )
( c cdot[z )
D. none of these
12
23 ( left|begin{array}{ccc}1+a & 1 & 1 \ 1 & 1+b & 1 \ 1 & 1 & 1+cend{array}right|= )
( boldsymbol{X} boldsymbol{a b c}left(1+frac{1}{boldsymbol{a}}+frac{1}{boldsymbol{b}}+frac{1}{c}right) . ) Find the value
of ( boldsymbol{X} )
12
24 if ( left|begin{array}{lll}boldsymbol{a} & boldsymbol{b} & mathbf{0} \ boldsymbol{0} & boldsymbol{a} & boldsymbol{b} \ boldsymbol{b} & boldsymbol{0} & boldsymbol{a}end{array}right|=mathbf{0} ) then
A. ( a ) is a cube root of 1
B. ( b ) is a cube root of 1
c. ( frac{a}{b} ) is a cube root of 1
D. ( frac{a}{b} ) is a cube root of -1
12
25 Find the values of ( K ) if Area of the
triangle is 4 sq. units and vertices are ( (k 0)(40)(02) ) using determinants.
12
26 If ( boldsymbol{A}=left[begin{array}{cc}-mathbf{4} & -mathbf{1} \ mathbf{3} & mathbf{1}end{array}right], ) then the determinant
of the matrix ( left(A^{2016}-2 A^{2015}-A^{2014}right) )
is:
A. -175
в. 2014
( c .2016 )
D. -25
12
27 If the value of ( left|begin{array}{ccc}mathbf{1} & mathbf{2} & mathbf{4} \ -mathbf{1} & mathbf{3} & mathbf{0} \ mathbf{4} & mathbf{1} & mathbf{0}end{array}right| ) is ( k, ) then find
( frac{-boldsymbol{k}}{mathbf{1 3}} )
12
28 f ( l, m, n ) are ( p^{t h}, q^{t h}, r^{t h} ) terms of ( G . P . ) al positive, then ( left|begin{array}{lll}log l & p & 1 \ log m & q & 1 \ log n & r & 1end{array}right| ) equals
A . –
B. 2
( c )
( D )
12
29 If ( boldsymbol{x}=boldsymbol{c} boldsymbol{y}+boldsymbol{b} boldsymbol{z}, boldsymbol{y}=boldsymbol{a} boldsymbol{z}+boldsymbol{c} boldsymbol{x}, boldsymbol{z}=boldsymbol{b} boldsymbol{x}+ )
( a y, ) where ( x, y, z ) are not all zero, then
the value of ( a^{2}+b^{2}+c^{2}+2 a b c )
( mathbf{A} cdot mathbf{0} )
B.
c. -1
D. None of these
12
30 Find the integral value of ( x, ) if ( left|begin{array}{ccc}boldsymbol{x}^{2} & boldsymbol{x} & mathbf{1} \ mathbf{0} & boldsymbol{2} & mathbf{1} \ boldsymbol{3} & boldsymbol{1} & boldsymbol{4}end{array}right|=mathbf{2} mathbf{8} ) 12
31 Let ( a, b, c in R ) be such that ( a+b+ )
( boldsymbol{c}>mathbf{0} ) and ( boldsymbol{a b c}=mathbf{2 .} ) Let ( boldsymbol{A}=left[begin{array}{lll}boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{b} & boldsymbol{c} & boldsymbol{a} \ boldsymbol{c} & boldsymbol{a} & boldsymbol{b}end{array}right] )
If ( A^{2}=I, ) then value of ( a^{3}+b^{3}+c^{3} ) is
( A cdot 7 )
B.
( c cdot 0 )
( D )
12
32 If the lines ( boldsymbol{a} boldsymbol{x}+boldsymbol{y}+mathbf{1}=mathbf{0}, boldsymbol{x}+boldsymbol{b} boldsymbol{y}+ )
( mathbf{1}=mathbf{0} & boldsymbol{x}+boldsymbol{y}+boldsymbol{c}=mathbf{0} ) where ( mathbf{a}, mathbf{b} & mathbf{c} )
are distinct real numbers different from
are concurrent, then the value of ( frac{1}{1-a}+frac{1}{1-b}+frac{1}{1-c}= )
( A cdot 4 )
B. 3
( c cdot 2 )
( D )
12
33 If ( a+b+c neq 0 ) and ( left|begin{array}{lll}a & b & c \ b & c & a \ c & a & bend{array}right|=0 )
then using properties of determinants, prove that ( boldsymbol{a}=boldsymbol{b}=boldsymbol{c} )
12
34 The system of linear equations:
( boldsymbol{lambda} boldsymbol{x}+boldsymbol{2} boldsymbol{y}+boldsymbol{2} boldsymbol{z}=mathbf{5} )
( 2 lambda x+3 y+5 z= )
( 4 x+lambda y+6 z=10 ) has
A. no solution when ( lambda=2 )
B. a unique solution when ( lambda=-8 )
c. infinitely many solutions when ( lambda=2 )
D. no solution when ( lambda=8 )
12
35 Find the value of determinant.
(i) ( left|begin{array}{cc}cos theta & -sin theta \ sin theta & cos thetaend{array}right| )
(ii) ( left|begin{array}{cc}x^{2}-x+1 & x-1 \ x+1 & x+1end{array}right| )
12
36 If ( alpha=beta+frac{2 pi}{3} . ) then ( A_{theta} ) is maximum
when ( gamma ) equals
A. ( alpha+pi / 3 )
в. ( alpha-pi / 3 )
c. ( alpha+2 pi / 3 )
D. none of these
12
37 begin{tabular}{c|ccc}
( mathbf{1} ) & ( mathbf{s i n Q} ) & ( mathbf{1} ) \
( -mathbf{s i n Q} ) & ( mathbf{1} ) & ( mathbf{s i n Q} ) \
( mathbf{- 1} ) & ( -mathbf{s i n Q} ) & ( mathbf{1} )
end{tabular}
12
38 ( left|begin{array}{ccc}mathbf{1} & mathbf{1} & mathbf{1} \ boldsymbol{x} & boldsymbol{y} & boldsymbol{z} \ boldsymbol{y} boldsymbol{z} & boldsymbol{z} boldsymbol{x} & boldsymbol{x} boldsymbol{y}end{array}right|=boldsymbol{b}(boldsymbol{x}-boldsymbol{y})(boldsymbol{y}- )
( z)(z-x) . ) Find
12
39 The points which are not collinear are:
A. (0,1),(8,3) and (6,7)
B. (4,3),(5,1) and (1,9)
C. (2,5),(-1,2) and (4,7)
D. (-3,2)(1,-2) and (9,-10)
12
40 ( fleft(a_{1}, a_{2}, a_{3} dots, a_{n} cdot ) are in G.P, then the right.
determinant ( Delta= ) ( left|begin{array}{ccc}log a_{n} & log a_{n+1} & log a_{n+2} \ log a_{n+3} & log a_{n+4} & log a_{n+5} \ log a_{n+6} & log a_{n+7} & log a_{n+8}end{array}right| ) is equal
to
A . 0
B. 1
( c cdot 2 )
D.
12
41 Consider the system of linear equations in x, y, z:
(sin 30) x-y+z=0
(cos 20) x + 4y + 3z=0
2x+ 7y+ 7z=0
Find the values of for which this system has nontrivial
solutions.
(1986 – 5 Marks)
12
42 ( fleft(alpha, beta text { are the roots of } x^{2}+x+1=0right. )
( operatorname{then}left|begin{array}{ccc}boldsymbol{y}+mathbf{1} & boldsymbol{beta} & boldsymbol{alpha} \ boldsymbol{beta} & boldsymbol{y}+boldsymbol{alpha} & mathbf{1} \ boldsymbol{alpha} & mathbf{1} & boldsymbol{y}+boldsymbol{beta}end{array}right|=? )
A ( cdot y^{2}-1 )
В . ( yleft(y^{2}-1right) )
c. ( y^{2}-y )
D. ( y^{3} )
12
43 What are the values of ( x ) that satisfy the
equation ( left|begin{array}{ccc}boldsymbol{x} & mathbf{0} & mathbf{2} \ mathbf{2} boldsymbol{x} & mathbf{2} & mathbf{1} \ mathbf{1} & mathbf{1} & mathbf{1}end{array}right|+left|begin{array}{ccc}mathbf{3} boldsymbol{x} & mathbf{0} & mathbf{2} \ boldsymbol{x}^{mathbf{2}} & boldsymbol{2} & mathbf{1} \ mathbf{0} & boldsymbol{1} & mathbf{1}end{array}right|=mathbf{0} ? )
B. ( -1 pm sqrt{3} )
c. ( -1 pm sqrt{6} )
( mathbf{D} cdot-2 pm sqrt{6} )
12
44 f ( x+y+z=pi ) and
( Delta=left|begin{array}{ccc}sin 3 x & sin 3 y & sin 3 z \ sin x & sin y & sin z \ cos x & cos y & cos zend{array}right| )
then ( Delta ) equals
( A )
в.
( c . )
( D )
12
45 f ( omega ) is a cube root of unity, then ( left|begin{array}{ccc}mathbf{1} & boldsymbol{omega} & boldsymbol{omega}^{2} \ boldsymbol{omega} & boldsymbol{omega}^{2} & boldsymbol{1} \ boldsymbol{omega}^{2} & boldsymbol{1} & boldsymbol{omega}end{array}right| ) is equal to
A .
B. ( omega )
( c cdot omega^{2} )
D.
12
46 An equilateral triangle has each of its
sides of length ( 6 mathrm{cm} . ) If ( left(x_{1}, y_{1}right) ;left(x_{2}, y_{2}right) )
( &left(x_{3}, y_{3}right) ) are its vertices then the value
of determinant (in nearest integer value) ( ,left|begin{array}{lll}boldsymbol{x}_{1} & boldsymbol{y}_{1} & mathbf{1} \ boldsymbol{x}_{2} & boldsymbol{y}_{2} & mathbf{1} \ boldsymbol{x}_{3} & boldsymbol{y}_{3} & mathbf{1}end{array}right| ) is equal to a then
find ( frac{boldsymbol{a}}{mathbf{3 1}} )
12
47 $
If A2 – A+I=0, then the inverse of A is
(2) A+1 (b) A (c) A-I
[2005]
(2) I-A
12
48 If ( boldsymbol{A}=left[begin{array}{ll}boldsymbol{alpha} & boldsymbol{2} \ boldsymbol{2} & boldsymbol{alpha}end{array}right] ) and ( |boldsymbol{A}|^{3}=125 ) then the
value of ( boldsymbol{alpha} ) is
A. ±1
B. ±2
( c .pm 3 )
D. ±5
12
49 39.
If the system of linear equations
x+ky+3z=0
3x + ky-2z=0
2x + 4y-3z=0
(JEE M 2018]
has a non-zero solution (x, y, z), then
XZ
2 is equal to :
(2) 10
(6) – 30
(c) 30
(2) – 10
12
50 Show that the following set of points are
collinear.
(2,5),(4,6) and (8,8)
12
51 The value of the determinant
( left|begin{array}{ccc}1-alpha & alpha-alpha^{2} & alpha^{2} \ 1-beta & beta-beta^{2} & beta^{2} \ 1-gamma & gamma-gamma^{2} & gamma^{2}end{array}right| ) is equal to
A ( cdot(alpha-beta)(beta-gamma)(alpha-gamma) )
В . ( (alpha-beta)(beta-gamma)(gamma-alpha) )
c. ( (alpha-beta)(beta-gamma)(alpha-gamma)(alpha+beta+gamma) )
D.
12
52 √3
1
15.
If P=
and A=
and Q = PAPT and
√3
(2005)
x=PTQ2005 P then x is equal to
fi 2005
(a) 0 1
[4+200573 6015 1
(b) 2005 4-2005/3]
12+√3 1
(c) A 1-1 2-13]
w 1 2005 2-√3]
(d) 4 2+13 2005
12
53 ( operatorname{Det}left{begin{array}{lll}2 & 45 & 55 \ 1 & 29 & 32 \ 3 & 68 & 87end{array}right}=dots dots )
A . 45
B. 64
( c cdot 54 )
D. 32
12
54 The least value of the product xyz for which the determinant ( left|begin{array}{lll}boldsymbol{x} & mathbf{1} & mathbf{1} \ mathbf{1} & boldsymbol{y} & mathbf{1} \ mathbf{1} & mathbf{1} & boldsymbol{z}end{array}right| ) is
non-negative, is :
begin{tabular}{l}
A ( -16 sqrt{2} ) \
hline
end{tabular}
В. ( -2 sqrt{2} )
( c cdot-1 )
D. – –
12
55 Solve for ( boldsymbol{lambda} ) if
( left|begin{array}{ccc}boldsymbol{a}^{2}+boldsymbol{lambda} & boldsymbol{a} boldsymbol{b} & boldsymbol{a c} \ boldsymbol{a b} & boldsymbol{b}^{2}+boldsymbol{lambda} & boldsymbol{b c} \ boldsymbol{a c} & boldsymbol{b c} & boldsymbol{c}^{2}+boldsymbol{lambda}end{array}right|=mathbf{0} )
12
56 ( left|begin{array}{ccc}boldsymbol{a}^{2}+mathbf{1} & boldsymbol{a} boldsymbol{b} & boldsymbol{a} boldsymbol{c} \ boldsymbol{a} boldsymbol{b} & boldsymbol{b}^{2}+mathbf{1} & boldsymbol{b} boldsymbol{c} \ boldsymbol{c} boldsymbol{a} & boldsymbol{c} boldsymbol{b} & boldsymbol{c}^{2}+mathbf{1}end{array}right|=mathbf{1}+boldsymbol{a}^{2}+ ) 12
57 The number of distinct real values of ( alpha )
for which the vectors ( boldsymbol{alpha}^{2} hat{boldsymbol{i}}-hat{boldsymbol{j}}-hat{boldsymbol{k}},-hat{boldsymbol{i}}- )
( boldsymbol{alpha}^{2} hat{boldsymbol{j}}-hat{boldsymbol{k}},-hat{boldsymbol{i}}-hat{boldsymbol{j}}-boldsymbol{alpha}^{2} hat{boldsymbol{k}} ) will lie in the
same place is
( A cdot 1 )
B . 2
( c .3 )
D.
12
58 ( fleft|begin{array}{ccc}mathbf{6} i & -mathbf{3} i & mathbf{1} \ mathbf{4} & mathbf{3} i & -mathbf{1} \ mathbf{2 0} & mathbf{3} & boldsymbol{i}end{array}right|=boldsymbol{x}+boldsymbol{i} boldsymbol{y}, ) then
A ( . x=3, y=1 )
B. ( x=1, y=3 )
c. ( x=0, y=3 )
D. ( x=0, y=0 )
12
59 Prove that ( mid begin{array}{ccc}b c-a^{2} & c a-b^{2} & a b \ -b c+c a+a b & b c-c a+a b & b c+c \ (a+b)(a+c) & (b+c)(b+a) & (c+aend{array} )
( mathbf{3} cdot(boldsymbol{b}-boldsymbol{c})(boldsymbol{c}-boldsymbol{a})(boldsymbol{a}-boldsymbol{b})(boldsymbol{a}+boldsymbol{b}+ )
( c(a b+b c+c a) )
12
60 The line ( A x+B y+C=0 ) cuts the
circle ( boldsymbol{x}^{2}+boldsymbol{y}^{2}+boldsymbol{a} boldsymbol{x}+boldsymbol{b} boldsymbol{y}+boldsymbol{c}=boldsymbol{0} ) in ( boldsymbol{P} )
and ( Q )
The line ( boldsymbol{A}^{prime} boldsymbol{x}+boldsymbol{B}^{prime} boldsymbol{y}+boldsymbol{C}^{prime}=mathbf{0} ) cuts the
circle ( boldsymbol{x}^{2}+boldsymbol{y}^{2}+boldsymbol{a}^{prime} boldsymbol{x}+boldsymbol{b}^{prime} boldsymbol{y}+boldsymbol{c}^{prime}=mathbf{0} ) in ( boldsymbol{R} )
and ( S ).
If ( P, Q, R, S ) are concyclic, then show
that
( left|begin{array}{ccc}boldsymbol{a}-boldsymbol{a}^{prime} & boldsymbol{b}-boldsymbol{b}^{prime} & boldsymbol{c}-boldsymbol{c}^{prime} \ boldsymbol{A} & boldsymbol{B} & boldsymbol{C} \ boldsymbol{A}^{prime} & boldsymbol{B}^{prime} & boldsymbol{C}^{prime}end{array}right|=mathbf{0} )
12
61 ( operatorname{Let} boldsymbol{A}(mathbf{1}, mathbf{3}), boldsymbol{B}(mathbf{0}, mathbf{0}) ) and ( boldsymbol{C}(boldsymbol{k}, boldsymbol{0}) ) be
vertices of a triangle ( A B C ) such that
area of ( triangle A B C ) is ( 3 . ) Find the value of ( k )
( A cdot pm 2 )
B. ±3
( c .pm 4 )
D. ±1
12
62 ( f(a, b, c)=left[begin{array}{ccc}a & 0 & 0 \ {[0.3 e m] 0} & b & 0 \ {[0.3 e m] 0} & 0 & cend{array}right] )
such that ( a b c neq 0 )
( operatorname{then} A^{-1}=operatorname{diag}left(frac{1}{a}, frac{1}{b}, frac{1}{c}right)= )
( left[begin{array}{ccc}frac{1}{a} & 0 & 0 \ {[0.3 e m] 0} & frac{1}{b} & 0 \ {[0.3 e m] 0} & 0 & frac{1}{c}end{array}right] )
12
63 If ( D_{x}=25, D=5 ) are the values of the
determinants for certain simultaneous
equations in ( x ) and ( y, ) find ( x )
12
64 The value of ( mid begin{array}{cc}mathbf{2} & boldsymbol{a}+boldsymbol{b}+boldsymbol{c}+boldsymbol{d} \ boldsymbol{a}+boldsymbol{b}+boldsymbol{c}+boldsymbol{d} & boldsymbol{2}(boldsymbol{a}+boldsymbol{b})(boldsymbol{c}+boldsymbol{d}) \ boldsymbol{a} boldsymbol{b}+boldsymbol{c} boldsymbol{d} & boldsymbol{a} boldsymbol{b}(boldsymbol{c}+boldsymbol{d})+boldsymbol{c} boldsymbol{d}(boldsymbol{a}+boldsymbol{b})end{array} )
( mathbf{A} cdot mathbf{0} )
B.
( c .-1 )
D. None of these
12
65 ( left|begin{array}{ccc}2 & -3 & 3 \ 2 & 2 & 3 \ 3 & -2 & 2end{array}right| ) 12
66 Using the properties of determinant and
without expanding, prove that:
( left|begin{array}{lll}2 & 7 & 65 \ 3 & 8 & 75 \ 5 & 9 & 86end{array}right|=0 )
12
67 ( (1,6),(3 .-2) ) and ( (-2, K) ) are collinear
points. What is ( boldsymbol{K} ) ?
A . -6
B. 2
c. 8
D. 10
E . 18
12
68 If 07,a2, az ……., an….. are in G.P., then the value of the
determinant
[2004]
logan log an+1 log an+2|
log An+3 log an+4 log an+5 . is
log an+6 log an+7 log an+8|
(a) -2 (b) 1 (c) 2
(d) 0
.
12
69 Find determinant ( boldsymbol{D} boldsymbol{c}=left|begin{array}{ccc}mathbf{1} & mathbf{1} & -mathbf{2} \ mathbf{1} & -mathbf{2} & mathbf{3} \ mathbf{2} & -mathbf{1} & -mathbf{1}end{array}right| ) 12
70 f ( (k, 2-2 k),(-k+1,2 k),(-4- )
( k, 6-2 k) ) are collinear, then ( k= )
( A cdot+1 )
B. –
( c cdot-2 )
( D .2 )
12
71 Prove that:
( left|begin{array}{ccc}mathbf{0} & boldsymbol{a} & -boldsymbol{b} \ -boldsymbol{a} & boldsymbol{0} & -boldsymbol{c} \ boldsymbol{b} & boldsymbol{c} & boldsymbol{0}end{array}right|=mathbf{0} )
12
72 13.
If
and|A = 125 then the value of a is
(2004S)
(a) #1
(b) +2
(c) +3
(d) 5
12
73 If the points ( boldsymbol{A}(boldsymbol{x}, mathbf{2}), boldsymbol{B}(-mathbf{3},-mathbf{4}) ) and
( C(7,-5) ) are collinear, then the value of ( x ) is :
A . -63
B. 63
( c .60 )
D. – 60
12
74 ( left|begin{array}{ccc}a^{2}+1 & a b & a c \ a b & b^{2}+1 & b c \ a c & b c & c^{2}+1end{array}right|= )
A . abc
B. atb+c
c. ( 1+a^{2}+b^{2}+c^{2} )
( D cdot a b c(1+a+b+c) )
12
75 f ( x, y, z ) are all different and if ( left|begin{array}{lll}boldsymbol{x} & boldsymbol{x}^{2} & boldsymbol{1}+boldsymbol{x}^{3} \ boldsymbol{y} & boldsymbol{y}^{2} & boldsymbol{1}+boldsymbol{y}^{3} \ boldsymbol{z} & boldsymbol{z}^{2} & boldsymbol{1}+boldsymbol{z}^{3}end{array}right|=boldsymbol{0} operatorname{then} 1+boldsymbol{x} boldsymbol{y} boldsymbol{z}= )
A . -1
B.
( c cdot 1 )
D. 2
12
76 Find the adjoint of matrix ( A= ) ( left[begin{array}{lll}1 & 1 & 2 \ 2 & 3 & 5 \ 2 & 0 & 1end{array}right] ) 12
77 In a triangle ( A B C, ) with usual notations, if ( left|begin{array}{ccc}1 & a & b \ 1 & c & a \ 1 & b & cend{array}right|=0, ) then ( 4 sin ^{2} A+ )
( 24 sin ^{2} B+36 sin ^{2} C ) is equal to
A . 48
B. 50
c. 44
D. 34
12
78 Let ( triangle boldsymbol{a}=left|begin{array}{ccc}boldsymbol{a}-mathbf{1} & boldsymbol{n} & boldsymbol{6} \ (boldsymbol{a}-mathbf{1})^{2} & boldsymbol{2} boldsymbol{n}^{2} & boldsymbol{4} boldsymbol{n}-boldsymbol{2} \ (boldsymbol{a}-mathbf{1})^{3} & boldsymbol{3} boldsymbol{n}^{3} & boldsymbol{3} boldsymbol{n}^{2}-boldsymbol{3} boldsymbol{n}end{array}right| )
Then ( sum_{a-1}^{n} triangle a ) is equal to
( mathbf{A} cdot mathbf{0} )
В ( cdot(a-1) sum n^{2} )
c. ( (a-1) n sum n )
D. None of these
12
79 If ( boldsymbol{A}_{mathbf{3} times mathbf{3}} ) and ( |boldsymbol{A}| neq mathbf{0} Rightarrow boldsymbol{A} boldsymbol{d} boldsymbol{j}(boldsymbol{A} boldsymbol{d} boldsymbol{j} boldsymbol{A})= )
( mathbf{A} cdot|A|^{2} A )
B ( cdot|A| A )
c. ( frac{A}{|A|} )
D. ( frac{A}{|A|^{2}} )
12
80 Find the value of the following
determinant:
( left|begin{array}{cc}1.2 & 0.03 \ 0.57 & -0.23end{array}right| )
A. -0.266
B. -0.2471
c. -0.2381
D. -0.2931
12
81 ( left|begin{array}{ccc}1+sin ^{2} theta & sin ^{2} theta & sin ^{2} theta \ cos ^{2} theta & 1+cos ^{2} theta & cos ^{2} theta \ 4 sin 4 theta & 4 sin 4 theta & 1+4 sin 4 thetaend{array}right|= )
( 0, ) then ( sin 4 theta ) equals to
A. ( 1 / 2 )
B.
( c cdot-1 / 2 )
D. -1
12
82 The roots of the equation ( left|begin{array}{ccc}boldsymbol{x}-mathbf{1} & mathbf{1} & mathbf{1} \ mathbf{1} & boldsymbol{x}-mathbf{1} & mathbf{1} \ mathbf{1} & mathbf{1} & boldsymbol{x}-mathbf{1}end{array}right|=mathbf{0} operatorname{are} )
A. 1,2
в. -1,2
c. -1,-2
D. 1,-2
12
83 Find the value of ( left|begin{array}{ccc}53 & 106 & 159 \ 52 & 65 & 91 \ 102 & 153 & 221end{array}right| ) 12
84 The value of the determinant
( left|begin{array}{lll}k a & k^{2}+a^{2} & 1 \ k b & k^{2}+b^{2} & 1 \ k c & k^{2}+c^{2} & 1end{array}right| ) is
A. ( k(a+b)(b+c)(c+a) )
B. ( k a b cleft(a^{2}+b^{2}+c^{2}right) )
c. ( k(a-b)(b-c)(c-a) )
D. ( k(a+b-c)(b+c-a)(c+a-b) )
12
85 If ( boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{1} & mathbf{1} \ mathbf{1} & mathbf{1}+boldsymbol{x} & mathbf{1} \ mathbf{1} & mathbf{1} & mathbf{1}+boldsymbol{y}end{array}right] ) for ( boldsymbol{x} neq )
( mathbf{0}, boldsymbol{y} neq mathbf{0}, ) then ( boldsymbol{D} ) is:
A. divisible by neither ( x ) nor ( y )
B. divisible by both ( x ) nor ( y )
c. divisible by ( x ) but not ( y )
D. divisible by ( y ) but not ( x )
12
86 ( mathbf{f} mathbf{Delta}=left|begin{array}{lll}boldsymbol{b}^{2}-boldsymbol{a} boldsymbol{b} & boldsymbol{b}-boldsymbol{c} & boldsymbol{b} boldsymbol{c}-boldsymbol{a} boldsymbol{c} \ boldsymbol{a} boldsymbol{b}-boldsymbol{a}^{2} & boldsymbol{a}-boldsymbol{b} & boldsymbol{b}^{2}-boldsymbol{a} boldsymbol{b} \ boldsymbol{b} boldsymbol{c}-boldsymbol{a} boldsymbol{c} & boldsymbol{c}-boldsymbol{a} & boldsymbol{a} boldsymbol{b}-boldsymbol{a}^{2}end{array}right| ) then
( Delta ) equals
A ( cdot(b-c)(c-a)(a-b) )
B. ( a b c(b-c)(c-a)(a-b) )
c. ( (a+b+c)(b-c)(c-a)(a-b) )
D.
12
87 ( mathbf{f}_{mathbf{0}}=left|begin{array}{lll}boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{x} & boldsymbol{y} & boldsymbol{z} \ boldsymbol{p} & boldsymbol{q} & boldsymbol{r}end{array}right| ) and ( boldsymbol{Delta}_{2}=left|begin{array}{lll}boldsymbol{y} & boldsymbol{b} & boldsymbol{q} \ boldsymbol{x} & boldsymbol{a} & boldsymbol{p} \ boldsymbol{z} & boldsymbol{c} & boldsymbol{r}end{array}right| )
then ( Delta_{1} ) is equal to
( A cdot 2 Delta_{2} )
в. ( Delta_{2} )
( c cdot-Delta_{2} )
D. none of these
12
88 The value of the determinant
( left|begin{array}{ccc}cos alpha & -sin alpha & 1 \ sin alpha & cos alpha & 1 \ cos (alpha+beta) & -sin (alpha+beta) & 1end{array}right| )
A. Independent of ( alpha )
B. Independent of ( beta )
c. Independent of ( alpha ) and ( beta )
D. None of the above
12
89 f ( a neq b neq c, ) prove that the points
( left(a, a^{2}right),left(b, b^{2}right),left(c, c^{2}right) ) can never be
collinear.
12
90 ( mathbf{f}left(mathbf{1}+boldsymbol{a} boldsymbol{x}+boldsymbol{b} boldsymbol{x}^{2}right)^{4}=boldsymbol{a}_{mathbf{0}}+boldsymbol{a}_{1} boldsymbol{x}+ )
( a_{2} x^{2}+ldots ldots . .+a_{8} x^{8}, ) where
( a, b, a_{0}, a_{1}, dots dots a_{8} in R ) such that ( a_{0}+ )
( a_{1}+a_{2} neq 0 ) and ( left|begin{array}{lll}a_{0} & a_{1} & a_{2} \ a_{1} & a_{2} & a_{0} \ a_{2} & a_{0} & a_{1}end{array}right|=0, ) then
the value of ( frac{mathbf{5} a}{b} ) is
12
91 If ( k ) is a scalar and ( A ) is an ( n times n ) square
matrix, then ( |boldsymbol{k} boldsymbol{A}|= )
A ( cdot k|A|^{n} )
в. ( k|A| )
c ( cdot k^{n}left|A^{n}right| )
D ( cdot k^{n}|A| )
12
92 The value of the determinant
( left|begin{array}{c}b^{2}-a b b-c b c-a c \ a b-a^{2} a-b b^{2}-a b \ b c-a c c-a a b-a^{2}end{array}right|= )
A ( . a b c )
B. ( a+b+c )
c. 0
( mathbf{D} cdot a b+b c+c a )
12
93 If ( A ) is a ( 3- ) rowed square matrix and
( |boldsymbol{A}|=mathbf{4} ) then ( boldsymbol{a} boldsymbol{d} boldsymbol{j}(boldsymbol{a} boldsymbol{d} boldsymbol{j} boldsymbol{A})=? )
A ( .4 A )
в. ( 16 A )
( c cdot 64 A )
D. None of these
12
94 Prove that ( left|begin{array}{lll}boldsymbol{a}+boldsymbol{b} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{b}+boldsymbol{c} & boldsymbol{c} & boldsymbol{a} \ boldsymbol{c}+boldsymbol{a} & boldsymbol{a} & boldsymbol{b}end{array}right|=mathbf{3} boldsymbol{a} boldsymbol{b} boldsymbol{c}-boldsymbol{1} )
( boldsymbol{a}^{3}-boldsymbol{b}^{3}-boldsymbol{c}^{3} )
12
95 If ( A=left|begin{array}{ll}2 & 3 \ 6 & 9end{array}right| ) then ( |A|= )
A .
B. 1
( c cdot 2 )
D.
12
96 ( fleft|begin{array}{lll}boldsymbol{x}_{1} & boldsymbol{y}_{1} & mathbf{1} \ boldsymbol{x}_{2} & boldsymbol{y}_{2} & mathbf{1} \ boldsymbol{x}_{3} & boldsymbol{y}_{3} & 1end{array}right|=left|begin{array}{lll}boldsymbol{a}_{1} & boldsymbol{b}_{1} & mathbf{1} \ boldsymbol{a}_{2} & boldsymbol{b}_{2} & mathbf{1} \ boldsymbol{a}_{3} & boldsymbol{b}_{3} & 1end{array}right| ), then the
two triangles with vertices ( left(x_{1}, y_{1}right),left(x_{2}, y_{2}right),left(x_{3}, y_{3}right) ) and ( left(a_{1}, b_{1}right) )
( left(a_{2}, b_{2}right),left(a_{3}, b_{3}right) ) must be congruent
A. True
B. False
12
97 The value of ( left|begin{array}{ccc}(boldsymbol{a}+boldsymbol{d})(boldsymbol{a}+mathbf{2} boldsymbol{d}) & boldsymbol{a}+mathbf{2} boldsymbol{d} & boldsymbol{a} \ mathbf{2} boldsymbol{d}(boldsymbol{a}+mathbf{2} boldsymbol{d}) & boldsymbol{d} & boldsymbol{d} \ mathbf{2} boldsymbol{d}(boldsymbol{a}+mathbf{3} boldsymbol{d}) & boldsymbol{d} & boldsymbol{d}end{array}right| )
A . ( 4 d )
B ( .4 d^{2} )
( c cdot 4 d^{3} )
D. ( 4 d^{4} )
12
98 f ( a, b, c ) are non-zero and different from
1, then the value of
( left|begin{array}{ccc}log _{a} 1 & log _{a} b & log _{a} c \ log _{a}left(frac{1}{b}right) & log _{b} 1 & log _{a}left(frac{1}{c}right) \ log _{a}left(frac{1}{c}right) & log _{a} c & log _{c} 1end{array}right| )
A . 0
B. ( 1+log _{a}(a+b+c) )
( mathbf{c} cdot log _{a}(a b+b c+c a) )
D. 1
( E cdot log _{a}(a+b+c) )
12
99 9.
Which of the following values of a satisfy the equation
| (1+a)2 (1+2a)2 (1+3a)2
(2+ a)2 (2+2a)2 (2+3a)2 = -648a ?
|(3+a)? (3+2a)2 (3+3a)2 |
(a)
4
(6)
9
(c)
9
(JEE Adv. 2015)
(d) 4
12
100 Solve:
( left|begin{array}{ccc}mathbf{0} & -mathbf{3} & boldsymbol{x} \ boldsymbol{x}+mathbf{1} & mathbf{3} & mathbf{1} \ mathbf{4} & mathbf{1} & mathbf{5}end{array}right|=mathbf{0} )
12
101 solve: ( left|begin{array}{ccc}boldsymbol{x} & boldsymbol{2} & -mathbf{1} \ boldsymbol{3} & -boldsymbol{1} & boldsymbol{4} \ boldsymbol{x} & boldsymbol{2} & -boldsymbol{5}end{array}right|=mathbf{0} ) 12
102 Assertion
Let ( A ) be a ( 2 times 2 ) matrix with real
entries. Let ( I ) be the ( 2 times 2 ) identity
matrix. Denote by ( t r(A), ) the sum of
diagonal entries of ( A ). Assume that
( boldsymbol{A}^{2}=boldsymbol{I} )
If ( boldsymbol{A} neq boldsymbol{I} ) and ( boldsymbol{A} neq-boldsymbol{I}, ) then ( operatorname{det}(boldsymbol{A})=-mathbf{1} )
Reason
If ( boldsymbol{A} neq boldsymbol{I} ) and ( boldsymbol{A} neq-boldsymbol{I}, ) then ( boldsymbol{t r}(boldsymbol{A}) neq mathbf{0} )
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
103 Evaluate: ( left|begin{array}{ccc}mathbf{1} & boldsymbol{b} boldsymbol{c} & boldsymbol{a}(boldsymbol{b}+boldsymbol{c}) \ mathbf{1} & boldsymbol{c} boldsymbol{a} & boldsymbol{b}(boldsymbol{c}+boldsymbol{a}) \ boldsymbol{1} & boldsymbol{a} boldsymbol{b} & boldsymbol{c}(boldsymbol{a}+boldsymbol{b})end{array}right| )
( mathbf{A} cdot mathbf{0} )
B.
( c . a b c )
D. ( a^{2}+b^{2}+c^{2} )
12
104 x + 4y + z = 0;
If the system of linear equations
x + 2ay + az = 0 ; x + 3y + bz = 0 ; x + 4y + CZ
has a non – zero solution, then a, b, c.
(a) satisfy a + 2b + 3c = 0 (b) are in A.P
(c) are in G.P
(d) are in H.P.
12
105 Find the value of following determinant. ( left|begin{array}{ll}frac{7}{3} & frac{5}{3} \ frac{3}{2} & frac{1}{2}end{array}right| ) 12
106 If the points ( boldsymbol{A}(-2,1), B(a, b) ) and ( C(4,-1) ) are collinear and ( a-b=1 )
find the values of ( a ) and ( b )
A ( . a=1, b=5 )
В. ( a=1, b=0 )
c. ( a=2, b=0 )
D. None of these
12
107 Using properties of determinants, prove
the following
[
mid begin{array}{ccc}
1+a^{2}-b^{2} & 2 a b & -2 b \
2 a b & 1-a^{2}+b^{2} & 2 a \
2 b & -2 a & 1-a^{2}-b^{2}
end{array}
]
( left(1+a^{2}+b^{2}right)^{3} )
12
108 ( left|begin{array}{ccc}mathbf{1}+boldsymbol{x} & mathbf{2} & mathbf{3} \ mathbf{1} & mathbf{2}+boldsymbol{x} & mathbf{3} \ mathbf{1} & mathbf{2} & mathbf{3}+boldsymbol{x}end{array}right|=mathbf{0} operatorname{then} boldsymbol{x}= )
( A )
B. -1
( c .-6 )
( D )
12
109 Let ( n ) be a positive integer and ( Delta_{r}= ) ( left|begin{array}{ccc}2 r-1 & n & C_{r} & 1 \ n^{2}-1 & 2^{n} & n+1 \ cos ^{2}left(n^{2}right) & cos ^{2} n & cos ^{2}(n+1)end{array}right| ) then
( sum_{r=0}^{n} Delta_{r}=dots )
( A cdot 0 )
B.
( c cdot 2 )
( D )
12
110 (i) Solve the equation ( left|begin{array}{ccc}boldsymbol{x}-mathbf{1} & boldsymbol{2} & boldsymbol{3} \ mathbf{0} & boldsymbol{x}-boldsymbol{2} & boldsymbol{4} \ boldsymbol{0} & boldsymbol{0} & boldsymbol{x}-boldsymbol{3}end{array}right| )
(ii) Show that ( left|begin{array}{ccc}mathbf{1} & boldsymbol{a} & boldsymbol{b} boldsymbol{c} \ mathbf{1} & boldsymbol{b} & boldsymbol{c a} \ mathbf{1} & boldsymbol{c} & boldsymbol{a b}end{array}right|=left|begin{array}{ccc}mathbf{1} & boldsymbol{a} & boldsymbol{a}^{2} \ mathbf{1} & boldsymbol{b} & boldsymbol{b}^{2} \ mathbf{1} & boldsymbol{c} & boldsymbol{c}^{2}end{array}right| )
12
111 Prove the following:
( left|begin{array}{lll}boldsymbol{b}+boldsymbol{c} & boldsymbol{c}+boldsymbol{a} & boldsymbol{a}+boldsymbol{b} \ boldsymbol{a}+boldsymbol{b} & boldsymbol{b}+boldsymbol{c} & boldsymbol{c}+boldsymbol{a} \ boldsymbol{c}+boldsymbol{a} & boldsymbol{a}+boldsymbol{b} & boldsymbol{b}+boldsymbol{c}end{array}right|=mathbf{2}left|begin{array}{lll}boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{c} & boldsymbol{a} & boldsymbol{b} \ boldsymbol{b} & boldsymbol{c} & boldsymbol{a}end{array}right| )
12
112 ( mathbf{a}=left|begin{array}{lll}boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{c} & boldsymbol{a} & boldsymbol{b} \ boldsymbol{b} & boldsymbol{c} & boldsymbol{a}end{array}right|, ) then the value of
( left|begin{array}{ccc}a^{2}-b c & b^{2}-c a & c^{2}-a b \ c^{2}-a b & a^{2}-b c & b^{2}-c a \ b^{2}-c a & c^{2}-a b & a^{2}-b cend{array}right| )
( A cdot Delta^{2} )
B ( .2 Delta^{2} )
( c cdot Delta^{3} )
D. none of these
12
113 Solve: Value of ( boldsymbol{D}=left|begin{array}{ccc}mathbf{1} & mathbf{1} & mathbf{1} \ boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{a}^{2} & boldsymbol{b}^{2} & boldsymbol{c}^{2}end{array}right| ) is 12
114 If ( a neq 6, b, c ) satisfy ( left|begin{array}{ccc}a & 2 b & 2 c \ 3 & b & c \ 4 & a & bend{array}right|=0 )
then ( a b c= )
( mathbf{A} cdot a+b+c )
B.
( c cdot b^{3} )
( mathbf{D} cdot a b+b-c )
12
115 Using properties of determinants, show that triangle ( A B C ) is isosceles, if :
( mid begin{array}{ccc}mathbf{1} & mathbf{1} \ mathbf{1}+cos boldsymbol{A} & mathbf{1}+cos boldsymbol{B} & mathbf{1} \ cos ^{2} boldsymbol{A}+cos boldsymbol{A} & cos ^{2} boldsymbol{B}+cos boldsymbol{B} & mathbf{c o s}^{2}end{array} )
0
12
116 Let ( S ) be the sample space of all ( 3 times 3 )
matrices with entries from the set
( {0,1} . ) Let the events ( E_{1}= )
( {A in S: operatorname{det} A=0} ) and ( E_{2}= )
( {A in S: S u m text { of entries of } A text { is } 7} )
If a matrix is chosen at random from ( boldsymbol{S} )
then the conditional probability
( boldsymbol{P}left(boldsymbol{E}_{1} mid boldsymbol{E}_{2}right) ) equals.
12
117 ( left|begin{array}{lll}mathbf{a}+mathbf{b} & mathbf{a} & mathbf{b} \ mathbf{a} & mathbf{a}+mathbf{c} & mathbf{c} \ mathbf{b} & mathbf{c} & mathbf{b}+mathbf{c}end{array}right|= )
( A cdot 4 ) abc
B. abç
( c cdot 2 a^{2} b^{2} c^{2} )
D. ( 4 a^{2} b^{2} c^{2} )
12
118 ( boldsymbol{D}=left|begin{array}{ccc}mathbf{1 8} & mathbf{4 0} & mathbf{8 9} \ mathbf{4 0} & mathbf{8 9} & mathbf{1 9 8} \ mathbf{8 9} & mathbf{1 9 8} & mathbf{4 4 0}end{array}right|= )
( A )
B. –
c. zero
( D )
12
119 Let
( boldsymbol{f}(boldsymbol{x})=left|begin{array}{ccc}2 cot boldsymbol{x} & -mathbf{1} & mathbf{0} \ mathbf{1} & cot boldsymbol{x} & -mathbf{1} \ mathbf{0} & mathbf{1} & mathbf{2} cot boldsymbol{x}end{array}right| )
then
12
120 ( left|begin{array}{lll}a^{2}+lambda^{2} & a b+c lambda & c a-b lambda \ a b-c lambda & b^{2}+lambda^{2} & b c+a lambda \ c a+b lambda & b c-a lambda & c^{2}+lambda^{2}end{array}right| mid begin{array}{cc}lambda & c \ -c & lambda \ b & -aend{array} )
( left(1+a^{2}+b^{2}+c^{2}right)^{3}, ) then the value of ( lambda )
is
( A cdot 8 )
в. 2
( c )
( D )
12
121 Using properties of deteminants, prove
[
operatorname{that}left|begin{array}{ccc}
frac{(boldsymbol{a}+boldsymbol{b})^{2}}{boldsymbol{c}} & boldsymbol{c} & boldsymbol{c} \
boldsymbol{a} & frac{(boldsymbol{b}+boldsymbol{c})^{2}}{boldsymbol{a}} & boldsymbol{a} \
boldsymbol{b} & boldsymbol{b} & frac{(boldsymbol{c}+boldsymbol{a})^{2}}{boldsymbol{b}}
end{array}right|
]
( 2(a+b+c)^{3} )
12
122 Show that ( left|begin{array}{ccc}boldsymbol{b}+boldsymbol{c} & boldsymbol{c}+boldsymbol{a} & boldsymbol{a}+boldsymbol{b} \ boldsymbol{a}+boldsymbol{b} & boldsymbol{b}+boldsymbol{c} & boldsymbol{c}+boldsymbol{a} \ boldsymbol{a} & boldsymbol{b} & boldsymbol{c}end{array}right|= )
( boldsymbol{a}^{3}+boldsymbol{b}^{3}+boldsymbol{c}^{3}-boldsymbol{3} boldsymbol{a} boldsymbol{b} boldsymbol{c} )
12
123 Assertion If ( A=left(begin{array}{ll}cos alpha & sin alpha \ cos alpha & sin alphaend{array}right) ) and ( B= )
( left(begin{array}{cc}cos alpha & cos alpha \ sin alpha & sin alphaend{array}right) ) then ( A B neq I )
Reason
The product of two matrices can never be equal to an identity matrix
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
124 Solve: ( left|begin{array}{ccc}-1 & x & 2 \ 3 & 4 & -2 \ 4 & x & -3end{array}right|=0 ) 12
125 ( fleft|begin{array}{ccc}6 i & -3 i & 1 \ 4 & 3 i & -1 \ 20 & 3 & iend{array}right|=x+i y ) then
A ( . x=3, y=1 )
B. ( x=1, y=3 )
c. ( x=0, y=3 )
D. ( x=0, y=0 )
12
126 The value of ( left|begin{array}{ll}cos 15^{circ} & sin 15^{circ} \ sin 75^{circ} & cos 75^{circ}end{array}right| ) 12
127 If ( A=left[begin{array}{cc}-8 & 5 \ 2 & 4end{array}right] ) satisfies the equation ( x^{2}+4 x-p=0, ) then ( p= )
A . 64
B. 42
( c . ) 36
D. 24
12
128 ( mathrm{ft} mathrm{A}=left[begin{array}{ccc}1 & 0 & 1 \ 0 & 1 & 2 \ 0 & 0 & 4end{array}right] ) then show that
( |3 A|=27|A| )
12
129 If the points ( (-3,6),(-9, a) ) and (0,15) are collinear, then find a 12
130 f ( boldsymbol{A}=left[begin{array}{ll}boldsymbol{4} & boldsymbol{7} \ boldsymbol{6} & boldsymbol{5}end{array}right], ) find ( |boldsymbol{3} boldsymbol{A}| ) 12
131 If ( A A^{T}=I ) and ( operatorname{det}(A)=1, ) then
A. Every element of ( A ) is equal to it’s co-factor
B. Every element of A and it’s co-factor are additive inverse of each other
C. Every element of A and it’s co-factor are multiplicative inverse of each other
D. None of these
12
132 Find the non-zero roots of the equation ( boldsymbol{Delta}=left|begin{array}{ccc}boldsymbol{a} & boldsymbol{b} & boldsymbol{a} boldsymbol{x}+boldsymbol{b} \ boldsymbol{b} & boldsymbol{c} & boldsymbol{b} boldsymbol{x}+boldsymbol{c} \ boldsymbol{a} boldsymbol{x}+boldsymbol{b} & boldsymbol{b} boldsymbol{x}+boldsymbol{c} & boldsymbol{c}end{array}right|=0 ) 12
133 Find the values of ( x, ) if ( left|begin{array}{ll}mathbf{2} & mathbf{3} \ mathbf{4} & mathbf{5}end{array}right|=left|begin{array}{ll}boldsymbol{x} & mathbf{3} \ mathbf{2} boldsymbol{x} & mathbf{5}end{array}right| ) 12
134 3.
If 1,0,02 are the cube roots of unity, then
1
on
w2n|
A=0″
@21
1
is equal to
[2003]
(a) 0²
(b) 0
(0)
1
(d) w
12
135 ( boldsymbol{A}=left[begin{array}{ccc}mathbf{5} & mathbf{5} boldsymbol{a} & boldsymbol{a} \ mathbf{0} & boldsymbol{a} & mathbf{5} boldsymbol{a} \ mathbf{0} & mathbf{0} & mathbf{5}end{array}right] ) If ( left|boldsymbol{A}^{2}right|=mathbf{2 5} ) then
( |boldsymbol{a}|= )
( A cdot 5 )
В. ( 5^{2} )
( c )
( D )
12
136 f ( boldsymbol{y}=sin boldsymbol{p} boldsymbol{x} ) prove that
( boldsymbol{Delta}=left|begin{array}{lll}boldsymbol{y} & boldsymbol{y}_{1} & boldsymbol{y}_{2} \ boldsymbol{y}_{3} & boldsymbol{y}_{4} & boldsymbol{y}_{5} \ boldsymbol{y}_{6} & boldsymbol{y}_{7} & boldsymbol{y}_{8}end{array}right|=0 )
where ( y_{r}, ) means ( r-t h ) differential
coefficient of ( boldsymbol{y} )
12
137 ( f(x)=left|begin{array}{ccc}cos x & 1 & 0 \ 1 & cos x & 1 \ 0 & 1 & cos xend{array}right| ) the
( f^{prime}left(frac{pi}{3}right) ) equals
A ( frac{11 sqrt{3}}{8} )
( B cdot frac{5 sqrt{3}}{8} )
( c cdot-frac{5 sqrt{3}}{8} )
D. none of these
12
138 Find the relation between ( a ) and ( b ). If the
points ( boldsymbol{P}(1,2), Q(0,0) ) and ( P(a, b) ) are collinear
12
139 Show that ( left|begin{array}{lll}boldsymbol{x} & boldsymbol{x}^{2} & boldsymbol{y} boldsymbol{z} \ boldsymbol{y} & boldsymbol{y}^{2} & boldsymbol{z} boldsymbol{x} \ boldsymbol{z} & boldsymbol{z}^{2} & boldsymbol{x} boldsymbol{y}end{array}right|=(boldsymbol{x}-boldsymbol{y})(boldsymbol{y}- )
( z(z-x)(x y+y z+z x) )
12
140 If the points (0,4),(4,0) and ( (5, p) ) are collinear, then value of ( p ) is
A . -1
B. 7
( c cdot 6 )
D.
12
141 If ( boldsymbol{u}_{n}= )
then ( boldsymbol{u}_{n}=boldsymbol{a}_{n} boldsymbol{u}_{n-1}+boldsymbol{u}_{n-2} )
f true enter 1 else enter
12
142 Find the largest value of a third-order
determinant whose element are 0 or 1
12
143 Using properties of determinant, prove
[
text { that }left|begin{array}{lll}
boldsymbol{b}+boldsymbol{c} & boldsymbol{a}-boldsymbol{b} & boldsymbol{a} \
boldsymbol{c}+boldsymbol{a} & boldsymbol{b}-boldsymbol{c} & boldsymbol{b} \
boldsymbol{a}+boldsymbol{b} & boldsymbol{c}-boldsymbol{a} & boldsymbol{c}
end{array}right|=mathbf{3} boldsymbol{a} boldsymbol{b} boldsymbol{c}-boldsymbol{a}^{3}-
]
( b^{3}-c^{3} )
12
144 If ( boldsymbol{A}=[boldsymbol{a} boldsymbol{i} boldsymbol{j}] ) is a matrix of order ( 2 x boldsymbol{2} )
such that ( |A|=15 ) and cij represents
the co factor of aij then find ( a_{21} c_{21}+ )
( boldsymbol{a}_{22} boldsymbol{c}_{22} )
12
145 The value of ( left|begin{array}{ccc}boldsymbol{a}+boldsymbol{p} boldsymbol{d} & boldsymbol{a}+boldsymbol{q} boldsymbol{d} boldsymbol{a}+boldsymbol{r} boldsymbol{d} \ boldsymbol{p} & boldsymbol{q} & boldsymbol{r} \ boldsymbol{d} & boldsymbol{f} & boldsymbol{d}end{array}right| )
( mathbf{A} cdot mathbf{0} )
B. –
( c cdot 1 )
D. ( p+q+r )
12
146 The value of determinant ( left|begin{array}{ccc}mathbf{1 9} & mathbf{6} & mathbf{7} \ mathbf{2 1} & mathbf{3} & mathbf{1 5} \ mathbf{2 8} & mathbf{1 1} & mathbf{6}end{array}right| )
is :
A. 150
B. -110
( c cdot 0 )
D. None of these
12
147 Find the Adjoint matrix of the matrix ( left|begin{array}{lll}1 & 2 & 3 \ 2 & 3 & 2 \ 3 & 3 & 4end{array}right| ) 12
148 Find the value of the determinant with
out expanding:
( left|begin{array}{lll}5 & 2 & 3 \ 7 & 3 & 4 \ 9 & 4 & 5end{array}right| )
12
149 Let ( a, b, c ) be such that ( b(a+c) neq 0 ) ( left|begin{array}{ccc}boldsymbol{a} & boldsymbol{a}+mathbf{1} & boldsymbol{a}-mathbf{1} \ -boldsymbol{b} & boldsymbol{b}+mathbf{1} & boldsymbol{b}-mathbf{1} \ boldsymbol{c} & boldsymbol{c}-mathbf{1} & boldsymbol{c}+mathbf{1}end{array}right|+ )
( left|begin{array}{ccc}boldsymbol{a}+mathbf{1} & boldsymbol{b}+mathbf{1} & boldsymbol{c}-mathbf{1} \ boldsymbol{a}-mathbf{1} & boldsymbol{b}-mathbf{1} & boldsymbol{c}+mathbf{1} \ (-mathbf{1})^{n+mathbf{2}} boldsymbol{a} & (-mathbf{1})^{n+mathbf{1}} boldsymbol{b} & (-mathbf{1})^{n} boldsymbol{c}end{array}right|=mathbf{0} )
then the value of ( n ) is
A. Any integer
B. zero
c. Any even integer
D. Any odd integer
12
150 5
15. Let A = 10
Το
5α α
α 5α
O 5
then la equals
(a) 1/5
(6) 5
(0) 52
[2007]
(d) 1
12
151 f ( x, y z ) are all different and if ( left|begin{array}{lll}boldsymbol{x} & boldsymbol{x}^{boldsymbol{2}} & boldsymbol{1}+boldsymbol{x}^{boldsymbol{3}} \ boldsymbol{y} & boldsymbol{y}^{boldsymbol{2}} & boldsymbol{1}+boldsymbol{y}^{boldsymbol{3}} \ boldsymbol{z} & boldsymbol{z}^{boldsymbol{2}} & boldsymbol{1}+boldsymbol{z}^{boldsymbol{3}}end{array}right|=0 ) then ( mathbf{x y z}= )
A . -1
B.
( c . )
D. ±1
12
152 Evaluate the following determinant : ( = ) ( begin{array}{|lll|}mathbf{6 7} & mathbf{1 9} & mathbf{2 1} \ mathbf{3 9} & mathbf{1 3} & mathbf{1 4} \ mathbf{8 1} & mathbf{2 4} & mathbf{2 6}end{array} ) 12
153 ( begin{array}{|ccc|}text { If } boldsymbol{x}+boldsymbol{y}+boldsymbol{z}= & mathbf{0} text { ,find } \ boldsymbol{x} & boldsymbol{y} & boldsymbol{z} \ boldsymbol{x}^{mathbf{2}} & boldsymbol{y}^{mathbf{2}} & boldsymbol{z}^{mathbf{2}} \ boldsymbol{y}+boldsymbol{z} & boldsymbol{z}+boldsymbol{x} & boldsymbol{x}+boldsymbol{y}end{array} ) 12
154 Using the properties of determinant and without expanding, prove that:
( left|begin{array}{ccc}boldsymbol{x} & boldsymbol{a} & boldsymbol{x}+boldsymbol{a} \ boldsymbol{y} & boldsymbol{b} & boldsymbol{y}+boldsymbol{b} \ boldsymbol{z} & boldsymbol{c} & boldsymbol{z}+boldsymbol{c}end{array}right|=mathbf{0} )
12
155 Find ( ^{prime} x^{prime} ) if
( left|begin{array}{ccc}mathbf{4} & boldsymbol{x} & mathbf{6} \ mathbf{2} & mathbf{3} & mathbf{4} \ mathbf{1} & mathbf{1} & mathbf{1}end{array}right|=mathbf{1 0} )
12
156 If ( a>0, b>0, c>0 ) are respectively
the ( p^{t h}, q^{t h}, r^{t h} ) terms of a G.P., then the value of the deteminant ( left|begin{array}{lll}log boldsymbol{a} & boldsymbol{p} & 1 \ log boldsymbol{b} & boldsymbol{q} & 1 \ log boldsymbol{c} & boldsymbol{r} & 1end{array}right| ) is
A . 1
B. 0
( c cdot-1 )
D. None of these
12
157 Let ( 0<theta<pi / 2 ) and
( boldsymbol{Delta}(boldsymbol{x}, boldsymbol{theta})=left|begin{array}{ccc}boldsymbol{x} & tan boldsymbol{theta} & cot boldsymbol{theta} \ -tan boldsymbol{theta} & -boldsymbol{x} & mathbf{1} \ cot boldsymbol{theta} & boldsymbol{1} & boldsymbol{x}end{array}right| )
then
This question has multiple correct options
A ( cdot Delta(0, theta)=0 )
B. ( Deltaleft(x, frac{pi}{4}right)=x-x^{3} )
c. ( operatorname{Min}_{0<theta<pi / 2} Delta(1, theta)=0 )
D. ( Delta(x, theta) ) is independent of ( x )
12
158 The determinant ( left|begin{array}{ccc}sin alpha & cos alpha & 1 \ sin beta & cos beta & 1 \ sin gamma & cos gamma & 1end{array}right| ) is
equal to
This question has multiple correct options
( ^{mathrm{A}} cdot_{-4 sin } frac{alpha-beta}{2} sin frac{alpha-gamma}{2} sin frac{gamma-alpha}{2} )
( mathbf{B} cdot sin alpha+sin beta+sin gamma )
c. ( sin (alpha-beta)+sin (beta-gamma)+sin (gamma-alpha) )
D. none of these
12
159 The vertices of the triangle ( A B C ) are
( (2,1,1),(3,1,2),(-4,0,1) . ) The area of
triangle is
A ( cdot frac{3 sqrt{38}}{2} )
B. ( sqrt{38} )
c. ( frac{sqrt{38}}{2} )
D. 4
12
160 ( fleft(a_{1}, a_{2}, a_{3}, dots ) from a geometric right.
progression and ( a_{i}>0 ) for all ( i geq 1 )
[
operatorname{then}left|begin{array}{ccc}
log boldsymbol{a}_{m} & log boldsymbol{a}_{m+1} & log boldsymbol{a}_{m+2} \
log boldsymbol{a}_{m+3} & log boldsymbol{a}_{m+4} & log boldsymbol{a}_{m+5} \
log boldsymbol{a}_{m+boldsymbol{6}} & log boldsymbol{a}_{m+boldsymbol{7}} & log boldsymbol{a}_{m+8}
end{array}right| text { is }
]
equal to
( mathbf{A} cdot log a_{m+8}-log a_{m} )
B. ( log a_{m} )
( mathbf{c} cdot 2 log a_{m+1} )
D.
12
161 ( A_{3 * 3} ) is a non – singular matrix ( Rightarrow )
( boldsymbol{A}^{2}(boldsymbol{A} boldsymbol{d} boldsymbol{j} boldsymbol{A})= )
( mathbf{A} cdot|A| A )
B. ( I )
c. ( |A| )
D・ ( |A|^{2} I )
12
162 ( left|begin{array}{ccc}mathbf{0} & boldsymbol{p}-boldsymbol{q} & boldsymbol{p}-boldsymbol{r} \ boldsymbol{q}-boldsymbol{p} & boldsymbol{0} & boldsymbol{q}-boldsymbol{r} \ boldsymbol{r}-boldsymbol{p} & boldsymbol{r}-boldsymbol{q} & boldsymbol{0}end{array}right| ) is equal to
A. ( p+q+r )
B.
c. ( p-q-r )
( mathbf{D} cdot-p+q+r )
12
163 If ( t_{1}, t_{2} ) and ( t_{3} ) distinct. and the points
( left(t_{1} cdot 2 a t_{1}+a t_{1}^{3}right) cdotleft(t_{2} cdot 2 a t_{2}+a t_{2}^{3}right),left(t_{3} cdot 2 aright. )
are collinear, then ( t_{1}+t_{2}+t_{3}= )
A ( cdot t_{1} t_{2} t_{3}=-1 )
B ( cdot t_{1}+t_{2}+t_{3}=t_{1} t_{2} t )
( mathbf{c} cdot t_{1}+t_{2}+t_{3}=0 )
D. ( t_{1}+t_{2}+t_{3}=-1 )
12
164 Prove that ( :=(a-b)(b-c)(c-a)(a+ )
( b+c) )
[
begin{array}{ll}1 & a \ 1 & b \ 1 & c & a^{3}end{array}
]
12
165 Iet a b c be the real numbers. Then following system of
equations in x, y and z
(1995)
IN
NI
r2
y2 + 3 = 1 has
22
(a) no solution
(b) unique solution
(c) infinitely many solutions(d) finitely many solutions
Q2 x 22
62. x
12
166 f ( x, y, z ) are different and ( left|begin{array}{lll}boldsymbol{x} & boldsymbol{x}^{2} & boldsymbol{1}+boldsymbol{x}^{2} \ boldsymbol{y} & boldsymbol{y}^{2} & boldsymbol{1}+boldsymbol{y}^{2} \ boldsymbol{z} & boldsymbol{z}^{2} & boldsymbol{1}+boldsymbol{z}^{2}end{array}right|=mathbf{0} ) then prove that
( mathbf{1}+boldsymbol{x} boldsymbol{y} boldsymbol{z}=mathbf{0} )
12
167 Prove that
[
left|begin{array}{ccc}
mathbf{1} & mathbf{1} & mathbf{1}+mathbf{3} boldsymbol{x} \
mathbf{1}+mathbf{3} boldsymbol{y} & mathbf{1} & mathbf{1} \
mathbf{1} & mathbf{1}+mathbf{3} boldsymbol{z} & mathbf{1}
end{array}right|=mathbf{9}(mathbf{3} boldsymbol{x} boldsymbol{y} boldsymbol{z}+
]
( boldsymbol{x} boldsymbol{y}+boldsymbol{y} boldsymbol{z}+boldsymbol{z} boldsymbol{x} )
12
168 For what value of ( x ) the matrix ( A ) is
singular? ( boldsymbol{A}=left[begin{array}{ll}1+x & 7 \ 3-x & 8end{array}right] )
12
169 f ( boldsymbol{A}=left|begin{array}{cc}2 & mathbf{3} \ -mathbf{1} & mathbf{2}end{array}right| ) Find ( boldsymbol{A} ) 12
170 Consider the points ( boldsymbol{P}=(-sin (boldsymbol{beta}- )
( boldsymbol{alpha}),-cos beta), boldsymbol{Q}=(cos (beta-boldsymbol{alpha}), sin beta) )
and ( boldsymbol{R}=(cos (boldsymbol{beta}-boldsymbol{alpha}+boldsymbol{theta}), sin (boldsymbol{beta}-boldsymbol{theta})) )
where ( 0<alpha, beta<frac{pi}{4} ) then
A. ( P ) lies on the line segment ( R Q )
B. ( Q ) lies on the line segment ( P R )
c. ( R ) lies on the line segment ( Q P )
D. ( P, Q, R ) are non-collinear
12
171 ( mathbf{A}=left[begin{array}{lll}b^{2} c^{2} & b c & b+c \ c^{2} a^{2} & c a & c+a \ a^{2} b^{2} & a b & a+bend{array}right] ) then ( |A|=? )
A ( cdot a b c )
B . ( a b c-1 )
( c cdot a b c+1 )
D.
12
172 Evaluate:
( left|begin{array}{cc}x^{2}-x+1 & x-1 \ x+1 & x+1end{array}right| )
12
173 If ( A ) is a square matrix of order ( 3 times 3 )
such that ( |boldsymbol{A}|=mathbf{5}, ) then find ( |boldsymbol{4} boldsymbol{A}| )
12
174 ( A_{3 times 3} ) is a matrix such that ( |A|= ) ( boldsymbol{a}, boldsymbol{B}=(boldsymbol{a} boldsymbol{d} boldsymbol{j} boldsymbol{A}) ) such that ( |boldsymbol{B}|=boldsymbol{b} . ) Find
the value of ( frac{left(a b^{2}+a^{2} b+1right) S}{25} ) where ( frac{1}{2} S=frac{a}{b}+frac{a^{2}}{b^{3}}+frac{a^{3}}{b^{5}}+dots dots dots u p t o infty )
and ( boldsymbol{a}=mathbf{3} )
12
175 The number of ( boldsymbol{A} ) in ( boldsymbol{T}_{boldsymbol{p}} ) such that ( boldsymbol{A} ) is either symmetric or skew-symmetric or both, and det ( (A) ) divisible by ( p, ) is
A ( cdot(p-1)^{2} )
в. ( 2(p-1) )
c. ( (p-1)^{2}+1 )
D. ( 2 p-1 )
12
176 Consider the following statements in respect of the determinant ( left|begin{array}{cc}cos ^{2} frac{alpha}{2} & sin ^{2} frac{alpha}{2} \ sin ^{2} frac{beta}{2} & cos ^{2} frac{beta}{2}end{array}right| ) where ( alpha, beta ) are
complementary angles
1. The value of the determinant is ( frac{1}{sqrt{2}} cos left(frac{alpha-beta}{2}right) )
2. The maximum value of the determinant is ( frac{mathbf{1}}{sqrt{mathbf{2}}} )
Which of the above statements is/are
correct?
A. 1 only
B. 2 only
c. Both 1 and 2
D. Neither 1 nor 2
12
177 ( mathbf{a}=left|begin{array}{ccc}cos theta / 2 & 1 & 1 \ 1 & cos theta / 2 & -cos theta / 2 \ -cos theta / 2 & 1 & -1end{array}right| )
If the minimun of ( Delta ) is ( m_{1} ) and
maximum of ( Delta ) is ( m_{2}, ) then ( left[m_{1}, m_{2}right] ) are
related
A. [-4,-2]
B. [2, 4]
( c cdot[-4,0] )
D. [0, 2]
12
178 ( left|begin{array}{lll}1 & 2 & 3 \ 0 & 2 & 4 \ 0 & 0 & 5end{array}right| ) 12
179 Prove that ( left|begin{array}{lll}boldsymbol{b} boldsymbol{c} & boldsymbol{b} boldsymbol{c}^{prime}+boldsymbol{b}^{prime} boldsymbol{c} & boldsymbol{b}^{prime} boldsymbol{c}^{prime} \ boldsymbol{c} boldsymbol{a} & boldsymbol{c} boldsymbol{a}^{prime}+boldsymbol{c}^{prime} boldsymbol{a} & boldsymbol{c}^{prime} boldsymbol{a}^{prime} \ boldsymbol{a} boldsymbol{b} & boldsymbol{a} boldsymbol{b}^{prime}+boldsymbol{a}^{prime} boldsymbol{b} & boldsymbol{a}^{prime} boldsymbol{b}^{prime}end{array}right|= )
( left(boldsymbol{a} boldsymbol{b}^{prime}-boldsymbol{a}^{prime} boldsymbol{b}right)left(boldsymbol{b} boldsymbol{c}-boldsymbol{b}^{prime} boldsymbol{c}right)left(boldsymbol{c} boldsymbol{a}^{prime}-boldsymbol{c}^{prime} boldsymbol{a}right) )
12
180 Evaluate the determinant to the closest nteger: ( boldsymbol{A}=left[begin{array}{cc}log _{3} mathbf{5 1 2} & log _{4} mathbf{3} \ log _{3} mathbf{8} & log _{4} mathbf{9}end{array}right] ) 12
181 ( mathbf{f} mathbf{Delta}=left|begin{array}{ccc}mathbf{1} & mathbf{2} & mathbf{3} \ mathbf{2} & mathbf{0} & mathbf{1} \ mathbf{5} & mathbf{3} & mathbf{8}end{array}right|, ) write the minor of the
elements ( a_{22} )
12
182 ( fleft|begin{array}{lll}boldsymbol{x}+mathbf{1} & boldsymbol{x}+boldsymbol{2} & boldsymbol{x}+boldsymbol{a} \ boldsymbol{x}+mathbf{2} & boldsymbol{x}+boldsymbol{3} & boldsymbol{x}+boldsymbol{b} \ boldsymbol{x}+mathbf{3} & boldsymbol{x}+boldsymbol{4} & boldsymbol{x}+boldsymbol{c}end{array}right|=mathbf{0} ) then show
that ( a, b, c ) are in ( A . P )
12
183 ( mathrm{f}left|begin{array}{lll}boldsymbol{a}+boldsymbol{b} & boldsymbol{b}+boldsymbol{c} & boldsymbol{c}+boldsymbol{a} \ boldsymbol{c}+boldsymbol{a} & boldsymbol{a}+boldsymbol{b} & boldsymbol{b}+boldsymbol{c} \ boldsymbol{b}+boldsymbol{c} & boldsymbol{c}+boldsymbol{a} & boldsymbol{a}+boldsymbol{b}end{array}right|=boldsymbol{t} times operatorname{det} ) of
circulant matrix whose elements of first
column are ( a, b, c ) then ( t ) equals
A. 5
B. 6
( c cdot-2 )
( D )
12
184 (a) JL
How many 3 x 3 matrices M with entries from {0, 1, 2} are
there, for which the sum of the diagonal entries of M Mis
5?
(JEE Adv. 2017)
(a) 126 (b) 198 (c) 162 (d) 135
12
185 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{3} & mathbf{4}end{array}right], ) find ( |mathbf{2} boldsymbol{A}| ) 12
186 Evaluate the following deteminants:
i) ( left|begin{array}{cc}boldsymbol{x} & -mathbf{7} \ boldsymbol{x} & mathbf{5} boldsymbol{x}+mathbf{1}end{array}right| )
ii) ( mid begin{array}{cc}cos 15^{circ} & sin 15^{circ} \ sin 75^{circ} & cos 75^{circ}end{array} )
12
187 20. Let a, b, c be such that b(a + c)+ 0 if
[2009
a a+1 a 1 a+1 6+1 c-1
1-6 6+1 6-1 – a-1 6-1 ct1=0,
c c-1 c+1 |(–1)n+2a (+1) +1b (-1)” cl
then the value of n is :
(a) any even integer (b) any odd integer
(c) any integer
(d) zero
12
188 15.
(2003 – 2 Marks)
If Mis a 3 x 3 matrix, where det M=1 and MM=I, where I
is an identity matrix, prove that det (M-1)=0.
(2004 2 Maula
12
189 Let ( boldsymbol{d} in boldsymbol{R}, ) and ( boldsymbol{A}= )
( left[begin{array}{ccc}-2 & 4+d & (sin theta)-2 \ 1 & (sin theta)+2 & d \ 5 & (2 sin theta)-d & (-sin theta)+2+2 dend{array}right. )
( boldsymbol{theta} in[mathbf{0}, mathbf{2} boldsymbol{pi}] . ) If the minimum value of
( operatorname{det}(A) ) is ( 8, ) then a value of d is?
A . -7
B. ( 2(sqrt{2}+2) )
( c .-5 )
D. ( 2(sqrt{2}+1) )
12
190 ( mathbf{a}_{r}=left|begin{array}{ccc}mathbf{2}^{r}-mathbf{1} & mathbf{2} mathbf{.} mathbf{3}^{r}-mathbf{1} & mathbf{4} . mathbf{5}^{r}-mathbf{1} \ boldsymbol{alpha} & boldsymbol{beta} & boldsymbol{gamma} \ mathbf{2}^{n}-mathbf{1} & mathbf{3}^{n}-mathbf{1} & mathbf{5}^{n}-mathbf{1}end{array}right| )
then find the value of ( sum_{r=1}^{n} Delta_{r} )
( A )
в. ( alpha beta gamma )
( mathbf{c} cdot-alpha beta gamma )
( D )
12
191 If ( alpha, beta ) and ( gamma ) are the roots of the
equation ( x^{3}+p x+q=0 ) then the value of the determinant ( left|begin{array}{lll}boldsymbol{alpha} & boldsymbol{beta} & gamma \ boldsymbol{beta} & gamma & boldsymbol{alpha} \ boldsymbol{gamma} & boldsymbol{alpha} & boldsymbol{beta}end{array}right| ) is
2
( mathbf{A} cdot underline{p} )
B. ( q )
c. ( p^{2}-2 q )
( D )
12
192 One factor of ( Delta= )
( left|begin{array}{ccc}a^{2}+lambda & a b & a c \ a b & b^{2}+lambda & c b \ c a & c b & c^{2}+lambdaend{array}right| )
( A cdot lambda^{2} )
B. ( 1 / lambda )
c. ( left(a^{2}+lambdaright)left(b^{2}+lambdaright)left(c^{2}+lambdaright) )
D. none
12
193 Evaluate the following determinant:
( begin{array}{|ccc|}15 & 11 & 7 \ 11 & 17 & 14 \ 10 & 16 & 13end{array} )
12
194 Find the equation of line passing through the points (3,2) and (-1,3) by using determinants. 12
195 If ( A ) is a square matrix of order ( n times n )
and ( k ) is a scalar, then ( a d j(k A) ) is equal
to
A ( cdot k^{n-1} a d j A )
в. ( k^{n} ) adj ( A )
c. ( k^{n+1} a d j A )
D. kadj A
12
196 If ( f(x)=left|begin{array}{ccc}a & -1 & 0 \ a x & a & -1 \ a x^{2} & a x & aend{array}right|, ) using
properties of determinets find the value
of ( f(2 x)-f(x) ? )
12
197 The repeated factor of the determinant ( left|begin{array}{lll}boldsymbol{y}+boldsymbol{z} & boldsymbol{x} & boldsymbol{y} \ boldsymbol{z}+boldsymbol{x} & boldsymbol{z} & boldsymbol{x} \ boldsymbol{x}+boldsymbol{y} & boldsymbol{y} & boldsymbol{z}end{array}right| )
A. ( z-x )
в. ( x-y )
( mathbf{c} cdot y-z )
D. none of these
12
198 Evaluate ( left|begin{array}{ll}cos 65^{circ} & sin 65^{circ} \ sin 25^{circ} & cos 25^{circ}end{array}right| ) 12
199 ( operatorname{lf} Delta=left|begin{array}{lll}a_{1} & b_{1} & c_{1} \ a_{2} & b_{2} & c_{2} \ a_{3} & b_{3} & c_{3}end{array}right| ) and ( A_{2}, B_{2}, C_{2} ) are
respectively cofactors of ( a_{2}, b_{2}, c_{2} ) then
( a_{1} A_{2}+b_{1} B_{2}+c_{1} C_{2} ) is equal to
( A cdot-Delta )
B.
( c cdot Delta )
D. none of these
12
200 In the diagram on a lunar eilpse, if the positions od sun,Earth and moon are shown by ( (-4,6),(k,-2) ) and (5,-6)
respectively, then find the value of ( mathrm{k} )
12
201 If ( A=left[begin{array}{ll}alpha & 2 \ 2 & alphaend{array}right] ) and ( left|A^{3}right|=125 ) then ( alpha ) is
( mathbf{A} cdot pm 1 )
B. =2
( c .pm 3 )
D. ±5
12
202 Find the values of the following determinants
( left|begin{array}{cc}mathbf{1}+mathbf{3} i & boldsymbol{i}-mathbf{2} \ -boldsymbol{i}-mathbf{2} & mathbf{1}-mathbf{3} boldsymbol{i}end{array}right| )
where ( i=sqrt{-1} )
12
203 [2005]
10. If a? +62 +62 =-2 and
| 1+a’x (1+62)x (1+c%)x
f(x) = (1 + a²)x 1 +6²x (1+ c²)x.
(1 + a²)x (1+6²)x 1+ c²x
then f(x) is a polynomial of degree
(a) 1 (b) 0 (c) 3
(d) 2
12
204 ( left|begin{array}{ccc}left(a^{x}+a^{-x}right)^{2} & left(a^{x}-a^{-x}right)^{2} & 1 \ left(b^{x}+b^{-x}right)^{2} & left(b^{x}-b^{-x}right)^{2} & 1 \ left(c^{x}+c^{-x}right)^{2} & left(c^{x}-c^{-x}right)^{2} & 1end{array}right| ) is equa
to
( A )
B. ( 2 a b c )
( mathbf{c} cdot a^{2} b^{2} c^{2} )
D. None of these
12
205 ( mathbf{A}=left|begin{array}{lll}mathbf{5} & mathbf{3} & mathbf{8} \ mathbf{2} & mathbf{0} & mathbf{1} \ mathbf{1} & mathbf{2} & mathbf{3}end{array}right|, ) write the cofactor of
the element ( a_{32} )
12
206 ( |f(x)|=left[begin{array}{ccc}sin x & operatorname{cosec} x & tan x \ sec x & x sin x & x tan x \ x^{2}-1 & cos x & x^{2}+1end{array}right] )
then ( ldots int_{-a}^{a}|boldsymbol{f}(boldsymbol{x})| boldsymbol{d} ) equals
( A )
в.
( c cdot 2 a )
( D )
12
207 Let ( Delta= )
( left|begin{array}{ccc}sin theta cos phi & sin theta sin phi & cos theta \ cos theta cos phi & cos theta sin phi & -sin theta \ -sin theta sin phi & sin theta cos phi & 0end{array}right|, ) ther
( A cdot Delta ) is independent of ( theta )
B. ( Delta ) is independent of ( phi )
( c cdot Delta ) is a constant
D. ( Delta ) is dependent of ( phi )
12
208 Find the values of ( x, ) if ( left|begin{array}{cc}mathbf{2 x} & mathbf{5} \ mathbf{8} & boldsymbol{x}end{array}right|=left|begin{array}{ll}mathbf{6} & mathbf{5} \ mathbf{8} & mathbf{3}end{array}right| ) 12
209 ( left|begin{array}{ccc}mathbf{0} & boldsymbol{a} boldsymbol{b}^{2} & boldsymbol{a} boldsymbol{c}^{2} \ boldsymbol{a}^{2} boldsymbol{b} & boldsymbol{0} & boldsymbol{b} boldsymbol{c}^{2} \ boldsymbol{a} boldsymbol{c}^{2} & boldsymbol{b}^{2} boldsymbol{c} & boldsymbol{0}end{array}right|= )
( mathbf{A} cdot a^{3} b^{3} c^{3}+a^{2} b^{3} c^{4} )
B ( cdot a^{3} b^{3} c^{3} )
( mathbf{c} cdot 2 a^{3} b^{3} c^{3} )
D・ ( (2 a b c)^{3} )
12
210 ff ( boldsymbol{D}_{boldsymbol{r}}=left|begin{array}{ccc}boldsymbol{r}-mathbf{1} & boldsymbol{n} & boldsymbol{6} \ (boldsymbol{r}-mathbf{1})^{2} & boldsymbol{2} boldsymbol{n}^{2} & boldsymbol{4} boldsymbol{n}-boldsymbol{2} \ (boldsymbol{r}-mathbf{1})^{3} & boldsymbol{3} boldsymbol{n}^{3} & boldsymbol{3} boldsymbol{n}^{2}-boldsymbol{3} boldsymbol{n}end{array}right| )
then ( sum_{r=1}^{n} D_{r}= )
( A )
B.
c. ( frac{n(n-1)}{2}-r^{2} )
D. ( 2 n-n^{2} )
12
211 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{2} & mathbf{3}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{ll}mathbf{1} & mathbf{1} \ mathbf{0} & mathbf{0}end{array}right] )
then what is determinant of AB?
( mathbf{A} cdot mathbf{0} )
B.
c. 10
D. 20
12
212 12.
Given 2x – y + 2z=2, x – 2y+z=-4, x+y+nza
en the value of 2 such that the given system of equation
has NO solution, is
(a) 3 (6) 1 (c) 0 (d) -3
(2004S)
12
213 Consider the matrix ( boldsymbol{A}=left(begin{array}{ll}mathbf{3} & -mathbf{2} \ mathbf{4} & -mathbf{1}end{array}right) )
Then all possible values of ( lambda ) such that
the determinant of ( B=A-lambda I ) is 0
where ( boldsymbol{I}=left(begin{array}{ll}mathbf{1} & mathbf{0} \ mathbf{0} & mathbf{1}end{array}right) ) and ( i=sqrt{-mathbf{1}} )
( mathbf{A} cdot 1 pm 2 i )
B . ( 2 pm 3 i )
c. ( 3 pm 4 i )
( mathbf{D} cdot 5 pm 6 i )
12
214 Find the area of ( triangle P Q R ) whose vertices ( operatorname{are} boldsymbol{P}(mathbf{2}, mathbf{1}), boldsymbol{Q}(mathbf{3}, mathbf{4}) ) and ( boldsymbol{R}(mathbf{5}, mathbf{2}) ) 12
215 Evaluate ( left|begin{array}{ccc}boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ -mathbf{1} & mathbf{1} & -mathbf{1} \ mathbf{1} & mathbf{- 1} & mathbf{1}end{array}right| ) 12
216 The values of lying between O=0 and O=TJ2 and satisfying
the equation
(1988 – 2 Marks)
1+ sine cose
sin 1+ cos2 e
sin cos
4sin 40
4sin 40
1+4 sin 40
= 0 are
12
217 ( begin{array}{ccc}(boldsymbol{b}+boldsymbol{c})^{2} & boldsymbol{a}^{2} & boldsymbol{a}^{2} \ boldsymbol{b}^{2} & (boldsymbol{c}+boldsymbol{a})^{2} & boldsymbol{b}^{2} \ boldsymbol{c}^{2} & boldsymbol{c}^{2} & (boldsymbol{a}+boldsymbol{b})^{2}end{array} mid= ) 12
218 Find a value of ( boldsymbol{x} ) if ( left|begin{array}{cc}boldsymbol{x} & boldsymbol{2} \ boldsymbol{1} boldsymbol{8} & boldsymbol{x}end{array}right|=left|begin{array}{cc}boldsymbol{6} & boldsymbol{2} \ boldsymbol{1} boldsymbol{8} & boldsymbol{6}end{array}right| ) 12
219 Find the values of ( k, ) if the points ( boldsymbol{A}(boldsymbol{k}+ ) 1, ( 2 k ) ), ( B(3 k, 2 k+3) ) and ( C(5 k+1,5 k) )
are collinear.
This question has multiple correct options
A .
B. ( frac{1}{2} )
( c cdot 2 )
D. 2.
12
220 Prove that the points ( (a, 0),(0, b) ) and
(1,1) are collinear if ( left(frac{1}{a}+frac{1}{b}=1right) )
12
221 Let ( A ) be the matrix of order ( 3 times 3 ) such
that ( |boldsymbol{A}|=mathbf{1}, boldsymbol{B}=mathbf{2} boldsymbol{A}^{-1} ) and ( boldsymbol{C}=frac{(boldsymbol{a} d boldsymbol{j} boldsymbol{A})}{sqrt[3]{2}} )
then the value of ( left|A B^{2} . C^{3}right| ) is [Note : ( |A| ) represent determinant value of matrix A.]
12
222 The cofactors of elements in second row
of the determinant ( left|begin{array}{ccc}mathbf{2} & mathbf{- 1} & mathbf{4} \ mathbf{4} & mathbf{2} & mathbf{- 3} \ mathbf{1} & mathbf{1} & mathbf{2}end{array}right| ) are
( mathbf{A} cdot 5,6,4 )
В. 6,0,-3
c. 5,1,8
( mathbf{D} cdot 6,0,3 )
12
223 The number of positive integral
solutions of the equation ( left|begin{array}{ccc}boldsymbol{x}^{boldsymbol{3}}+mathbf{1} & boldsymbol{x}^{boldsymbol{2}} boldsymbol{y} & boldsymbol{x}^{boldsymbol{2}} boldsymbol{z} \ boldsymbol{x} boldsymbol{y}^{boldsymbol{2}} & boldsymbol{y}^{boldsymbol{3}}+boldsymbol{1} & boldsymbol{y}^{boldsymbol{2}} boldsymbol{z} \ boldsymbol{x} boldsymbol{z}^{boldsymbol{2}} & boldsymbol{y} boldsymbol{z}^{boldsymbol{2}} & boldsymbol{z}^{boldsymbol{3}}+mathbf{1}end{array}right|=mathbf{1 1} ) is
A.
B. 3
( c cdot 6 )
D. 12
12
224 Evaluate the following determinant:
( left|begin{array}{ccc}1 & 3 & 5 \ 2 & 6 & 10 \ 31 & 11 & 38end{array}right| )
12
225 ff ( Delta=left|begin{array}{ccc}mathbf{3} & mathbf{5} & mathbf{7} \ mathbf{2} & -mathbf{3} & mathbf{1} \ mathbf{1} & mathbf{1} & mathbf{2}end{array}right|, ) find it’s value 12
226 Let
( boldsymbol{f}(boldsymbol{x}, boldsymbol{y})=left|begin{array}{lll}boldsymbol{y}-boldsymbol{x}^{2} & boldsymbol{x}-boldsymbol{y}^{2} & boldsymbol{x} boldsymbol{y}-mathbf{1} \ boldsymbol{x}-boldsymbol{y}^{2} & boldsymbol{x} boldsymbol{y}-mathbf{1} & boldsymbol{y}-boldsymbol{x}^{2} \ boldsymbol{x} boldsymbol{y}-mathbf{1} & boldsymbol{y}-boldsymbol{x}^{2} & boldsymbol{x}-boldsymbol{y}^{2}end{array}right| )
find ( boldsymbol{f}(boldsymbol{2}, boldsymbol{2}) )
12
227 ( left|begin{array}{cc}boldsymbol{x}+mathbf{2} & boldsymbol{x} \ mathbf{4} & mathbf{3}end{array}right|=mathbf{0} ) find the value of ( mathbf{x} ) 12
228 If the points ( (3,-2),(x, 2) ) and (8,8) are collinear find ( 10 x ) using determinant 12
229 If ( (3,2),left(x, frac{22}{5}right),(8,8) ) lie on a line, then ( x ) is equal to
A . -5
B. 2
( c cdot 4 )
D. 5
12
230 Prove that:
( left|begin{array}{ccc}a^{2}+2 a & 2 a+1 & 1 \ 2 a+1 & a+2 & 1 \ 3 & 3 & 1end{array}right|=(a-1)^{2} )
12
231 Prove that ( :left|begin{array}{ccc}boldsymbol{a} & boldsymbol{c} & boldsymbol{a}+boldsymbol{c} \ boldsymbol{a}+boldsymbol{b} & boldsymbol{b} & boldsymbol{a} \ boldsymbol{b} & boldsymbol{b}+boldsymbol{c} & boldsymbol{c}end{array}right|= )
( 4 a b c )
12
232 Without expanding, show that the value of each of the following determinants is
zero:
( left|begin{array}{ccc}a & b & c \ a+2 x & b+2 y & c+2 z \ x & y & zend{array}right| )
12
233 Let ( omega neq 1 ) be a cube root of unity and ( S )
be the set of all non-singular matrices of the form ( left[begin{array}{ccc}mathbf{1} & boldsymbol{a} & boldsymbol{b} \ boldsymbol{omega} & boldsymbol{1} & boldsymbol{c} \ boldsymbol{omega}^{2} & boldsymbol{omega} & boldsymbol{1}end{array}right] ) Where each of
( a, b ) and ( c ) is either ( omega ) or ( omega^{2} ). Then the
number of distinct matrices in the set
( boldsymbol{S} ) is
( A cdot 2 )
B. 6
( c cdot 4 )
D. 8
12
234 How do I find
( boldsymbol{A}=left[begin{array}{ccc}1 & 2 & -2 \ -1 & 3 & 0 \ 0 & -2 & 1end{array}right]=|A|=? )
12
235 The number of solutions of equations ( left|begin{array}{ccc}sin 3 theta & -1 & 1 \ cos 2 theta & 4 & 3 \ 2 & 7 & 7end{array}right|=0 ) in ( [0,2 pi] ) is
A .2
B. 3
( c cdot 4 )
D. 5
12
236 If ( boldsymbol{u}=boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{2 b} boldsymbol{x} boldsymbol{y}+boldsymbol{c} boldsymbol{y}^{2}, boldsymbol{u}^{prime}=boldsymbol{a}^{prime} boldsymbol{x}^{2}+ )
( 2 b^{prime} x y+c^{prime} y^{2}, ) then prove that
( left|begin{array}{ccc}boldsymbol{y}^{2} & -boldsymbol{x} boldsymbol{y} & boldsymbol{x}^{2} \ boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{a}^{prime} & boldsymbol{b}^{prime} & boldsymbol{c}^{prime}end{array}right|= )
( left|begin{array}{cc}boldsymbol{a x}+boldsymbol{b y} & boldsymbol{b x}+boldsymbol{c y} \ boldsymbol{a}^{prime} boldsymbol{x}+boldsymbol{b}^{prime} boldsymbol{y} & boldsymbol{b}^{prime} boldsymbol{x}+boldsymbol{c}^{prime} boldsymbol{y}end{array}right|= )
( -frac{1}{y}left|begin{array}{cc}u & u^{prime} \ a x+b y & a^{prime} x+b^{prime} yend{array}right| )
12
237 zoaluate ( left|begin{array}{ccc}log _{x} x y z & log _{x} y & log _{x} z \ log _{y} x y z & 1 & log _{y} z \ log _{z} x y z & log _{z} y & 1end{array}right| ) 12
238 If ( 1, omega, omega^{2} ) are two cube roots of unity
then ( Delta=left|begin{array}{ccc}1 & omega^{n} & omega^{2 n} \ omega^{2 n} & 1 & omega^{n} \ omega^{n} omega^{2 n} & 1end{array}right| ) has the value
( mathbf{A} cdot mathbf{0} )
B.
( c cdot omega^{2} )
D.
12
239 If ( left.left.right|_{2} ^{4} 1right|^{2}=left|begin{array}{cc}3 & 2 \ 1 & xend{array}right|-left|begin{array}{cc}x & 3 \ -2 & 1end{array}right|, ) then ( x= )
( A cdot 6 )
B. 7
( c cdot 8 )
D. 16
12
240 ( left|begin{array}{cc}boldsymbol{x} & mathbf{2} \ mathbf{1 8} & boldsymbol{x}end{array}right|=left|begin{array}{cc}mathbf{6} & mathbf{2} \ mathbf{3} boldsymbol{x} & mathbf{6}end{array}right|, ) then ( boldsymbol{x} ) is equal to
( A cdot 6 )
B. ±6
( c .-6 )
( D )
12
241 ff ( x_{1}, x_{2}, x_{3} ) as well as ( y_{1}, y_{2}, y_{3} ) are in
G.P. with same common ratio, then the
points ( boldsymbol{P}left(boldsymbol{x}_{1}, boldsymbol{y}_{1}right), boldsymbol{Q}left(boldsymbol{x}_{2}, boldsymbol{y}_{2}right) ) and
( boldsymbol{R}left(boldsymbol{x}_{3}, boldsymbol{y}_{3}right) )
A. lies on a straight line
B. lie on an ellipse
c. lie on a circle
D. are vertices of a triangle
12
242 Show that:
( left|begin{array}{lll}boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{b} & boldsymbol{c} & boldsymbol{a} \ boldsymbol{c} & boldsymbol{a} & boldsymbol{b}end{array}right|^{2}= )
[
left|begin{array}{ccc}
2 b c-a^{2} & c^{2} & b^{2} \
c^{2} & 2 a c-b^{2} & a^{2} \
b^{2} & a^{2} & 2 a b-c^{2}
end{array}right|=
]
( left(a^{3}+b^{3}+c^{3}-3 a b cright)^{2} )
12
243 Find the value of: ( left|begin{array}{ll}-3 & -5 \ -2 & -1end{array}right| ) 12
244 <Tt,
1. The number of all possible values of 0, where 0 <
for which the system of equations
(y + 2) cos 30 = (xyz) sin 30
2 cos 30 2 sin 30
x sin 30= =
Y Z
(xyz) sin 30= (y + 2z) cos 30+ y sin30
has a solution (xo, Yo, zo) with yożo 70 is
– +-
12
245 ( operatorname{Let} boldsymbol{A}=left[begin{array}{lll}mathbf{1} & mathbf{0} & mathbf{0} \ mathbf{2} & mathbf{1} & mathbf{0} \ mathbf{3} & mathbf{2} & mathbf{1}end{array}right] ) and ( boldsymbol{U}_{1}, boldsymbol{U}_{2}, boldsymbol{U}_{3} ) be
column
matrices satisfying ( boldsymbol{A} boldsymbol{U}_{1}= ) ( left[begin{array}{l}1 \ 0 \ 0end{array}right], A U_{2}=left[begin{array}{l}2 \ 3 \ 0end{array}right], A U_{3}=left[begin{array}{l}2 \ 3 \ 1end{array}right] . ) If ( U ) is
( 3 times 3 ) matrix whose columns are
( boldsymbol{U}_{1}, boldsymbol{U}_{2}, boldsymbol{U}_{3}, ) then ( |boldsymbol{U}|= )
A . 3
B. -3
( c cdot frac{3}{2} )
( D )
12
246 Without expanding at any stage,
evaluate the value of determinant
12
247 ( left|begin{array}{ccc}boldsymbol{x}_{1} & boldsymbol{y}_{1} & mathbf{1} \ boldsymbol{x}_{2} & boldsymbol{y}_{2} & mathbf{1} \ boldsymbol{x}_{3} & boldsymbol{y}_{3} & 1end{array}right|=left|begin{array}{ccc}mathbf{1} & mathbf{1} & mathbf{1} \ boldsymbol{b}_{1} & boldsymbol{b}_{2} & boldsymbol{b}_{3} \ boldsymbol{a}_{1} & boldsymbol{a}_{2} & boldsymbol{a}_{3}end{array}right| ) then the
two triangles whose vertices are
( left(x_{1}, y_{1}right),left(x_{2}, y_{2}right),left(left(x_{3}, y_{3}right) ) and right.
( left(a_{1}, b_{1}right),left(a_{2}, b_{2}right),left(a_{13}, b_{3}right), ) are
A . congruent
B. similar
c. equal in area
D. none of these
12
248 If each row of a determinant of third
order of value ( Delta ) is multipled by ( 3, ) then the value of new determinant is
A. ( Delta )
B. ( 27 Delta )
c. ( 21 Delta )
D. ( 54 Delta )
12
249 Adj ( left[begin{array}{ccc}1 & 0 & 2 \ -1 & 5 & -2 \ 0 & 2 & 1end{array}right]= )
( left[begin{array}{ccc}mathbf{9} & boldsymbol{a} & mathbf{- 2} \ mathbf{- 1} & mathbf{1} & mathbf{0} \ mathbf{- 2} & mathbf{2} & boldsymbol{b}end{array}right] Rightarrowleft[begin{array}{ll}boldsymbol{a} & boldsymbol{b}end{array}right]= )
( left.begin{array}{ll}text { A. }[-4 & 5end{array}right] )
B ( cdotleft[begin{array}{ll}-4 & -1end{array}right] )
( mathbf{c} cdotleft[begin{array}{ll}4 & 1end{array}right] )
D・[4 -1
12
250 Find the value of ( x ) for which the points
( (x,-1),(2,1) ) and (4,5) are collinear.
12
251 If ( mathbf{A} ) is an unitary matrix then ( |boldsymbol{A}| ) is
equal to:
( mathbf{A} cdot mathbf{1} )
B. –
( c .pm 1 )
D. 2
12
252 begin{tabular}{|ccc}
If ( boldsymbol{f}(boldsymbol{x})= ) & \
( mathbf{1} ) & ( boldsymbol{x} ) & ( boldsymbol{x} ) \
( boldsymbol{2} boldsymbol{x} ) & ( boldsymbol{x}(boldsymbol{x}-mathbf{1}) ) & ( (boldsymbol{x}-1) ) \
( boldsymbol{3} boldsymbol{x}(boldsymbol{x}-mathbf{1}) ) & ( boldsymbol{x}(boldsymbol{x}-mathbf{1})(boldsymbol{x}-mathbf{2}) ) & ( (boldsymbol{x}+mathbf{1}) )
end{tabular}
then ( f(100) ) is equal to
A.
B.
( c cdot 100 )
D. -100
12
253 Let ( A ) be a ( 3 times 3 ) matrix and ( B ) be its
adjoint matrix. If ( |boldsymbol{B}|=mathbf{6 4}, ) then ( |boldsymbol{A}|= )
( A cdot pm 2 )
B. ±4
( c .pm 8 )
D. ±12
12
254 The value of determinant
( left|begin{array}{ccc}boldsymbol{x} & boldsymbol{x}+boldsymbol{y} & boldsymbol{x}+boldsymbol{2} boldsymbol{y} \ boldsymbol{x}+boldsymbol{2} boldsymbol{y} & boldsymbol{x} & boldsymbol{x}+boldsymbol{y} \ boldsymbol{x}+boldsymbol{y} & boldsymbol{x}+boldsymbol{2} boldsymbol{y} & boldsymbol{x}end{array}right| ) is:
A ( cdot 9 x^{2}(x+y) )
В ( cdot 9 y^{2}(x+y) )
c. ( 3 y^{2}(x+y) )
D. ( 7 x^{2}(x+y) )
12
255 If ( Delta_{1}=left|begin{array}{ll}mathbf{1} & mathbf{0} \ boldsymbol{a} & boldsymbol{b}end{array}right| ) and ( boldsymbol{Delta}_{2}=left|begin{array}{ll}mathbf{1} & mathbf{0} \ boldsymbol{c} & boldsymbol{d}end{array}right| ) then
( Delta_{2} Delta_{1} ) is equal to
( mathbf{A} cdot a c )
B. ( b d )
c. ( (b-a)(d-c) )
D. none of these
12
256 Evaluate ( sum_{n=1}^{N} U_{n} ) if
[
boldsymbol{U}_{boldsymbol{n}}=mid begin{array}{ccc}boldsymbol{n} & mathbf{1} & mathbf{5} \ boldsymbol{n}^{2} & boldsymbol{2} boldsymbol{N}+mathbf{1} & boldsymbol{2} boldsymbol{N}+mathbf{1} \ boldsymbol{n}^{boldsymbol{3}} & boldsymbol{3} boldsymbol{N}^{2} & boldsymbol{3} boldsymbol{N}end{array}
]
12
257 f ( x, y, z ) are complex numbers, then ( Delta= ) ( left|begin{array}{ccc}mathbf{0} & -boldsymbol{y} & -boldsymbol{z} \ overline{boldsymbol{y}} & mathbf{0} & -boldsymbol{x} \ overline{boldsymbol{z}} & overline{boldsymbol{x}} & boldsymbol{0}end{array}right| ) is equal
A. Purely real
B. Purely imaginary
c. complex
D.
12
258 ( operatorname{Let} A=left|begin{array}{lll}boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{p} & boldsymbol{q} & boldsymbol{r} \ boldsymbol{x} & boldsymbol{y} & boldsymbol{z}end{array}right| ) and suppose that
det. ( (A)=2 ) then the det.(B) equals, where ( boldsymbol{B}=left|begin{array}{ccc}mathbf{4} boldsymbol{x} & mathbf{2} boldsymbol{a} & -boldsymbol{p} \ mathbf{4} boldsymbol{y} & mathbf{2} boldsymbol{b} & -boldsymbol{q} \ boldsymbol{4} boldsymbol{z} & boldsymbol{2} boldsymbol{c} & -boldsymbol{t}end{array}right| )
A. ( operatorname{det}(B)=-2 )
B. ( operatorname{det}(B)=-8 )
c. ( operatorname{det}(B)=-16 )
D. ( operatorname{det}(B)=8 )
12
259 If ( a, b, c ) are real, then ( f(x)= ) ( left|begin{array}{ccc}boldsymbol{x}+boldsymbol{a}^{2} & boldsymbol{a} boldsymbol{b} & boldsymbol{a} boldsymbol{c} \ boldsymbol{a} boldsymbol{b} & boldsymbol{x}+boldsymbol{b}^{2} & boldsymbol{b} boldsymbol{c} \ boldsymbol{a} boldsymbol{c} & boldsymbol{b} boldsymbol{c} & boldsymbol{x}+boldsymbol{c}^{2}end{array}right| ) is decreasing
in
A ( cdotleft(-frac{2}{3}left(a^{2}+b^{2}+c^{2}right), 0right) )
B. ( left(0, frac{2}{3}left(a^{2}+b^{2}+c^{2}right)right) )
( left(frac{a^{2}+b^{2}+c^{2}}{3}, 0right) )
D. None of these
12
260 Using determinants show that points ( boldsymbol{A}(boldsymbol{a}, boldsymbol{b}+boldsymbol{c}), boldsymbol{B}(boldsymbol{b}, boldsymbol{c}+boldsymbol{a}) ) and ( boldsymbol{C}(boldsymbol{c}, boldsymbol{a}+boldsymbol{b}) )
are col-linear.
12
261 Evaluate the following determinant:
( left|begin{array}{lll}a & h & g \ h & b & f \ g & f & cend{array}right| )
12
262 Using properties of determinant, prove
[
text { that }left|begin{array}{lll}
boldsymbol{b}+boldsymbol{c} & boldsymbol{a}-boldsymbol{b} & boldsymbol{a} \
boldsymbol{c}+boldsymbol{a} & boldsymbol{b}-boldsymbol{c} & boldsymbol{b} \
boldsymbol{a}+boldsymbol{b} & boldsymbol{c}-boldsymbol{a} & boldsymbol{c}
end{array}right|=mathbf{3} boldsymbol{a} boldsymbol{b} boldsymbol{c}-boldsymbol{a}^{3}-
]
( b^{3}-c^{3} )
12
263 ( mathrm{ff}=left[begin{array}{ccc}boldsymbol{a} & mathbf{0} & mathbf{0} \ {[mathbf{0 . 3 e m}] mathbf{0}} & boldsymbol{a} & mathbf{0} \ {[mathbf{0 . 3 e m}] mathbf{0}} & mathbf{0} & boldsymbol{a}end{array}right], ) then the value
of |A| |Adj. A|
A ( cdot a^{3} )
в. ( a^{6} )
( c cdot a^{9} )
( mathbf{D} cdot a^{2} )
12
264 Solve ( left|begin{array}{ccc}mathbf{1} & mathbf{1} & mathbf{1} \ mathbf{1} & mathbf{1}+boldsymbol{x} & mathbf{1} \ mathbf{1} & mathbf{1} & mathbf{1}+boldsymbol{y}end{array}right| )
( A )
B.
( c )
D. ( x y )
12
265 Using the method of slope, show that the following points are collinear:
( (i) A(4,8), B(5,12), C(9,28) )
( (i i) A(16,-18), B(3,-6), C(-10,6) )
12
266 If ( A ) is a singular matrix, then ( A(operatorname{adj} A) )
is a
A. scalar matrix
B. zero matrix
c. identity matrix
D. orthogonal matrix
12
267 Find the value of ( mathbf{x} ) for which ( left|begin{array}{ll}3 & x \ x & 1end{array}right|= )
( left|begin{array}{ll}3 & 2 \ 8 & 1end{array}right| )
12
268 The line ( A x+B y+C=0 ) cuts the
circle ( boldsymbol{x}^{2}+boldsymbol{y}^{2}+boldsymbol{a} boldsymbol{x}+boldsymbol{b} boldsymbol{y}+boldsymbol{c}=boldsymbol{0} ) in ( boldsymbol{P} )
and ( Q )
The line ( boldsymbol{A}^{prime} boldsymbol{x}+boldsymbol{B}^{prime} boldsymbol{y}+boldsymbol{C}^{prime}=mathbf{0} ) cuts the
circle ( boldsymbol{x}^{2}+boldsymbol{y}^{2}+boldsymbol{a}^{prime} boldsymbol{x}+boldsymbol{b}^{prime} boldsymbol{y}+boldsymbol{c}^{prime}=mathbf{0} ) in ( boldsymbol{R} )
and ( S ).
If ( P, Q, R, S ) are concyclic, then show
that
( left|begin{array}{ccc}boldsymbol{a}-boldsymbol{a}^{prime} & boldsymbol{b}-boldsymbol{b}^{prime} & boldsymbol{c}-boldsymbol{c}^{prime} \ boldsymbol{A} & boldsymbol{B} & boldsymbol{C} \ boldsymbol{A}^{prime} & boldsymbol{B}^{prime} & boldsymbol{C}^{prime}end{array}right|=mathbf{0} )
12
269 Find the value of the following
determinant:
( left|begin{array}{cc}mathbf{3} sqrt{mathbf{6}} & -mathbf{4} sqrt{mathbf{2}} \ mathbf{5} sqrt{mathbf{3}} & mathbf{2}end{array}right| )
A. ( 20 sqrt{6} )
B. ( 16 sqrt{6} )
c. ( 26 sqrt{6} )
D. ( 10 sqrt{6} )
12
270 Evaluate ( Delta=mid begin{array}{cc}cos 2 alpha & sin 2 alpha \ -sin 3 alpha & cos 3 alphaend{array} ) 12
271 ff ( f(x), g(x), h(x) ) are polynomials in ( x ) of ( operatorname{degree} 2 ) and ( F(x)=left|begin{array}{ccc}boldsymbol{f} & boldsymbol{g} & boldsymbol{h} \ boldsymbol{f}^{prime} & boldsymbol{g}^{prime} & boldsymbol{h}^{prime} \ boldsymbol{f}^{prime prime} & boldsymbol{g}^{prime prime} & boldsymbol{h}^{prime prime}end{array}right|, ) then
( F(x) ) is equal to
( A cdot 1 )
B.
( c cdot-1 )
D. ( f(x) . g(x) . h(x) )
12
272 Let ( t ) be a positive integer and ( Delta_{t}= ) ( left|begin{array}{lll}mathbf{2} t-1 & boldsymbol{m}^{2}-mathbf{1} & cos ^{2}left(boldsymbol{m}^{2}right) \ boldsymbol{m}_{C_{t}} & mathbf{2}^{m} & cos ^{2}(boldsymbol{m}) \ mathbf{1} & boldsymbol{m}+mathbf{1} & cos left(boldsymbol{m}^{2}right)end{array}right| ) then the
value of ( sum_{t=0}^{m} Delta_{t} ) is equal to:
( A cdot 2^{m} )
B.
( mathbf{c} cdot 2^{m} cos ^{2}left(2^{m}right) )
D. ( m^{2} )
12
273 The point ( (-a,-b),(0,0),(a, b) ) and
( left(a^{2}, a bright) ) are
A. collinear
B. concyclic
c. vertices of a rectangle
D. vertices of a parallelogram
12
274 ff ( f(x)= )
( f(x-a)(x-b)(x-c)(x-d) ) then
prove that
( Delta=left|begin{array}{cccc}boldsymbol{a} & boldsymbol{x} & boldsymbol{x} & boldsymbol{x} \ boldsymbol{x} & boldsymbol{b} & boldsymbol{x} & boldsymbol{x} \ boldsymbol{x} & boldsymbol{x} & boldsymbol{c} & boldsymbol{x} \ boldsymbol{x} & boldsymbol{x} & boldsymbol{x} & boldsymbol{d}end{array}right|=boldsymbol{f}(boldsymbol{x})-boldsymbol{x} boldsymbol{f}^{prime}(boldsymbol{x}) )
12
275 Let ( m ) be a positive integer and ( Delta_{r}= ) ( left|begin{array}{ccc}mathbf{2 r}-mathbf{1} & boldsymbol{m} boldsymbol{C}_{boldsymbol{r}} & mathbf{1} \ boldsymbol{m}^{mathbf{2}}-mathbf{1} & mathbf{2}^{boldsymbol{m}} & boldsymbol{m}+mathbf{1} \ sin ^{2}left(boldsymbol{m}^{2}right) & sin ^{2}(boldsymbol{m}) & sin ^{2}(boldsymbol{m}+mathbf{1})end{array}right| )
( boldsymbol{r} leq boldsymbol{m}) . ) Then the value of ( sum_{r=0}^{m} boldsymbol{Delta}_{boldsymbol{r}} )
( A . )
B. ( m^{2}-1 )
( c cdot 2^{m} )
D ( cdot 2^{m} sin ^{2}left(2^{m}right) )
12
276 Using the property of determinants and with out expanding prove that ( left|begin{array}{lll}boldsymbol{a}-boldsymbol{b} & boldsymbol{b}-boldsymbol{c} & boldsymbol{c}-boldsymbol{a} \ boldsymbol{b}-boldsymbol{c} & boldsymbol{c}-boldsymbol{a} & boldsymbol{a}-boldsymbol{b} \ boldsymbol{c}-boldsymbol{a} & boldsymbol{a}-boldsymbol{b} & boldsymbol{b}-boldsymbol{c}end{array}right|=mathbf{0} ) 12
277 a b c]
14. If matrix A= b c a where a, b, c are real positive
Lc a b
numbers, abc = 1 and ATA = 1, then find the value of
a3 + b3 + c3
(2003 – 2 Marks)
12
278 ( f(x)=left|begin{array}{ccc}2 x & x^{2} & x^{3} \ x^{2}+2 x & 1 & 3 x+1 \ 2 x & 1-3 x^{2} & 5 xend{array}right| )
then find ( boldsymbol{f}^{prime}(mathbf{1}) )
12
279 ( left|begin{array}{ccc}mathbf{1} & boldsymbol{w} & boldsymbol{w}^{2} \ boldsymbol{w} & boldsymbol{w}^{2} & mathbf{1} \ boldsymbol{w}^{2} & boldsymbol{w} & mathbf{1}end{array}right| ) Where ( mathbf{w} ) is a complex
cube root of unity
12
280 Prove
[
left|begin{array}{ccc}
x^{2} & y^{2} & z^{2} \
(x+1)^{2} & (y+1)^{2} & (z+1)^{2} \
(x-1)^{2} & (y-1)^{2} & (z-1)^{2}
end{array}right|=
]
( -4(x-y)(y-z)(x-z) )
12
281 If ( f(x)= ) ( left|begin{array}{ccc}(mathbf{1}+mathbf{3} boldsymbol{x})^{2 boldsymbol{a}} & (mathbf{1}+mathbf{2} boldsymbol{x})^{boldsymbol{3} boldsymbol{b}} & mathbf{1} \ mathbf{1} & (mathbf{1}+mathbf{3} boldsymbol{x})^{2 boldsymbol{a}} & (mathbf{1}+mathbf{2} boldsymbol{x})^{boldsymbol{3} boldsymbol{b}} \ (mathbf{1}+mathbf{2} boldsymbol{x})^{boldsymbol{3} boldsymbol{b}} & mathbf{1} & (mathbf{1}+mathbf{3} boldsymbol{x})^{mathbf{2} boldsymbol{a}}end{array}right| )
then
A. ( f(x) ) has constant term 1
B. constant term is ( 2 a-3 b )
c. coefficient of ( x ) in ( f(x) ) is zero
D. constant term is ( 2 a+3 b )
12
282 f ( a, b, c ) are real numbers such that ( left|begin{array}{ccc}boldsymbol{b}+boldsymbol{c} & boldsymbol{c}+boldsymbol{a} & boldsymbol{a}+boldsymbol{b} \ boldsymbol{c}+boldsymbol{a} & boldsymbol{a}+boldsymbol{b} & boldsymbol{b}+boldsymbol{c} \ boldsymbol{a}+boldsymbol{b} & boldsymbol{b}+boldsymbol{c} & boldsymbol{c}+boldsymbol{a}end{array}right|=0, ) then show
that either ( boldsymbol{a}+boldsymbol{b}+boldsymbol{c}=boldsymbol{0} ) or, ( boldsymbol{a}=boldsymbol{b}=boldsymbol{c} )
12
283 1.
If a > 0 and discriminant of ax2+2bx+c is-ve,
[2001
a
b
ax+b
b
c
bx+c
ax + bl.
bx+c is equal to
o
(b) (ac-b2)(ax2 +2bx+c)
(d) 0
(a) +ve
(C) -ve
12
284 Assertion
If ( A ) is skew symmetric matrix of order 3 then its determinant should be zero
Reason

If ( A ) is square matrix, then ( d e t A= )
( operatorname{det} A^{prime}=operatorname{det}left(-A^{prime}right) )
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
285 ( begin{array}{lllll}mathbf{0} & & mathbf{c o s} boldsymbol{alpha} & mathbf{c o s} & boldsymbol{beta} \ mathbf{c o s} & boldsymbol{alpha} & mathbf{0} & mathbf{c o s} & gamma \ mathbf{c o s} & boldsymbol{beta} & mathbf{c o s} boldsymbol{gamma} & mathbf{0} & end{array} mid= )
A ( cdot cos alpha+cos beta+cos gamma )
( mathbf{B} cdot cos alpha cos beta cos gamma )
c. ( 2 cos alpha cos beta cos gamma )
D. ( 2 sum cos alpha cos beta )
12
286 Solve for ( boldsymbol{x} )
( left|begin{array}{ccc}1-x & 2 & 3 \ 0 & x & 0 \ 0 & 0 & xend{array}right|=0 )
( mathbf{A} cdot 1,0,0 )
B. 1,1,0
c. 1,1,1
D. 0,0,0
12
287 Points (1,5),(2,3) and (-2,-11) are
A. Non-collinear
B. Collinear
c. vertices of equilateral triangle
D. Vertices of right angle triangle
12
288 Verify whether the points (1,5),(2,3) ,and (-2,-1) are collinear or not. 12
289 If ( left.A=left[begin{array}{ll}a & b \ c & dend{array}right] text { (where } b neq cright) ) and satisfies the equation ( A^{2}+k I=0, ) then
( mathbf{A} cdot a+d=0 )
B . ( k=-|A| )
c. ( k=|A| )
D. None of the above
12
290 Evaluate the following:
( left|begin{array}{ccc}1 & a & b c \ 1 & b & c a \ 1 & c & a bend{array}right| )
12
291 Find the equation of the line joining ( A(1,3) ) and ( B(0,0) ) using determinants. 12
292 ( left|begin{array}{ccc}boldsymbol{x}+mathbf{2} & mathbf{2} boldsymbol{x}+mathbf{3} & mathbf{3} boldsymbol{x}+mathbf{4} \ mathbf{2} boldsymbol{x}+mathbf{3} & mathbf{3} boldsymbol{x}+mathbf{4} & mathbf{4} boldsymbol{x}+mathbf{5} \ mathbf{3} boldsymbol{x}+mathbf{5} & mathbf{5} boldsymbol{x}+mathbf{8} & mathbf{1 0} boldsymbol{x}+mathbf{1 7}end{array}right|=mathbf{0} ) then
( x ) is equal to
A ( .-1,-2 )
в. 1,2
( c cdot 1,-2 )
D. -1,2
12
293 If ( A ) is a matrix of order ( 3 times 3 ) then find
( |boldsymbol{a} boldsymbol{d} boldsymbol{j} boldsymbol{A}| ) where ( |boldsymbol{A}|=boldsymbol{2} )
12
294 Show that
( begin{array}{l}sin 10^{circ} quad-cos 10^{circ} \ sin 80^{circ} quad cos 80^{circ}end{array} mid=1 )
12
295 Solve
( left|begin{array}{ccc}1 & 1 & -1 \ 6 & 4 & -5 \ -4 & -2 & 3end{array}right| )
12
296 ( begin{array}{ccc}(boldsymbol{b}+boldsymbol{c})^{2} & boldsymbol{a}^{2} & boldsymbol{a}^{2} \ boldsymbol{b}^{2} & (boldsymbol{c}+boldsymbol{a})^{2} & boldsymbol{b}^{2} \ boldsymbol{c}^{2} & boldsymbol{c}^{2} & (boldsymbol{a}+boldsymbol{b})^{2}end{array} mid= ) 12
297 Using properties of determinants, find
the following:
( left|begin{array}{ccc}boldsymbol{alpha} & boldsymbol{beta} & boldsymbol{gamma} \ boldsymbol{alpha}^{2} & boldsymbol{beta}^{2} & boldsymbol{gamma}^{2} \ boldsymbol{beta}+boldsymbol{gamma} & boldsymbol{gamma}+boldsymbol{alpha} & boldsymbol{alpha}+boldsymbol{beta}end{array}right| )
A ( cdot(alpha+beta)(beta+gamma)(gamma-alpha)(alpha+beta+gamma) )
B . ( (alpha-beta)(beta-gamma)(gamma-alpha)(alpha+beta+gamma) )
c.
D. None of these
12
298 ( left|begin{array}{ccc}1 & sin theta & 1 \ -sin theta & 1 & sin theta \ -1 & -sin theta & 1end{array}right| ) then
( mathbf{A} cdot Delta=0 )
B. ( Delta in(0, infty) )
c. ( Delta in[-1,2] )
D. ( Delta in[2,4] )
12
299 The number of ordered triplets of positive integral solutions of ( left|begin{array}{ccc}boldsymbol{y}^{3}+mathbf{1} & boldsymbol{y}^{2} boldsymbol{z} & boldsymbol{y}^{2} boldsymbol{x} \ boldsymbol{y} boldsymbol{z}^{2} & boldsymbol{z}^{3}+mathbf{1} & boldsymbol{z}^{2} boldsymbol{x} \ boldsymbol{y} boldsymbol{x}^{2} & boldsymbol{x}^{2} boldsymbol{z} & boldsymbol{x}^{3}+mathbf{1}end{array}right|=mathbf{1 1} ) 12
300 If ( omega ) is a non-real cube root of unity and
n is not a multiple of ( 3, ) then ( Delta= ) ( left|begin{array}{ccc}mathbf{1} & boldsymbol{omega}^{n} & boldsymbol{omega}^{2 n} \ boldsymbol{omega}^{2 boldsymbol{n}} & boldsymbol{1} & boldsymbol{omega}^{n} \ boldsymbol{omega}^{boldsymbol{n}} & boldsymbol{omega}^{2 boldsymbol{n}} & boldsymbol{1}end{array}right| ) is equal to
( mathbf{A} cdot mathbf{0} )
B. ( omega )
( c cdot omega^{2} )
D.
12
301 ( mathbf{f} boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{2} & mathbf{- 1} \ -mathbf{1} & mathbf{1} & mathbf{2} \ mathbf{2} & mathbf{- 1} & mathbf{1}end{array}right], ) then
( operatorname{det}(operatorname{adj}(operatorname{adj} A)) ) is equal to
A ( cdot 14^{4} )
B. ( 14^{text {? }} )
( c cdot 14^{2} )
D. 14
12
302 Show that ( (x-a) ) is a factor of
( left|begin{array}{lll}boldsymbol{x} & boldsymbol{a} & boldsymbol{a} \ boldsymbol{a} & boldsymbol{x} & boldsymbol{a} \ boldsymbol{a} & boldsymbol{a} & boldsymbol{x}end{array}right| )
12
303 Write minors and cofactors of the
elements of following determinants
(i) ( left|begin{array}{cc}2 & -4 \ 0 & 3end{array}right| )
(ii) ( left|begin{array}{ll}boldsymbol{a} & boldsymbol{c} \ boldsymbol{b} & boldsymbol{d}end{array}right| )
12
304 ( fleft(begin{array}{lll}1 & 0 & 1 \ 0 & 1 & 2 \ 0 & 0 & 4end{array}right], ) then show that
[
|mathbf{3} boldsymbol{A}|=mathbf{2 7}|boldsymbol{A}|
]
12
305 If ( A=left[begin{array}{cc}2 & -1 \ -1 & 2end{array}right], ) then the general
solution of ( sin theta=left|A^{2}-4 A+3 Iright| ) is
A ( . n pi )
B ( cdot 2 n+1 frac{pi}{2} )
c. ( n pi+(-1)^{n} frac{pi}{2} )
D. ( 2 n pi, n epsilon z )
12
306 ( operatorname{Det}left[begin{array}{ccc}mathbf{4 3} & mathbf{1} & mathbf{6} \ mathbf{3 5} & mathbf{7} & mathbf{4} \ mathbf{1 7} & mathbf{3} & mathbf{2}end{array}right]=dots )
( A )
B. – –
( c cdot 0 )
( D )
12
307 Prove that ( left|begin{array}{ccc}boldsymbol{y} boldsymbol{z}-boldsymbol{x}^{2} & boldsymbol{z} boldsymbol{x}-boldsymbol{y}^{2} & boldsymbol{x} boldsymbol{y}-boldsymbol{z}^{2} \ boldsymbol{z} boldsymbol{x}-boldsymbol{y}^{2} & boldsymbol{x} boldsymbol{y}-boldsymbol{z}^{2} & boldsymbol{y} boldsymbol{z}-boldsymbol{x}^{2} \ boldsymbol{x} boldsymbol{y}-boldsymbol{z}^{2} & boldsymbol{y} boldsymbol{z}-boldsymbol{x}^{2} & boldsymbol{z} boldsymbol{x}-boldsymbol{y}^{2}end{array}right| )
divisible by ( (x+y+z) ) and hence find
the quotient
12
308 The remainder when the determinant
( left|begin{array}{lll}2014^{2014} & 2015^{2015} & 2016^{2016} \ 2017^{2017} & 2018^{2018} & 2019^{2019} \ 2020^{2020} & 2021^{2021} & 2022^{2022}end{array}right| )
is divided by 5 is.
A . 1
B . 2
( c cdot 4 )
( D )
12
309 Prove that: ( left|begin{array}{ccc}mathbf{1} & boldsymbol{x} & boldsymbol{y}+boldsymbol{z} \ mathbf{1} & boldsymbol{y} & boldsymbol{z}+boldsymbol{x} \ mathbf{1} & boldsymbol{z} & boldsymbol{x}+boldsymbol{y}end{array}right|=mathbf{0} ) 12
310 Let ( boldsymbol{A}=left[boldsymbol{a}_{i j}right]_{n times n} ) be a square matirx and
let ( c_{i j} ) be cofactor of ( a_{i j} ) in A. If ( C=left[c_{i j}right] )
then
B . ( |C|=|A|^{n-1} )
c. ( |C|=|A|^{n-2} )
D. none of these
12
311 ( left|begin{array}{ccc}mathbf{1} & boldsymbol{a} & boldsymbol{b} boldsymbol{c} \ mathbf{1} & boldsymbol{b} & boldsymbol{c a} \ mathbf{1} & boldsymbol{c} & boldsymbol{a b}end{array}right|=boldsymbol{lambda}left|begin{array}{ccc}boldsymbol{a}^{2} & boldsymbol{b}^{2} & boldsymbol{c}^{2} \ boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{1} & boldsymbol{1} & boldsymbol{1}end{array}right| ) then
( lambda ) is equal to
( A )
B . –
( c )
( D .- )
12
312 Calculate the values of the
determinants:
( left|begin{array}{cccc}mathbf{0} & boldsymbol{x} & boldsymbol{y} & boldsymbol{z} \ -boldsymbol{x} & boldsymbol{0} & boldsymbol{c} & boldsymbol{b} \ -boldsymbol{y} & -boldsymbol{c} & boldsymbol{0} & boldsymbol{a} \ -boldsymbol{z} & -boldsymbol{b} & -boldsymbol{a} & boldsymbol{0}end{array}right| )
12
313 If ( boldsymbol{A}=left[begin{array}{ccc}boldsymbol{x} & mathbf{1} & -boldsymbol{x} \ mathbf{0} & mathbf{1} & -mathbf{1} \ boldsymbol{x} & mathbf{0} & mathbf{7}end{array}right] ) and ( operatorname{det}(boldsymbol{A})= )
( left|begin{array}{ccc}mathbf{3} & mathbf{0} & mathbf{1} \ mathbf{2} & mathbf{- 1} & mathbf{0} \ mathbf{0} & mathbf{6} & mathbf{7}end{array}right| ) then the value of ( boldsymbol{x} ) is
A . -3
B. 3
( c cdot 2 )
D. -8
E. -2
12
314 Evaluate
( left|begin{array}{ccc}265 & 240 & 219 \ 240 & 225 & 198 \ 219 & 198 & 181end{array}right| )
12
315 Find the values of ( x, ) if
( left|begin{array}{ll}boldsymbol{x}+mathbf{1} & boldsymbol{x}-mathbf{1} \ boldsymbol{x}-mathbf{3} & boldsymbol{x}+mathbf{2}end{array}right|=left|begin{array}{ll}boldsymbol{4} & -mathbf{1} \ mathbf{1} & mathbf{3}end{array}right| )
12
316 f ( a, b, c>1, Delta= )
( left|begin{array}{ccc}log _{a}(a b c) & log _{a} b & log _{a} c \ log _{b}(a b c) & 1 & log _{b} c \ log _{c}(a b c) & log _{c} b & 1end{array}right| ) is
( mathbf{A} cdot mathbf{0} )
B ( cdot log _{a} b+log _{b} c+log _{c} a )
( mathbf{c} cdot log _{a b c}(a+b+c) )
D. none of these
12
317 The value of the determinant
( left|begin{array}{ccc}left(a^{x}+a^{-x}right)^{2} & left(a^{x}-a^{-x}right)^{2} & 1 \ left(b^{x}+b^{-x}right)^{2} & left(b^{x}-b^{-x}right)^{2} & 1 \ left(c^{x}+c^{-x}right)^{2} & left(c^{x}-c^{-x}right)^{2} & 1end{array}right| ) is
( mathbf{A} cdot mathbf{0} )
B. 2abc
( mathbf{c} cdot a^{2} b^{2} c^{2} )
D. None of these
12
318 what is the value of
( left|begin{array}{ccc}boldsymbol{a} & boldsymbol{a}^{2} & boldsymbol{a}^{boldsymbol{3}}-mathbf{1} \ boldsymbol{b} & boldsymbol{b}^{2} & boldsymbol{b}^{3}-mathbf{1} \ boldsymbol{c} & boldsymbol{c}^{2} & boldsymbol{c}^{boldsymbol{3}}-mathbf{1}end{array}right|=? )
12
319 If ( A ) is a square matrix of order 3 such
that ( |boldsymbol{a} boldsymbol{d} boldsymbol{j} cdot boldsymbol{A}|=boldsymbol{3} boldsymbol{6}, ) find ( |boldsymbol{A}| )
12
320 Without expanding, show that the value
of the following determinant is zero:
( left|begin{array}{ccc}a & b & c \ a+2 x & b+2 y & c+2 z \ x & y & zend{array}right| )
12
321 Prove that
[
left|begin{array}{ccc}
boldsymbol{x}+boldsymbol{y}+mathbf{2} boldsymbol{z} & boldsymbol{x} & boldsymbol{y} \
boldsymbol{z} & boldsymbol{y}+boldsymbol{z}+mathbf{2} boldsymbol{x} & boldsymbol{y} \
boldsymbol{z} & boldsymbol{x} & boldsymbol{z}+boldsymbol{x}+boldsymbol{2} boldsymbol{y}
end{array}right|
]
( 2(x+y+z)^{3} )
12
322 The adjoint of the matrix ( boldsymbol{A}= )
[
left[begin{array}{ccc}
1 & 1 & 1 \
2 & 1 & -3 \
-1 & 2 & 3
end{array}right] text { is }
]
( A )
[
left.begin{array}{l|lcc}
& 9 & -1 & -4 \
frac{1}{11} & -3 & 4 & 5 \
& 5 & -3 & -1
end{array}right]
]
B.
[
left[begin{array}{ccc}
9 & 1 & -4 \
3 & 4 & -5 \
5 & 3 & -1
end{array}right]
]
( c )
[
left[begin{array}{ccc}
9 & -3 & 5 \
-1 & 4 & -3 \
-4 & 5 & -1
end{array}right]
]
D.
[
left[begin{array}{ccc}
9 & -1 & -4 \
-3 & 4 & 5 \
5 & -3 & -1
end{array}right]
]
12
323 If ( boldsymbol{A}=int_{1}^{sin theta} frac{boldsymbol{t}}{1+boldsymbol{t}^{2}} boldsymbol{d} boldsymbol{t} ) and
( B=int_{1}^{operatorname{cosec} theta} frac{1}{tleft(1+t^{2}right)} d t, ) then the value
of determinant ( left|begin{array}{ccc}boldsymbol{A} & boldsymbol{A}^{2} & boldsymbol{B} \ boldsymbol{e}^{boldsymbol{A}+boldsymbol{B}} & boldsymbol{B}^{2} & -mathbf{1} \ mathbf{1} & boldsymbol{A}^{2}+boldsymbol{B}^{2} & -mathbf{1}end{array}right| )
is
( A cdot sin theta )
B. ( operatorname{cosec} theta )
( c )
D.
12
324 If ( left(begin{array}{lll}1 & 2 & 4 \ 1 & 3 & 5 \ 1 & 4 & aend{array}right) ) is singular, the value
of ( a ) is
A ( . a=-6 )
B. ( a=5 )
c. ( a=-5 )
( mathbf{D} cdot a=6 )
12
325 The value of the determinant ( left|begin{array}{lll}boldsymbol{a} & boldsymbol{b} & mathbf{0} \ boldsymbol{0} & boldsymbol{a} & boldsymbol{b} \ boldsymbol{b} & boldsymbol{0} & boldsymbol{a}end{array}right| )
is equal to
A ( cdot a^{3}-b^{3} )
В . ( a^{3}+b^{3} )
( c cdot 0 )
D. none of these
12
326 For a positive numbers ( x, y ) and ( z ) the numerical value of the determinant ( left[begin{array}{ccc}1 & log _{x} y & log _{x} z \ log _{y} x & 1 & log _{y} z \ log _{z} x & log _{z} y & 1end{array}right] ) is:
A. 0
B. 1
( mathbf{c} cdot log _{e} x y z )
( mathbf{D} cdot-log _{e} x y z )
12
327 ( mathbf{A}=left|begin{array}{ccc}cos frac{theta}{2} & mathbf{1} & mathbf{1} \ mathbf{1} & cos frac{theta}{2} & -cos frac{theta}{2} \ -cos frac{theta}{2} & mathbf{1} & -mathbf{1}end{array}right| )
the minimum of ( Delta ) is ( m_{1} ) and
maximum of ( Delta ) is ( m_{2} ) then ( left[m_{1}, m_{2}right] ) are
related to
A. [4, 2]
B. [2,4]
( c cdot[4,0] )
D. [0,2]
12
328 ( left|begin{array}{ccc}boldsymbol{a}+boldsymbol{x} & boldsymbol{a}-boldsymbol{x} & boldsymbol{a}-boldsymbol{x} \ boldsymbol{a}-boldsymbol{x} & boldsymbol{a}+boldsymbol{x} & boldsymbol{a}-boldsymbol{x} \ boldsymbol{a}-boldsymbol{x} & boldsymbol{a}-boldsymbol{x} & boldsymbol{a}+boldsymbol{x}end{array}right|=mathbf{0} ) then
the non-zero value of ( x=dots )
( A )
B. ( 3 a )
( c .2 a )
D. ( 4 a )
12
329 Calculate the values of the
determinants:
( left|begin{array}{lll}boldsymbol{a} & boldsymbol{h} & boldsymbol{g} \ boldsymbol{h} & boldsymbol{b} & boldsymbol{f} \ boldsymbol{g} & boldsymbol{f} & boldsymbol{c}end{array}right| )
12
330 Let ( triangle_{1}, triangle_{2}, triangle_{3}, dots, triangle_{k} ) be the set of
third-order determinants that can be
made with the distinct nonzero real
numbers ( a_{1}, a_{2}, a_{3}, dots a_{9}, ) and ( Sigma a_{i}=0 )
Then
This question has multiple correct options
( mathbf{A} cdot k=9 ! )
B . ( sum_{i=1}^{k} Delta_{i}=0 )
C . at least one ( Delta_{i}=0 )
D. none of these
12
331 ( left|begin{array}{cc}mathbf{0} & boldsymbol{a}-boldsymbol{b} \ -boldsymbol{a} mathbf{0}-boldsymbol{c} \ boldsymbol{b} & boldsymbol{c} mathbf{0}end{array}right|=mathbf{0} ) 12
332 The value of the determinant
[
left|begin{array}{ccc}
mathbf{1} & cos (boldsymbol{alpha}-boldsymbol{beta}) & mathbf{c o s} boldsymbol{alpha} \
cos (boldsymbol{alpha}-boldsymbol{beta}) & mathbf{1} & cos beta \
cos boldsymbol{alpha} & cos beta & mathbf{1}
end{array}right|
]
A ( cdot alpha^{2}+beta^{2} )
B ( cdot alpha^{2}-beta^{2} )
( c .1 )
( D )
12
333 frue Enter ( ^{prime} 1^{prime} ) else ( ^{prime} 0^{prime} )
( left|begin{array}{ccc}boldsymbol{x} & boldsymbol{y} & boldsymbol{x}+boldsymbol{y} \ boldsymbol{y} & boldsymbol{x}+boldsymbol{y} & boldsymbol{x} \ boldsymbol{x}+boldsymbol{y} & boldsymbol{x} & boldsymbol{y}end{array}right|=-boldsymbol{2}(boldsymbol{x}+ )
( boldsymbol{y})left(boldsymbol{x}^{2}+boldsymbol{y}^{2}-boldsymbol{x} boldsymbol{y}right) )
12
334 Let ( Delta(x)= )
( left|begin{array}{ccc}(boldsymbol{x}-mathbf{2}) & (boldsymbol{x}-mathbf{1})^{2} & boldsymbol{x}^{mathbf{3}} \ (boldsymbol{x}-mathbf{1}) & boldsymbol{x}^{2} & (boldsymbol{x}+mathbf{1})^{3} \ boldsymbol{x} & (boldsymbol{x}+mathbf{1})^{2} & (boldsymbol{x}+mathbf{2})^{3}end{array}right| ) Then the
coefficient of ( x ) in ( Delta(x) ) is ( -k . ) Find ( k )
( A cdot-1 )
B. 2
( c cdot-2 )
( D )
12
335 ( mathbf{f} boldsymbol{u}_{boldsymbol{r}}=boldsymbol{a}_{boldsymbol{r}} boldsymbol{x}+boldsymbol{b}_{boldsymbol{r}} boldsymbol{y}+, boldsymbol{c}_{boldsymbol{r}}=mathbf{0}(r=1,2,3 )
be the three sides of a triangle them the equations of the circumcircle of this triangle is ( mid begin{array}{ccc}frac{mathbf{1}}{boldsymbol{u}_{1}} & frac{mathbf{1}}{boldsymbol{u}_{2}} & frac{mathbf{1}}{boldsymbol{u}_{3}} \ boldsymbol{a}_{2} boldsymbol{a}_{3}-boldsymbol{b}_{2} boldsymbol{b}_{3} & boldsymbol{a}_{3} boldsymbol{a}_{1}-boldsymbol{b}_{3} boldsymbol{b}_{1} & boldsymbol{a}_{1} boldsymbol{a}_{2}-boldsymbol{b}_{4} \ boldsymbol{a}_{2} boldsymbol{b}_{3}+boldsymbol{a}_{3} boldsymbol{b}_{2} & boldsymbol{a}_{3} boldsymbol{b}_{1}+boldsymbol{a}_{1} boldsymbol{b}_{3} & boldsymbol{a}_{1} boldsymbol{b}_{2}+boldsymbol{a}_{2}end{array} )
( =0 )
12
336 If ( A ) is a skew-symmetric matrix of
order 3 then find ( |A| )
12
337 14.
11
If D= 1
1
1
1+ x
1
1
1
1+y
for x = 0, y 70, then D is
(a) divisible by x but not y
(b) divisible by y but not x
(c) divisible by neither x nor y
(d) divisible by both x and y
12
338 ( f(x)= )
[
begin{array}{ccc}
cos ^{x} & cos x sin x & -sin x \
cos x sin x & sin ^{x} & cos x \
sin x & -cos x & 0
end{array}
]
Prove that ( f(x)=1 ) is an identity
12
339 Prove that ( triangle= )
( left|begin{array}{ccc}boldsymbol{a} & boldsymbol{b} & boldsymbol{a} boldsymbol{alpha}+boldsymbol{b} \ boldsymbol{b} & boldsymbol{c} & boldsymbol{b} boldsymbol{alpha}+boldsymbol{c} \ boldsymbol{a} boldsymbol{alpha}+boldsymbol{b} & boldsymbol{b} boldsymbol{alpha}+boldsymbol{c} & boldsymbol{0}end{array}right|=boldsymbol{0} ) if ( mathbf{a}, mathbf{b}, mathbf{c} )
( operatorname{arcin} mathrm{G} . mathrm{P} )
12
340 ( mathbf{f}left|begin{array}{ccc}boldsymbol{a}+boldsymbol{b} & boldsymbol{b}+boldsymbol{c} & boldsymbol{c}+boldsymbol{a} \ boldsymbol{c}+boldsymbol{a} & boldsymbol{a}+boldsymbol{b} & boldsymbol{b}+boldsymbol{c} \ boldsymbol{b}+boldsymbol{c} & boldsymbol{c}+boldsymbol{a} & boldsymbol{a}+boldsymbol{b}end{array}right|=boldsymbol{t} times operatorname{det} ) of
circulant matrix whose elements of first
column are ( a, b, c )
then ‘t’ equals
A . 5
B. 6
( c cdot-2 )
( D )
12
341 Three lines ( boldsymbol{p} boldsymbol{x}+boldsymbol{q} boldsymbol{y}+boldsymbol{r}=mathbf{0}, boldsymbol{q} boldsymbol{x}+boldsymbol{r} boldsymbol{y}+ )
( boldsymbol{p}=mathbf{0} ) and ( boldsymbol{r} boldsymbol{x}+boldsymbol{p} boldsymbol{y}+boldsymbol{q}=mathbf{0} ) are
concurrent if
This question has multiple correct options
A ( . p+q+r=0 )
B . ( p^{2}+q^{2}+r^{2}=p r+q r+p q )
c. ( p^{3}+q^{3}+r^{3}=3 p q r )
D. none of these
12
342 ( operatorname{Solve}left|begin{array}{ccc}mathbf{0} & boldsymbol{p}-boldsymbol{q} & boldsymbol{p}-boldsymbol{r} \ boldsymbol{q}-boldsymbol{p} & boldsymbol{0} & boldsymbol{q}-boldsymbol{r} \ boldsymbol{r}-boldsymbol{p} & boldsymbol{r}-boldsymbol{q} & boldsymbol{0}end{array}right|= )
( mathbf{A} cdot p+q+r )
в. ( p q+q r+r p )
( c cdot c )
D. ( p^{2}+q^{2}+r^{2} )
12
343 Without expanding prove that the determinant
( left|begin{array}{ccc}sin A & operatorname{Cos} A & sin (A+theta) \ sin B & cos B & sin (B+theta) \ sin C & cos C & sin (c+theta)end{array}right|=0 )
12
344 If planes ( boldsymbol{x}-boldsymbol{c} boldsymbol{y}-boldsymbol{b} boldsymbol{z}=boldsymbol{0}, boldsymbol{c} boldsymbol{x}-boldsymbol{y}+ )
( boldsymbol{a} boldsymbol{z}=mathbf{0} ) and ( boldsymbol{b} boldsymbol{x}+boldsymbol{a} boldsymbol{y}-boldsymbol{z}=mathbf{0} ) pass
through a straight line then ( a^{2}+b^{2}+ )
( boldsymbol{c}^{2}= )
A. ( 1-a b c )
B . ( a b c-1 )
c. ( 1-2 a b c )
D. ( 2 a b c-1 )
12
345 1.
Consider the set A of all determinants of order 3 with entries
0 or 1 only. Let B be the subset of A consisting of all
determinants with value 1. Let C be the subset of A consisting
of all determinants with value-1. Then
(a C is empty
(1981 – 2 Marks)
(b) B has as many elements as C
(c) A= BUC
(d) B has twice as many elements as elements as C
12
346 ( A=left[begin{array}{lll}3 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 3end{array}right], ) then ( operatorname{Adj}(A) )
( A cdot 3 A )
B. ( 6 A )
( mathrm{c} cdot 9 A^{T} )
D. ( 2 A^{text {? }} )
12
347 9.
(1992 – 4
For a fixed positive integer n, if
n! (n+1)! (n + 2)!|
D= (n+1)! (n+ 2)! (n+3)!
|(n+2)! (n+3)! (n+4)!
then show that

-4
is divisible by n.
n13
12
348 Find the values of ( a ) and ( b ) so that the
points ( (boldsymbol{a}, boldsymbol{b}, mathbf{3}),(mathbf{2}, mathbf{0},-mathbf{1}) ) and
(1,-1,-3) are collinear. This question has multiple correct options
A ( . a=4, b=2 )
В. ( a=0, b=2 )
c. ( a=4, b=-2 )
begin{tabular}{l}
D. ( a=-4, b=-2 ) \
hline
end{tabular}
12
349 1.
For what value of k do the following system of equations
possess a non trivial (i.e., not all zero) solution over the set
of rationals Q?
x + ky + 3z=0
3x + ky-2z=0
2x+3y – 4z=0
For that value of k, find all the solutions for the system.
12
350 Prove that ( left|begin{array}{lll}mathbf{1} & boldsymbol{x} & boldsymbol{x}^{3} \ mathbf{1} & boldsymbol{y} & boldsymbol{y}^{3} \ mathbf{1} & boldsymbol{z} & boldsymbol{z}^{3}end{array}right|=(boldsymbol{x}+boldsymbol{y}+ )
( z(x-y)(y-z)(z-x) )
12
351 If ( boldsymbol{A}=left[begin{array}{cc}mathbf{1} & mathbf{2} \ mathbf{2} & mathbf{1}end{array}right] ) then ( boldsymbol{a} boldsymbol{d} boldsymbol{j}(boldsymbol{A})=? )
( mathbf{A} cdotleft[begin{array}{cc}1 & -2 \ -2 & 1end{array}right] )
в. ( left[begin{array}{cc}2 & 1 \ 1 & 1end{array}right] )
c. ( left[begin{array}{cc}1 & -2 \ -2 & -1end{array}right] )
D. ( left[begin{array}{cc}-1 & 2 \ 2 & -1end{array}right] )
12
352 Solve:det ( left|begin{array}{lll}boldsymbol{x} & boldsymbol{x}^{2} & boldsymbol{y} boldsymbol{z} \ boldsymbol{y} & boldsymbol{y}^{2} & boldsymbol{z} boldsymbol{x} \ boldsymbol{z} & boldsymbol{z}^{2} & boldsymbol{x} boldsymbol{y}end{array}right| ) 12
353 If ( |boldsymbol{A}|=mathbf{3} ) and ( boldsymbol{A}^{-1}=left[begin{array}{cc}mathbf{3} & -mathbf{1} \ -mathbf{5} & mathbf{2} \ hline mathbf{3} & mathbf{3}end{array}right] ) then
( boldsymbol{a} boldsymbol{d} boldsymbol{j} boldsymbol{A}=? )
( mathbf{A} cdotleft[begin{array}{cc}9 & 3 \ -5 & -2end{array}right] )
В. ( left[begin{array}{cc}9 & -3 \ -5 & 2end{array}right] )
begin{tabular}{lll}
C. ( left[begin{array}{cc}-9 & 3 \
5 & -2end{array}right] ) \
hline
end{tabular}
D. ( left[begin{array}{cc}9 & -3 \ 5 & -2end{array}right] )
12
354 The value of ( Delta=left|begin{array}{llll}1^{2} & 2^{2} & 3^{2} & 4^{2} \ 2^{2} & 3^{2} & 4^{2} & 5^{2} \ 3^{2} & 4^{2} & 5^{2} & 6^{2} \ 4^{2} & 5^{2} & 6^{2} & 7^{2}end{array}right| )
equals to
( mathbf{A} cdot mathbf{0} )
B ( cdot 1^{2}+2^{2}+3^{2}+4^{2}+ldots+7^{2} )
( c cdot 1 )
D. –
12
355 The area of the triangle whose vertices ( operatorname{are} A(1,2,3), B(2,-1,1) ) and
( boldsymbol{C}(mathbf{1}, mathbf{2},-mathbf{4}) ) is
A . ( 7 sqrt{10} ) sq units
B. ( frac{1}{2} sqrt{10} ) sq units
c. ( frac{7}{2} sqrt{10} ) sq units
D. None of these
12
356 If ( left(x_{1}-x_{2}right)^{2}+left(y_{1}-y_{2}right)^{2}=a^{2} )
( left(x_{2}-x_{3}right)^{2}+left(y_{2}-y_{3}right)^{2}=b^{2} )
( left(x_{3}-x_{1}right)^{2}+left(y_{3}-y_{1}right)^{2}=c^{2} ) and
( kleft|begin{array}{lll}boldsymbol{x}_{1} & boldsymbol{y}_{1} & mathbf{1} \ boldsymbol{x}_{2} & boldsymbol{y}_{2} & mathbf{1} \ boldsymbol{x}_{3} & boldsymbol{y}_{3} & mathbf{1}end{array}right|=(boldsymbol{a}+boldsymbol{b}+boldsymbol{c})(boldsymbol{b}+boldsymbol{c}- )
( a)(c+a-b) times(a+b-c), ) then the
value of ( k ) is
( A )
в. 2
( c )
D. none of these
12
357 ( left|begin{array}{ccc}x^{2}+x & x+1 & x-2 \ 2 x^{2}+3 x-1 & 3 x & 3 x-3 \ x^{2}+2 x+3 & 2 x-1 & 2 x-1end{array}right|= )
( A x+B ) then
A. A and B are independent of ( x )
B. A and B are dependent of ( x )
C. A dependent on x but B does not depend on x
D. B depends on x but A does not depend on x
12
358 Let ( 0(0,0), P(3,4), Q(6,0) ) be the vertices of the triangle OPQ. The point R inside the triangle OPQ is such that the triangles OPR,PQR, OQR are of equal area. The coordinates of ( mathrm{R} ) are
( ^{mathrm{A}} cdotleft(frac{4}{3}, 3right) )
в. ( left(3, frac{2}{3}right) )
c. ( left(3, frac{4}{3}right) )
D ( cdotleft(frac{4}{3}, frac{2}{3}right) )
12
359 If ( omega ) is a complex cube root of unity, the ( left|begin{array}{ccc}mathbf{1} & boldsymbol{omega} & boldsymbol{omega}^{2} \ boldsymbol{omega} & boldsymbol{omega}^{2} & boldsymbol{1} \ boldsymbol{omega}^{2} & boldsymbol{1} & boldsymbol{omega}end{array}right| ) is equal to
A . -1
B.
( c cdot 0 )
D.
12
360 ( f(x)=left|begin{array}{ccc}cos x & x & 1 \ 2 sin x & x^{2} & 2 x \ tan x & x & 1end{array}right| ) then
( lim _{x rightarrow 0} f(x)= )
( A cdot O )
B.
( c cdot-2 )
( D )
12
361 If ( f(x)=left|begin{array}{cc}x & lambda \ 2 lambda & xend{array}right|, ) then ( f(lambda x)-f(x) ) is
equal to:
A ( cdot xleft(lambda^{2}-1right) )
B ( cdot 2 lambdaleft(x^{2}-1right) )
C ( cdot lambda^{2}left(x^{2}-1right) )
D. ( lambdaleft(x^{2}-1right) )
E ( cdot x^{2}left(lambda^{2}-1right) )
12
362 Value of ( left|begin{array}{lll}sin alpha & cos alpha & sin alpha+cos beta \ sin beta & cos alpha & sin beta+cos beta \ sin gamma & cos alpha & sin gamma+cos betaend{array}right| ) is
( A cdot sin alpha+sin beta+sin gamma )
B. ( cos alpha+cos beta+cos gamma )
( mathbf{c} cdot sin alpha-sin (alpha+beta)-cos alpha+cos (gamma+beta) )
( D )
12
363 Find the of ( lambda, ) so that the matrix ( left[begin{array}{cc}mathbf{5}-boldsymbol{lambda} & boldsymbol{lambda}+mathbf{1} \ mathbf{2} & mathbf{4}end{array}right] ) may be singular 12
364 28. If P=
1 a 37
1 3 3 is the adjoint of a 3 x 3 matrix A and
2 4 4
JA) = 4, then a is equal to :
(a) 4 (6) 11
[JEEM 2013
(d) 0
(C) 5
12
365 U
TIUTUU
ay = 0, az + y = 0 and
If the system of equations x + ay = 0, az + y =
ax + z = 0 has infinite solutions, then the value of a 15
(2003)
(a) -1
(b) 1
(c) o
(d) no real values
12
366 ff ( alpha, beta ) are the roots of ( left|begin{array}{lll}x & 1 & 2 \ 0 & 1 & 1 \ 1 & x & 2end{array}right|=0 )
( operatorname{then} boldsymbol{alpha}^{boldsymbol{n}}+boldsymbol{beta}^{boldsymbol{n}}=? )
A.
в.
( c cdot 2 )
D. 2n
12
367 If ( D_{p}=left|begin{array}{ccc}boldsymbol{p} & mathbf{1 5} & mathbf{8} \ boldsymbol{p}^{2} & mathbf{2 5} & mathbf{9} \ boldsymbol{p}^{mathbf{3}} & mathbf{4 5} & mathbf{1 0}end{array}right|, ) then ( operatorname{det}left(boldsymbol{D}_{1}+right. )
( D_{2} ) ) is equal to
( mathbf{A} cdot mathbf{0} )
B . 25
( c .625 )
D. None of these
12
368 Find the values of ( x ) for which ( left|begin{array}{ll}3 & x \ x & 1end{array}right|= ) ( left|begin{array}{ll}mathbf{3} & mathbf{2} \ mathbf{4} & mathbf{1}end{array}right| ) 12
369 6.
x +1
If fx) = 2x x (x-1) (x+1)x then
3x(x – 1) x(x – 1) (x – 2) (x+1) x(x – 1)|
f(100) is equal to
(1999 – 2 Marks)
(a) o (6) 1 (c) 100 (d) -100
12
370 The number of distinct real roots of the
quation ( left|begin{array}{ccc}cos x & sin x & sin x \ sin x & cos x & sin x \ sin x & sin x & cos xend{array}right|=0 ) in
the interval ( left[-frac{pi}{4}, frac{pi}{4}right] ) is
( A )
B. 4
( c cdot 2 )
( D )
12
371 ( fleft|begin{array}{ccc}6 i & -3 i & 1 \ 4 & 3 i & -1 \ 20 & 3 & iend{array}right|=x+i y ) then
A ( . x=3, y=1 )
B. ( x=1, y=3 )
c. ( x=0, y=3 )
D. ( x=0, y=0 )
12
372 Let ( A, B, C, D, E ) be the interior angles of
convex pentagon and ( boldsymbol{Delta}=left|begin{array}{ccc}cos boldsymbol{A} & sin boldsymbol{A} & sin (boldsymbol{A}+boldsymbol{D}+boldsymbol{E}) \ cos boldsymbol{B} & sin boldsymbol{B} & sin (boldsymbol{B}+boldsymbol{D}+boldsymbol{E}) \ cos boldsymbol{C} & sin boldsymbol{C} & sin (boldsymbol{C}+boldsymbol{D}+boldsymbol{E})end{array}right| )
find ( Delta(pi / 3)+Delta^{prime}(pi / 6) )
12
373 Find the area of the triangle formed by
the lines ( x=3, y=2 ) and ( 3 x+4 y=29 )
12
374 ( left|begin{array}{ccc}mathbf{1} & boldsymbol{a} & boldsymbol{a}^{2} \ mathbf{1} & boldsymbol{b} & boldsymbol{b}^{2} \ mathbf{1} & boldsymbol{c} & boldsymbol{c}^{2}end{array}right|=(boldsymbol{a}-boldsymbol{b})(boldsymbol{b}-boldsymbol{c})(boldsymbol{c}-boldsymbol{a}) ) 12
375 ( fleft(begin{array}{ccc}-1 & -3 & -3 \ 3 & 1 & -3 \ 3 & -3 & 1end{array}right] ) then adj ( (A) ) is
A.
[
=4left[begin{array}{ccc}-2 & 3 & 3 \ -3 & 2 & -3 \ -3 & 3 & 2end{array}right]
]
B. ( quadleft[begin{array}{ccc}-2 & 3 & 3 \ 3 & 2 & -3 \ -3 & -3 & 2end{array}right] )
C.
[
=4left[begin{array}{ccc}-2 & -3 & 3 \ -3 & 2 & -3 \ -3 & -3 & 2end{array}right]
]
D.
[
=4left[begin{array}{ccc}-2 & 3 & 3 \ -3 & 2 & -3 \ -3 & -3 & 2end{array}right]
]
12
376 ( left|begin{array}{ccc}boldsymbol{a} & boldsymbol{b} & boldsymbol{a}+boldsymbol{b} \ boldsymbol{b} & boldsymbol{c} & boldsymbol{b}+boldsymbol{c} \ boldsymbol{a}+boldsymbol{b} & boldsymbol{b}+boldsymbol{c} & boldsymbol{0}end{array}right|=mathbf{0}, ) then ( boldsymbol{a}, boldsymbol{b}, boldsymbol{c} )
( operatorname{are} ) in
A. A.P
B. G.P.
( c . ) н.
D. None of these
12
377 Prove the following:
( left|begin{array}{ccc}boldsymbol{a}^{2}+boldsymbol{b}^{2} & boldsymbol{c} & boldsymbol{c} \ boldsymbol{c} & frac{boldsymbol{b}^{2}+boldsymbol{c}^{2}}{boldsymbol{a}} & boldsymbol{a} \ boldsymbol{b} & boldsymbol{b} & frac{boldsymbol{c}^{2}+boldsymbol{a}^{2}}{boldsymbol{b}}end{array}right|=boldsymbol{4 a b c} )
12
378 ( operatorname{Matrix} boldsymbol{A}=left[begin{array}{ccc}boldsymbol{x} & boldsymbol{3} & boldsymbol{2} \ boldsymbol{1} & boldsymbol{y} & boldsymbol{4} \ boldsymbol{2} & boldsymbol{2} & boldsymbol{z}end{array}right], ) if ( boldsymbol{x} boldsymbol{y} boldsymbol{z}=boldsymbol{6} mathbf{0} )
and ( 8 x+4 y+3 z=20, ) then ( A(a d j A) )
is equal to
A. ( left|begin{array}{lll}64 & 0 & 0 \ 0 & 64 & 0 \ 0 & 0 & 64end{array}right| )
B. ( mid begin{array}{ccc}68 & 0 & 0 \ 0 & 68 & 0 \ 0 & 0 & 68end{array} )
begin{tabular}{l|lll}
38 & 0 & 0 \
0 & 38 & 0 \
0 & 0 & 38
end{tabular}
D. ( mid begin{array}{ccc}32 & 0 & 0 \ 0 & 32 & 0 \ 0 & 0 & 32end{array} )
12
379 Find the values of ( x, ) if ( left|begin{array}{cc}mathbf{3 x} & mathbf{7} \ mathbf{2} & mathbf{4}end{array}right|=mathbf{1 0} ) 12
380 If ( boldsymbol{D}_{boldsymbol{r}}=left|begin{array}{ccc}boldsymbol{r} & boldsymbol{x} & frac{boldsymbol{n}(boldsymbol{n}+mathbf{1})}{mathbf{2}} mid \ boldsymbol{2 r}-boldsymbol{1} & boldsymbol{y} & boldsymbol{n}^{2} \ boldsymbol{3 r}-boldsymbol{2} & boldsymbol{z} & frac{boldsymbol{n}(boldsymbol{3} boldsymbol{n}-mathbf{1})}{mathbf{2}}end{array}right|, ) then
( sum_{r=1}^{n} D_{r} ) is equal to
A ( cdot frac{1}{6} n(n+1)(2 n+1) )
B. ( frac{1}{4} n^{2}(n+1)^{2} )
( c cdot 0 )
D. none of these
12
381 The co-ordinates of the vertices ( A, B, C )
of a triangle are ( (mathbf{6}, mathbf{3}),(-mathbf{3}, mathbf{5}),(mathbf{4},-mathbf{2}) )
respectively and ( P ) is any point ( (x, y) )
then the ratio of areas of triangles PBC and ABC is
A ( cdot|x-y-2|: 7 )
B cdot | ( x+y+2 mid: 7 )
c. ( |x+y-2|: 7 )
D. None of these
12
382 For what value of ( x ) the matrix ( A ) is
singular? ( boldsymbol{A}=left[begin{array}{ll}mathbf{1}+boldsymbol{x} & mathbf{7} \ mathbf{3}-boldsymbol{x} & mathbf{8}end{array}right] )
A ( cdot frac{12}{15} )
в. ( frac{13}{15} )
c. ( frac{14}{15} )
D. none of these
12
383 The value of the determinant
( Delta=left|begin{array}{lll}log x & log y & log z \ log 2 x & log 2 y & log 2 z \ log 3 x & log 3 y & log 3 zend{array}right| )
( A cdot 0 )
( mathbf{B} cdot log (x y z) )
( mathbf{C} cdot log (6 x y z) )
( mathbf{D} cdot 6 log (x y z) )
12
384 If ( f(x)= )
( left|begin{array}{ccc}cos ^{2} x & cos x cdot sin x & -sin x \ cos x cdot sin x & sin ^{2} x & cos x \ sin x & -cos x & 0end{array}right|, ) then
for all ( x epsilon R, ) the value of ( f(x)= )
( mathbf{A} cdot mathbf{0} )
B.
( c )
D. None of the above
12
385 For what values of ( m ) will the expression ( y^{2}+2 x y+2 x+m y-3 ) be capable of
resolution into two rational factors?
12
386 The maximum value of ( left|begin{array}{ccc}1+sin ^{2} x & cos ^{2} x & 4 cos 2 x \ sin ^{2} x & 1+cos ^{2} x & 4 sin 2 x \ sin ^{2} x & cos ^{2} x & 1+4 sin 2 xend{array}right| )
( A )
B.
( c .5 )
( D )
12
387 If ( D=left|begin{array}{cc}3 sqrt{5} & 6 \ 5 & mend{array}right|=0 )
Find the value of ( boldsymbol{m} )
A . ( sqrt{5} )
B. ( 4 sqrt{5} )
( c cdot sqrt{3} )
D. ( 2 sqrt{5} )
12
388 13. Prove that for all values of o,
sine
cose
sin20
| sin ( 1 + 2 ) cos( 1+ 2+ sin(20 + 4) = 0
sin(0-20) cos(0 – 21) sin( 20 – 49)
COS
12
389 Find the value of ( x, ) if ( left|begin{array}{cc}boldsymbol{x}+mathbf{2} & boldsymbol{x} \ boldsymbol{x}-boldsymbol{4} & boldsymbol{x}+mathbf{3}end{array}right|=left|begin{array}{cc}boldsymbol{4} & boldsymbol{2} \ -boldsymbol{2} & boldsymbol{5}end{array}right| ) 12
390 The value of ( left|begin{array}{lllll}mathbf{1} & mathbf{0} & mathbf{0} & mathbf{0} & mathbf{0} \ mathbf{2} & mathbf{2} & mathbf{0} & mathbf{0} & mathbf{0} \ mathbf{4} & mathbf{4} & mathbf{3} & mathbf{0} & mathbf{0} \ mathbf{5} & mathbf{5} & mathbf{5} & mathbf{4} & mathbf{0} \ mathbf{6} & mathbf{6} & mathbf{6} & mathbf{6} & mathbf{5}end{array}right| )
( A cdot 6 )
B. 5
c. ( 1.2^{2} .3 .4^{3} .5^{4} .6^{4} )
D. None of these
12
391 Find the value of following determinant. ( left|begin{array}{cc}-1 & 7 \ 2 & 4end{array}right| ) 12
392 Find the equation of line joining (1,2) and (3,6) using determinants. 12
393 If ( operatorname{in} ) a ( Delta A B C ; frac{cos A}{7}=frac{cos B}{19}= )
( frac{cos C}{25}=k, ) then
( left|begin{array}{ccc}-1 / k & 25 & 19 \ 25 & -1 / k & 7 \ 19 & 7 & -1 / kend{array}right|= )
( A )
B.
( c cdot 2 )
( D )
12
394 The number of distinct real roots of ( left|begin{array}{lll}sin x & cos x & cos x \ cos x & sin x & cos x \ cos x & cos x & sin xend{array}right|=0 ) in the
nterval ( -frac{pi}{4}2 )
12
395 Find values of ( k ) if area of triangle is 4
sq. units and vertices are
(i) ( (k, 0),(4,0),(0,2) )
(ii) ( (2,0),(0,4),(0, k) )
12
396 A determinant of second order is made
with the elements 0 and ( 1 . ) Find the
number of determinants with non-
negative values.
12
397 ( left|begin{array}{ccc}mathbf{1} & mathbf{1} & mathbf{1} \ boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{a}^{2}-boldsymbol{b} boldsymbol{c} & boldsymbol{b}^{2}-boldsymbol{c} boldsymbol{a} & boldsymbol{c}^{2}-boldsymbol{a} boldsymbol{b}end{array}right|= )
( A . )
( B )
( c . a b c )
D. ( (a-b),(b-c),(c-a) )
12
398 Consider three points ( boldsymbol{P}=(-sin (boldsymbol{beta}- )
( boldsymbol{alpha}),-cos beta), boldsymbol{Q}=(cos (beta-boldsymbol{alpha}), sin beta) )
and ( boldsymbol{R}=(cos (boldsymbol{beta}-boldsymbol{alpha}+boldsymbol{theta}), sin (boldsymbol{beta}-boldsymbol{theta})) )
where ( 0<alpha, beta, theta<frac{pi}{4} . ) Then
A. ( P ) lies on the line segment ( R Q )
B. ( Q ) lies on the line segment ( P R )
c. ( R ) lies on the line segment ( Q P )
D. ( P, Q, R ) are non-collinear
12
399 The value of ( left|begin{array}{ccc}boldsymbol{y} boldsymbol{z} & boldsymbol{z} boldsymbol{x} & boldsymbol{x} boldsymbol{y} \ boldsymbol{p} & boldsymbol{2} boldsymbol{q} & boldsymbol{3} boldsymbol{r} \ boldsymbol{1} & boldsymbol{1} & boldsymbol{1}end{array}right| ) where ( boldsymbol{x}, boldsymbol{y}, boldsymbol{z} )
are respectively, ( p t h,(2 q) t h, a n d(3 r) t h )
terms of an H.P., is
A . -1
B.
c. 1
D. none of these
12
400 Calculate the value of the following
determinant:
( left|begin{array}{cccc}mathbf{1}+boldsymbol{a} & mathbf{1} & mathbf{1} & mathbf{1} \ mathbf{1} & mathbf{1}+boldsymbol{b} & mathbf{1} & mathbf{1} \ mathbf{1} & mathbf{1} & mathbf{1}+boldsymbol{c} & mathbf{1} \ mathbf{1} & mathbf{1} & mathbf{1} & mathbf{1}+boldsymbol{d}end{array}right| )
12
401 If ( boldsymbol{A}=left[begin{array}{cc}mathbf{1} & log _{boldsymbol{b}} boldsymbol{a} \ log _{boldsymbol{a}} boldsymbol{b} & mathbf{1}end{array}right] ) then ( |boldsymbol{A}| ) is equal
to
A. 0
B ( cdot log _{a} b )
( c cdot-1 )
( mathbf{D} cdot log _{b} a )
12
402 Prove that ( left|begin{array}{lll}1 ! & 2 ! & 3 ! \ 2 ! & 3 ! & 4 ! \ 3 ! & 4 ! & 5 !end{array}right|=4 ! ) 12
403 Let ( A ) be a square matrix of order 3 write the value of ( |2 A|, ) where ( |A|=4 ) 12
404 Find determinant of ( left|begin{array}{lll}mathbf{1} & mathbf{0} & mathbf{2} \ mathbf{2} & mathbf{1} & mathbf{0} \ mathbf{3} & mathbf{2} & mathbf{1}end{array}right| ) 12
405 ( mathbf{f} mathbf{Delta}=left|begin{array}{ccc}-boldsymbol{a} & mathbf{2} boldsymbol{b} & mathbf{0} \ mathbf{0} & -boldsymbol{a} & mathbf{2} boldsymbol{b} \ mathbf{2} boldsymbol{b} & mathbf{0} & -boldsymbol{a}end{array}right|=mathbf{0}, ) then
A ( cdot frac{1}{b} ) is a cube root of unity
B. ( a ) is one of the cube roots of unity
( mathrm{c} . b ) is one of the cube roots of 8
D. ( frac{a}{b} ) is a cube root of 8
12
406 Prove the following:
[
left|begin{array}{ccc}
boldsymbol{a}^{2} & boldsymbol{b} boldsymbol{c} & boldsymbol{a} boldsymbol{c}+boldsymbol{c}^{2} \
boldsymbol{a}^{2}+boldsymbol{a} boldsymbol{b} & boldsymbol{b}^{2} & boldsymbol{a} boldsymbol{c} \
boldsymbol{a b} & boldsymbol{b}^{2}+boldsymbol{b} boldsymbol{c} & boldsymbol{c}^{2}
end{array}right|=boldsymbol{4} boldsymbol{a}^{2} boldsymbol{b}^{2} boldsymbol{c}^{2}
]
12
407 ( left|begin{array}{lll}boldsymbol{a}+boldsymbol{b} & boldsymbol{b}+boldsymbol{c} & boldsymbol{c}+boldsymbol{a} \ boldsymbol{b}+boldsymbol{c} & boldsymbol{c}+boldsymbol{a} & boldsymbol{a}+boldsymbol{b} \ boldsymbol{c}+boldsymbol{a} & boldsymbol{a}+boldsymbol{b} & boldsymbol{b}+boldsymbol{c}end{array}right|=boldsymbol{K}left|begin{array}{lll}boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{b} & boldsymbol{c} & boldsymbol{a} \ boldsymbol{c} & boldsymbol{a} & boldsymbol{b}end{array}right| )
( operatorname{then} K= )
( A )
B. 2
( c )
( D )
12
408 f ( alpha, beta, gamma ) are the roots of the equation
( x^{3}+p x+q=0 ) then the value of the
determinant ( left|begin{array}{lll}boldsymbol{alpha} & boldsymbol{beta} & gamma \ boldsymbol{beta} & gamma & boldsymbol{alpha} \ boldsymbol{gamma} & boldsymbol{alpha} & boldsymbol{beta}end{array}right| ) is
( mathbf{A} cdot mathbf{q} )
B.
c. ( p )
D. ( p^{2}-2 q )
12
409 ( fleft(a_{1}, a_{2}, a_{3}, dots . . ) are in G.P. then the value right.
of determinant ( left|begin{array}{ccc}log a_{n} & log a_{n+1} & log a_{n+2} \ log a_{n+3} & log a_{n+4} & log a_{n+5} \ log a_{n+6} & log a_{n+7} & log a_{n+8}end{array}right| ) equal
( mathbf{A} cdot mathbf{0} )
B. 1
( c cdot 2 )
D. 4
12
410 What is the area of the triangle formed by the points ( (a, c+a),(a, c) ) and
( (-a, c-a) ? )
A. ( -a^{2} )
B. ( frac{1}{a^{2}} )
c. ( a^{2}+a )
D. zero
12
411 f the determinant ( D= )
[
left|begin{array}{ccc}
mathbf{1} & mathbf{1} & mathbf{1} \
boldsymbol{alpha}+boldsymbol{beta} & boldsymbol{alpha}^{2}+boldsymbol{beta}^{2} & mathbf{2} boldsymbol{alpha} boldsymbol{beta} \
boldsymbol{alpha}+boldsymbol{beta} & mathbf{2} boldsymbol{alpha} boldsymbol{beta} & boldsymbol{alpha}^{2}+boldsymbol{beta}^{2}
end{array}right| text { and }
]
( boldsymbol{D}_{1}=left|begin{array}{ccc}mathbf{1} & mathbf{0} & mathbf{0} \ mathbf{0} & boldsymbol{alpha} & boldsymbol{beta} \ mathbf{0} & boldsymbol{beta} & boldsymbol{alpha}end{array}right|, ) then find the
determinant of ( D_{2} ) such that ( D_{2}=frac{D}{D_{1}} )
12
412 f ( boldsymbol{A}+boldsymbol{B}+boldsymbol{C}=boldsymbol{pi}, ) then
( left|begin{array}{ccc}tan (boldsymbol{A}+boldsymbol{B}+boldsymbol{C}) & tan boldsymbol{B} & tan C \ tan (boldsymbol{A}+boldsymbol{C}) & boldsymbol{0} & tan boldsymbol{A} \ tan (boldsymbol{A}+boldsymbol{B}) & -tan boldsymbol{A} & boldsymbol{0}end{array}right| )
equal to
( mathbf{A} cdot mathbf{0} )
B.
( c . ) tan AtanBtan
D. -2
12
413 With out expanding show that ( left|begin{array}{ccc}mathbf{1} & boldsymbol{a} & boldsymbol{a}^{2} \ mathbf{1} & boldsymbol{b} & boldsymbol{b}^{2} \ mathbf{1} & boldsymbol{c} & boldsymbol{c}^{2}end{array}right|=(boldsymbol{a}-boldsymbol{b})(boldsymbol{b}-boldsymbol{c})(boldsymbol{c}-boldsymbol{a}) ) 12
414 If ( a, b, c ) are positive and are the ( p ) th ( , q t h )
and ( r t h ) terms, respectively, of a G.P. ( operatorname{then} Delta=left|begin{array}{lll}log a & p & 1 \ log b & q & 1 \ log c & r & 1end{array}right| ) is
( A )
B. ( log (a b c) )
c. ( -(p+q+r) )
D. none of these
12
415 Find area of triangle with vertices at the point given in each of the following
( (mathrm{i})(1,0),(6,0),(4,3) )
( (i i)(2,7),(1,1),(10,8) )
( (text { iii) }(-2,-3),(3,2),(-1,-8) )
12
416 23. Consider the system of linear equations ;
x₂ + 2x₂ + x₃ = 3
2x + 3x₂+xz=3
3x, + 5×2 + 2×3 = 1
The system has
(a) exactly 3 solutions
(b) a unique solution
(C) no solution
(d) infinite number of solutions
12
417 begin{tabular}{|ccc}
if ( boldsymbol{f}(boldsymbol{x})= ) & \
( mathbf{1} ) & ( boldsymbol{x} ) & ( boldsymbol{x}+ ) \
( boldsymbol{2} boldsymbol{x} ) & ( boldsymbol{x}(boldsymbol{x}-mathbf{1}) ) & ( (boldsymbol{x}+ ) \
( boldsymbol{3} boldsymbol{x}(boldsymbol{x}-mathbf{1}) ) & ( boldsymbol{x}(boldsymbol{x}-mathbf{1})(boldsymbol{x}-mathbf{2}) ) & ( (boldsymbol{x}+mathbf{1}) boldsymbol{x} )
end{tabular}
then ( f(100) ) is equal to
( mathbf{A} cdot mathbf{0} )
B. 1
( c .100 )
D. –
12
418 Using the properties of determinants & without expanding ( left[begin{array}{ccc}boldsymbol{b}+boldsymbol{c} & boldsymbol{q}+boldsymbol{r} & boldsymbol{y}+boldsymbol{z} \ boldsymbol{c}+boldsymbol{a} & boldsymbol{r}+boldsymbol{p} & boldsymbol{z}+boldsymbol{x} \ boldsymbol{a}+boldsymbol{b} & boldsymbol{p}+boldsymbol{q} & boldsymbol{x}+boldsymbol{y}end{array}right]=boldsymbol{2}left[begin{array}{ccc}boldsymbol{a} & boldsymbol{p} & boldsymbol{x} \ boldsymbol{b} & boldsymbol{q} & boldsymbol{y} \ boldsymbol{c} & boldsymbol{r} & boldsymbol{z}end{array}right] ) 12
419 If each element in a row of a
determinant is multiplied by the same
factor ( r, ) then the value of the
determinant:
A . Is multiplied by ( r^{3} )
B. Is increased by ( 3 r )
c. Remains unchanged
D. Is multiplied by ( r )
12
420 16. Let A be a 2 x 2 matrix with real entries. Let I be the 2 x 2
identity matrix. Denote by tr(A), the sum of diagonal entries
of a. Assume that A2 = 1.
[2008]
Statement-1: IfA #I and A#-I, then det(A)=-1
Statement-2: If A #I and A#-I, then tr (A) +0.
(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
(c) 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
421 STATEMENT 1: In a ( Delta A B C, a, b, c ) denotes lengths of the sides and ( left|begin{array}{lll}boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{b} & boldsymbol{c} & boldsymbol{a} \ boldsymbol{c} & boldsymbol{a} & boldsymbol{b}end{array}right|=mathbf{0} ) then the triangle is
equilateral triangle.
STATEMENT 2: Sum of three non-
negative numbers ( =0 Rightarrow ) each number
is zero.
A. Statement-1 is true, Statement-2 is true, Statement-2 is correct explanation of Statement-
B. Statement-1 is true, Statement-2 is true, Statement-2 is not correct explanation for Statement-
c. Statement-1 is true, Statement-2 is false
D. Statement-1 is false, Statement-2 is true
12
422 ff ( x_{i}=a_{i} b_{i} c_{i}, i=1,2,3 ) are three-digit
positive integers such that each ( x_{i} ) is a
multiple of ( 19, ) then for some integer ( n ) prove that ( left|begin{array}{lll}boldsymbol{a}_{1} & boldsymbol{a}_{2} & boldsymbol{a}_{3} \ boldsymbol{b}_{1} & boldsymbol{b}_{2} & boldsymbol{b}_{3} \ boldsymbol{c}_{1} & boldsymbol{c}_{2} & boldsymbol{c}_{3}end{array}right| ) is divisible
by 19
12
423 Evaluate the determinant: ( left|begin{array}{ccc}mathbf{8} & mathbf{2} & mathbf{7} \ mathbf{1 2} & mathbf{3} & mathbf{5} \ mathbf{1 6} & mathbf{4} & mathbf{3}end{array}right| ) 12
424 2. Let a>0, d>0. Find the value of the determinant
(1996 – 5 Marks)
a(a +d)
(a+d)(a +2d)
(a + d)
(a +d)(a +2d)
(a +2d)(a +3d)
(a + 2d)
(a +2d)(a +3d)
(a +3d)(a +40)
12
425 Evaluate :
[
boldsymbol{Delta}=left|begin{array}{ccc}
cos alpha cos beta & cos alpha sin beta & -sin alpha \
-sin beta & cos beta & 0 \
sin alpha cos beta & sin alpha sin beta & cos alpha
end{array}right|
]
12
426 ( left|begin{array}{ccc}-mathbf{2} boldsymbol{a} & boldsymbol{a}+boldsymbol{b} & boldsymbol{a}+boldsymbol{c} \ boldsymbol{b}+boldsymbol{a} & -boldsymbol{2 b} & boldsymbol{b}+boldsymbol{c} \ boldsymbol{c}+boldsymbol{a} & boldsymbol{c}+boldsymbol{b} & -boldsymbol{2 c}end{array}right|=boldsymbol{4}(boldsymbol{b}+boldsymbol{c})(boldsymbol{c}+ )
( boldsymbol{a})(boldsymbol{a}+boldsymbol{b}) )
12
427 ( left|begin{array}{ccc}mathbf{8} & mathbf{- 5} & mathbf{1} \ mathbf{5} & boldsymbol{x} & mathbf{3} \ mathbf{6} & mathbf{3} & mathbf{1}end{array}right|=mathbf{2} ) then what is the
value of ( x ) ?
A . 44
B. 55
c. 61
D. 84
12
428 ( boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{- 1} & mathbf{1} \ mathbf{2} & mathbf{1} & -mathbf{3} \ mathbf{1} & mathbf{1} & mathbf{1}end{array}right] ) and ( boldsymbol{B}= )
( left[begin{array}{ccc}4 & 2 & 2 \ -5 & 0 & alpha \ 1 & -2 & 3end{array}right] ) If ( B ) is the adjoint of ( A )
then ( alpha ) equals
A .2
в.
( c cdot-2 )
D.
12
429 If ( A ) is ( 3 x 3 ) matrix and ( operatorname{det} operatorname{adj}(A)=k )
then
( operatorname{det}(operatorname{adj} 2 A)= )
( A cdot 2 k )
в. 8k
( c cdot 16 k )
D. ( 64 k^{2} )
12
430 First row of the matrix ( boldsymbol{A} ) is ( left[begin{array}{lll}mathbf{1} & mathbf{3} & mathbf{2}end{array}right] . )
[
begin{array}{l}
boldsymbol{a d j}(boldsymbol{A})= \
{left[begin{array}{ccc}
-mathbf{2} & mathbf{4} & boldsymbol{a} \
-mathbf{1} & mathbf{2} & mathbf{1} \
mathbf{3} boldsymbol{a} & mathbf{- 5} & mathbf{- 2}
end{array}right]}
end{array}
]
then a possible value of ( operatorname{det}(boldsymbol{A}) ) is
A . 1
B . 2
( c cdot-1 )
D. -2
12
431 33. The system of linear equations
x+2y-z=0
ax-y-z=0
xty-dz=0
has a non-trivial solution for:
(a) exactly two values of 2.
(b) exactly three values of .
(c) infinitely many values of 2.
(d) exactly one value of 2.
12
432 f ( 2 s=a+b+c, ) prove that ( left|begin{array}{ccc}a^{2} & (s-a)^{2} & (s-a)^{2} \ (s-b)^{2} & b^{2} & (s-b)^{2} \ (s-c)^{2} & (s-c)^{2} & c^{2}end{array}right| )
( =2 s^{3}(s-a)(s-b)(s-c) )
12
433 If ( boldsymbol{A}=left[begin{array}{ccc}5 & 5 x & x \ 0 & x & 5 x \ 0 & 0 & 5end{array}right] ) and ( left|A^{2}right|=25 )
then ( |x| ) is equal to?
A ( cdot frac{1}{5} )
B. 5
( c cdot 5^{2} )
D.
12
434 Find the ratio in which the line segment
joining the points ( P(x, 2) ) divides the line segment joining the points ( boldsymbol{A}(mathbf{1} mathbf{2}, mathbf{5}) ) and ( B(4,-3) . ) Also find the value of ( x )
12
435 The value of the determinant
( left|begin{array}{ccc}5 & 5 & 14 \ ^{5} C_{1} & ^{5} C_{4} & 1 \ ^{5} C_{2} & ^{5} C_{5} & 1end{array}right| ) is
( mathbf{A} cdot mathbf{0} )
B. -576
c. 80
D. none of these
12
436 Find the values of the following determinants where ( i=sqrt{-1} )
( (i)left|begin{array}{cc}2 i & -3 i \ i^{3} & -2 i^{5}end{array}right| )
( (i i)left|begin{array}{cc}mathbf{1}+mathbf{3} i & boldsymbol{i}-mathbf{2} \ boldsymbol{i}+mathbf{2} & mathbf{1}-mathbf{3} iend{array}right| )
12
437 f ( a, b, c epsilon R, ) find the number of real
roots of the equation given by ( Delta=0 )
where ( boldsymbol{Delta}=left|begin{array}{ccc}boldsymbol{x} & boldsymbol{c} & boldsymbol{- b} \ -boldsymbol{c} & boldsymbol{x} & boldsymbol{a} \ boldsymbol{b} & boldsymbol{-} boldsymbol{a} & boldsymbol{x}end{array}right| )
A . 0
B.
( c cdot 2 )
D. 3
12
438 ( left|begin{array}{cc}boldsymbol{x}+mathbf{1} & boldsymbol{x}-mathbf{1} \ boldsymbol{x}-mathbf{3} & boldsymbol{x}+mathbf{2}end{array}right|=left|begin{array}{cc}mathbf{4} & -mathbf{1} \ mathbf{1} & mathbf{3}end{array}right|, ) then write
the value of ( x )
12
439 If the points ( (k, 2 k),(3 k, 3 k) ) and (3,1) are collinear then the value of ( k ) is
A ( cdot frac{7}{9} )
B. ( frac{2}{3} )
c. ( frac{-2}{3} )
D. ( frac{-1}{3} )
12
440 Let
( boldsymbol{Delta}_{1}=left|begin{array}{ccc}mathbf{1} & cos boldsymbol{alpha} & cos beta \ cos boldsymbol{alpha} & boldsymbol{1} & cos gamma \ cos beta & cos gamma & 1end{array}right| )
and ( Delta_{2}=left|begin{array}{ccc}0 & cos alpha & cos beta \ cos alpha & 0 & cos gamma \ cos beta & cos gamma & 0end{array}right| )
If ( Delta_{1}=Delta_{2}, ) find ( sin ^{2} alpha+sin ^{2} beta+sin ^{2} gamma )
12
441 Solve this ( left|begin{array}{ccc}boldsymbol{a}-boldsymbol{b}-boldsymbol{c} & boldsymbol{2} boldsymbol{a} & boldsymbol{2} boldsymbol{a} \ boldsymbol{2} boldsymbol{b} & boldsymbol{b}-boldsymbol{c}-boldsymbol{a} & boldsymbol{2} boldsymbol{b} \ boldsymbol{2} boldsymbol{c} & boldsymbol{2} boldsymbol{c} & boldsymbol{c}-boldsymbol{a}-boldsymbol{b}end{array}right| ) 12
442 Assertion
The area of the triangle formed by the
points ( boldsymbol{A}(mathbf{2 0 0 7}, mathbf{2 0 0 9}), boldsymbol{B}(mathbf{2 0 0 8}, mathbf{2 0 1 1}) )
( C(2009,2010) ) will be same as the
area formed by the points ( boldsymbol{P}(mathbf{0}, mathbf{0}) )
( Q(1,2), R(2,1) )
Reason
The area of the triangle remains same
w.r.t to transition of co-ordinate axes.
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
443 ( operatorname{det}left{begin{array}{lll}18 & 40 & 89 \ 40 & 89 & 198 \ 89 & 198 & 440end{array}right}= )
A . -8
B. –
( c .-1 )
( D )
12
444 Solve:
( mid begin{array}{ccc}x^{2}-1 & x^{2}+2 x+1 & 2 x^{2}+3 x \ 2 x^{2}+x-1 & 2 x^{2}+5 x-3 & 4 x^{2}+4 x \ 6 x^{2}-x-2 & 6 x^{2}-7 x+2 & 12 x^{2}-5end{array} )
0. Let sum of all values of ( x ) be ( k ). Find
( -2 k )
12
445 Find the minor of ( left[begin{array}{ccc}2 & 7 & 3 \ -4 & 3 & -1 \ 0 & -3 & 7end{array}right] ) 12
446 If ( (8,1),(k,-4),(2,-5) ) are collinaer,
then ( k= )
( mathbf{A} cdot mathbf{1} )
B . 2
( c .3 )
D. 4
12
447 The value of
( left|begin{array}{ccc}a_{1} x+b_{1} y & a_{2} x+b_{2} y & a_{3} x+b_{3} y \ b_{1} x+a_{1} y & b_{2} x+a_{2} y & b_{3} x+a_{3} y \ b_{1} x+a_{1} & b_{2} x+a_{2} & b_{3} x+a_{3}end{array}right| ) is
equal to
A ( cdot x^{2}+y^{2} )
B.
( mathbf{c} cdot a_{1} a_{2} a_{3} x^{2}+b_{1} b_{2} b_{3} y^{2} )
D. none of these
12
448 Solve
( left|begin{array}{ccc}boldsymbol{a}^{2}+mathbf{1} & boldsymbol{a} boldsymbol{b} & boldsymbol{a} boldsymbol{c} \ boldsymbol{a} boldsymbol{b} & boldsymbol{b}^{2}+mathbf{1} & boldsymbol{b} boldsymbol{c} \ boldsymbol{a} boldsymbol{c} & boldsymbol{b} boldsymbol{c} & boldsymbol{c}^{2}+1end{array}right|=boldsymbol{a}^{2}+boldsymbol{b}^{2}+ )
( c^{2}+1 )
12
449 If ( boldsymbol{A}=left(begin{array}{ccc}1 & 2 & 1 \ -1 & 0 & 3 \ 2 & -1 & 1end{array}right) ) then
characteristic equation is given by
A. ( -lambda^{3}+2 lambda^{2}-4 lambda+18=0 )
0
B . ( lambda^{3}+2 lambda^{2}+4 lambda+18=0 )
c. ( 2 lambda^{3}-lambda^{2}+6 lambda-2=0 )
D. None of these
12
450 Write the value of the determinant ( left|begin{array}{cc}mathbf{3} & mathbf{- 1} \ mathbf{2} & mathbf{1}end{array}right| ) 12
451 If ( f(x)=left|begin{array}{ccc}a & -1 & 0 \ a x & a & -1 \ a x^{2} & a x & aend{array}right|, ) by using
properties of determinants, find the
value ( boldsymbol{f}(mathbf{2 x})-boldsymbol{f}(boldsymbol{x}) )
12
452 If the determinant
( left|begin{array}{ccc}boldsymbol{b}-boldsymbol{c} & boldsymbol{c}-boldsymbol{a} & boldsymbol{a}-boldsymbol{b} \ boldsymbol{b}^{prime}-boldsymbol{c}^{prime} & boldsymbol{c}^{prime}-boldsymbol{a}^{prime} & boldsymbol{a}^{prime}-boldsymbol{b}^{prime} \ boldsymbol{b}^{prime prime}-boldsymbol{c}^{prime prime} & boldsymbol{c}^{prime prime}-boldsymbol{a}^{prime prime} & boldsymbol{a}^{prime prime}-boldsymbol{b}^{prime prime}end{array}right| )
is expressible as ( mleft|begin{array}{lll}a & b & c \ a^{prime} & b^{prime} & c^{prime} \ a^{prime prime} & b^{prime prime} & c^{prime prime}end{array}right|, ) then
the value of ( mathrm{m} ) is
A . –
B.
( c )
( D )
12
453 Find ( left|begin{array}{lll}log e & log e^{2} & log e^{3} \ log e^{2} & log e^{3} & log e^{4} \ log e^{3} & log e^{4} & log e^{5}end{array}right| )
( A )
B.
( c .4 log e )
( mathbf{D} cdot 5 log e )
12
454 If ( boldsymbol{A}=left[begin{array}{lll}boldsymbol{a} & boldsymbol{0} & boldsymbol{0} \ boldsymbol{0} & boldsymbol{a} & boldsymbol{0} \ boldsymbol{0} & boldsymbol{0} & boldsymbol{a}end{array}right], ) then the value of
( |boldsymbol{A}||boldsymbol{a} boldsymbol{d} boldsymbol{j} boldsymbol{A}| ) is ( boldsymbol{a}^{boldsymbol{k}} )
What is k?
12
455 (
19, 1U MOHON
17.
The number of 3 x 3 matrices A whose entries are either 0 or
1 and for which the system
A y = 0
has exactly two
(2010)
distinct solutions, is
(a) 0 (b) 22–1
(c) 168
(d) 2
10
Lot 1
1
12
456 ( left|begin{array}{ccc}boldsymbol{x} & boldsymbol{y} & boldsymbol{z} \ boldsymbol{x}^{2} & boldsymbol{y}^{2} & boldsymbol{z}^{2} \ boldsymbol{x}^{boldsymbol{3}} & boldsymbol{y}^{boldsymbol{3}} & boldsymbol{z}^{boldsymbol{3}}end{array}right|=boldsymbol{x} boldsymbol{y} boldsymbol{z}(boldsymbol{x}-boldsymbol{y})(boldsymbol{y}-boldsymbol{z})(boldsymbol{z} )
( boldsymbol{x}) )
A. True
B. Falss
12
457 ff ( a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} in R ) and are such
( operatorname{that} a_{i} b_{j} neq 1 ) for ( 1 leq i, j leq 3, ) then
( begin{array}{|lll|}frac{1-a_{1}^{3} b_{1}^{3}}{1-a_{1} b_{1}} & frac{1-a_{1}^{3} b_{2}^{3}}{1-a_{1} b_{2}} & frac{1-a_{1}^{3} b_{3}^{3}}{1-a_{1} b_{3}} \ frac{1-a_{2}^{3} b_{1}^{3}}{1-a_{2} b_{1}} & frac{1-a_{2}^{3} b_{2}^{3}}{1-a_{2} b_{2}} & frac{1-a_{2}^{3} b_{3}^{3}}{1-a_{2} b_{3}} \ frac{1-a_{3}^{3} b_{1}^{3}}{1-a_{3} b_{1}} & frac{1-a_{3}^{3} b_{2}^{3}}{1-a_{3} b_{2}} & frac{1-a_{3}^{3} b_{3}^{3}}{1-a_{3} b_{3}}end{array} )
either ( a_{1}<a_{2}<a_{3} ) and ( b_{1}<b_{2}a_{2} a_{3} ) and
( boldsymbol{b}_{1}>boldsymbol{b}_{2}>boldsymbol{b}_{3} )
then show ( left(a_{1}-a_{2}right)left(a_{2}-a_{3}right)left(a_{3}-right. )
( left.boldsymbol{a}_{1}right)left(boldsymbol{b}_{1}-boldsymbol{b}_{2}right)left(boldsymbol{b}_{2}-boldsymbol{b}_{3}right)left(boldsymbol{b}_{3}-boldsymbol{b}_{1}right)<mathbf{0} )
12
458 Find the solution set of
( left|begin{array}{ccc}2+x & 2-x & 2-x \ 2-x & 2+x & 2-x \ 2-x & 2-x & 2+xend{array}right|=0 )
12
459 ( mathrm{ff}=left[begin{array}{ccc}boldsymbol{a} & mathbf{0} & mathbf{0} \ {[mathbf{0 . 3 e m}] mathbf{0}} & boldsymbol{a} & mathbf{0} \ {[mathbf{0 . 3 e m}] mathbf{0}} & mathbf{0} & boldsymbol{a}end{array}right], ) then the value
of IAdj. Al is equal to
A ( cdot a^{3} )
в. ( a^{6} )
( c cdot a^{9} )
( mathbf{D} cdot a^{2} )
12
460 ( A ) and ( B ) are two points and ( C ) is any
point collinear with ( A ) and ( B . ) IF ( A B= )
( mathbf{1 0}, boldsymbol{B C}=mathbf{5}, ) then ( boldsymbol{A} boldsymbol{C} ) is equal to:
A. either 15 or 5
B. necessarily 5
c. necessarily 16
D. none of these
12
461 If ( boldsymbol{A}=left[begin{array}{cc}mathbf{1} & mathbf{2} \ mathbf{3} & mathbf{4}end{array}right], ) then ( boldsymbol{A}^{-mathbf{1}}= )
A ( cdot frac{-1}{2}left[begin{array}{cc}4 & -2 \ -3 & 1end{array}right] )
в. ( frac{1}{2}left[begin{array}{cc}4 & -2 \ -3 & 1end{array}right] )
c. ( left[begin{array}{cc}-2 & 4 \ 1 & 3end{array}right] )
D. ( left[begin{array}{ll}2 & 4 \ 1 & 3end{array}right] )
12
462 If ( f(x) ) and ( g(x) ) are functions such that ( boldsymbol{f}(boldsymbol{x}+)=boldsymbol{f}(boldsymbol{x}) boldsymbol{g}(boldsymbol{y})+boldsymbol{g}(boldsymbol{x}) boldsymbol{f}(boldsymbol{y}), ) then the
value of ( left|begin{array}{lll}boldsymbol{f}(boldsymbol{alpha}) & boldsymbol{g}(boldsymbol{alpha}) & boldsymbol{f}(boldsymbol{alpha}+boldsymbol{theta}) \ boldsymbol{f}(boldsymbol{beta}) & boldsymbol{g}(boldsymbol{beta}) & boldsymbol{f}(boldsymbol{beta}+boldsymbol{theta}) \ boldsymbol{f}(boldsymbol{gamma}) & boldsymbol{g}(boldsymbol{gamma}) & boldsymbol{f}(boldsymbol{gamma}+boldsymbol{theta})end{array}right| )
A ( cdot f(alpha) cdot f(beta) cdot f(gamma) )
B.
( mathbf{c} cdot g(alpha) cdot g(beta) cdot g(gamma) )
D. None of these
12
463 ( operatorname{Let} A=left|begin{array}{ll}mathbf{5} & mathbf{0} \ mathbf{1} & mathbf{0}end{array}right| ) and ( boldsymbol{B}=left|begin{array}{ll}mathbf{2 0} & mathbf{5} mid \ -mathbf{1} & mathbf{0}end{array}right| )
( mathbf{4} boldsymbol{A}+mathbf{5} boldsymbol{B}-boldsymbol{C}=mathbf{0}, ) then ( boldsymbol{C} ) is
( mathbf{A} cdotleft|begin{array}{cc}5 & 25 \ -1 & 0end{array}right| )
( mathbf{B} cdotleft|begin{array}{cc}120 & 25 \ -1 & 0end{array}right| )
( mathbf{c} cdotleft|begin{array}{cc}5 & -1 \ 0 & 25end{array}right| )
( mathbf{D} cdotleft|begin{array}{cc}5 & 25 \ -1 & 5end{array}right| )
12
464 ( operatorname{Let} boldsymbol{A}=left(begin{array}{cc}boldsymbol{x}+mathbf{2} & mathbf{3} boldsymbol{x} \ boldsymbol{3} & boldsymbol{x}+mathbf{2}end{array}right), boldsymbol{B}= )
( left(begin{array}{cc}boldsymbol{x} & mathbf{0} \ mathbf{5} & boldsymbol{x}+mathbf{2}end{array}right) . ) Then all solutions of the
equation ( operatorname{det}(A B)=0 ) is
A. 1,-1,0,2
B. 1,4,0,-2
c. 1,-1,4,3
D. -1,4,0,3
12
465 The value of the determinant ( mid begin{array}{ccc}1+a^{2}-b^{2} & 2 a b & -2 b \ 2 a b & 1-a^{2}+b^{2} & 2 a \ 2 b & -2 a & 1+a^{2}-b^{2}end{array} )
is equal to
( A )
B ( cdotleft(1+a^{2}+b^{2}right) )
c. ( left(1+a^{2}+b^{2}right)^{2} )
D. ( left(1+a^{2}+b^{2}right)^{3} )
12
466 ( mathbf{f} A=left[begin{array}{cc}frac{-mathbf{1}+i sqrt{mathbf{3}}}{mathbf{2} i} & frac{-mathbf{1}-i sqrt{mathbf{3}}}{mathbf{2} i} \ frac{mathbf{1}+i sqrt{mathbf{3}}}{mathbf{2} i} & frac{mathbf{1}-i sqrt{mathbf{3}}}{mathbf{2} i}end{array}right], i= )
( sqrt{-i} ) and ( f(x)=x^{2}+2, ) then ( f(A) ) is
equal to
( ^{A} cdotleft(frac{5-i sqrt{3}}{2}right)left[begin{array}{ll}1 & 0 \ 0 & 1end{array}right] )
( ^{text {В }} cdotleft(frac{3-i sqrt{3}}{2}right)left[begin{array}{ll}1 & 0 \ 0 & 1end{array}right] )
c. ( left[begin{array}{ll}1 & 0 \ 0 & 1end{array}right] )
D. ( (2+i sqrt{3})left[begin{array}{ll}1 & 0 \ 0 & 1end{array}right] )
12
467 ( left|begin{array}{lll}a^{2}+2 a & 2 a+1 & 1 \ 2 a+1 & a+2 & 1 \ 3 & 3 & 1end{array}right|= )
( A cdot(1-a)^{3} )
В – ( (a-1)^{2} )
c. ( (a-1)^{3} )
D・ ( (a+1)^{2} )
12
468 Let ( A ) be a square matrix of order ( 3 times 3 )
then ( |k A| ) is equal to
( mathbf{A} cdot k mid A )
в. ( k^{2}|A| )
c. ( k^{3}|A| )
D. ( 3 k|A| )
12
469 Evaluate the following ( begin{array}{|ccc|}15 & 11 & 7 \ 11 & 17 & 14 \ 10 & 16 & 13end{array} ) 12
470 Evaluate the following determinant:
( left|begin{array}{ccc}1 & 4 & 9 \ 4 & 9 & 16 \ 9 & 16 & 25end{array}right| )
12
471 If ( omega ) is an imaginary cube root of
unity,then the value of ( left|begin{array}{ccc}boldsymbol{a} & boldsymbol{b} boldsymbol{omega}^{2} & boldsymbol{a} boldsymbol{omega} \ boldsymbol{b} boldsymbol{c} & boldsymbol{c} & boldsymbol{b} boldsymbol{omega}^{2} \ boldsymbol{c} boldsymbol{omega}^{2} & boldsymbol{a} boldsymbol{omega} & boldsymbol{c}end{array}right|, ) is
( mathbf{A} cdot a^{3}+b^{3}+c^{3} )
B . ( a^{2} b-b^{2} c )
( c cdot 0 )
D. ( a^{3} b+b^{3}+3 a b c )
12
472 ( left|begin{array}{ccc}text { If } & \ cos (boldsymbol{A}+boldsymbol{B}) & -sin (boldsymbol{A}+boldsymbol{B}) & cos 2 boldsymbol{B} \ sin boldsymbol{operatorname { s o s }} boldsymbol{operatorname { s i n }} boldsymbol{A} & boldsymbol{operatorname { s i n } boldsymbol { operatorname { m o s } }}end{array}right| )
( =0 ) then ( B= )
( A cdot(2 n+1) frac{pi}{2} )
B. ( n pi )
( c cdot(2 n+1) pi )
D. 2 nn
12
473 If the lines ( boldsymbol{p}_{1} boldsymbol{x}+boldsymbol{q}_{1} boldsymbol{y}=mathbf{1}, boldsymbol{p}_{2} boldsymbol{x}+boldsymbol{q}_{2} boldsymbol{y}= )
1 and ( p_{3} x+q_{3} y=1 ) be concurrent
show that the points ( left(p_{1}, q_{1}right),left(p_{2}, q_{2}right) )
and ( left(p_{3}, q_{3}right) ) are collinear
A. vertices of right angle triangle
B. vertices of an equilateral triangle
c. vertices of an isosceles triangle
D. Collinear
12
474 If points ( (a, 0),(0, b) ) and ( (x, y) ) are
collinear, prove that ( frac{x}{a}+frac{y}{b}=1 )
12
475 If ( left|begin{array}{ccc}2 a & x_{1} & y_{1} \ 2 b & x_{2} & y_{2} \ 2 c & x_{3} & y_{3}end{array}right|=frac{a b c}{2} neq 0, ) then the
area of the triangle whose vertices are ( left(frac{x_{1}}{a}, frac{y_{1}}{a}right),left(frac{x_{2}}{b}, frac{y_{2}}{b}right) ) and ( left(frac{x_{3}}{c}, frac{y_{3}}{c}right) ) is
A ( cdot frac{1}{4} a b c )
B. ( frac{1}{8} a b c )
( c cdot frac{1}{4} )
D. ( frac{1}{8} )
E ( cdot frac{1}{12} )
12
476 If ( omega ) is cube root of unity, then ( boldsymbol{Delta}=left|begin{array}{cccc}boldsymbol{x}+mathbf{1} & boldsymbol{omega} & boldsymbol{omega}^{2} \ boldsymbol{omega} & boldsymbol{x}+boldsymbol{omega}^{2} & mathbf{1} \ boldsymbol{omega}^{2} & boldsymbol{1} & boldsymbol{x}+boldsymbol{omega}end{array}right|= )
( mathbf{A} cdot x^{3}+1 )
B. ( x^{3}+omega )
c. ( x^{3}+omega^{2} )
D. ( x^{3} )
12
477 If ( boldsymbol{A}=left|begin{array}{ll}mathbf{0} & mathbf{0} \ mathbf{1} & mathbf{1}end{array}right| ) then the value of ( boldsymbol{A}+ )
( boldsymbol{A}^{2}+boldsymbol{A}^{3}+ldots+boldsymbol{A}^{n}=? )
( A cdot A )
B. nA
c. ( (n+1) A )
( D )
12
478 Find the values of ( x, ) if ( left|begin{array}{ll}mathbf{2} & mathbf{3} \ mathbf{4} & mathbf{5}end{array}right|=left|begin{array}{ll}boldsymbol{x} & mathbf{3} \ mathbf{2} boldsymbol{x} & mathbf{5}end{array}right| ) 12
479 76. Consider the system of linear equations in x, y, and z:
(sin 30) x – y + z = 0
(cos 20) x + 4y + 3z = 0
2 x + 7y + z = 0
Which of the following can be the values of O for which
the system has a non-trivial solution?
a. nit+ (-1)” īc/6, Vnez
b. nn + (-1)” 7/3, ne z
c. nit+ (-1)” Tt/9, ne z
d. none of these
12
480 Consider the system of linear equations in x, y, z:
(sin 30) x-y+z=0
(cos 20) x+4y + 3z=0
2x+7y+z=0
12
481 f maximum and minimum values of ( boldsymbol{D}=left|begin{array}{ccc}mathbf{1} & -cos boldsymbol{theta} & mathbf{1} \ cos boldsymbol{theta} & mathbf{1} & -cos boldsymbol{theta} \ mathbf{1} & cos boldsymbol{theta} & mathbf{1}end{array}right| ) are ( p ) and
respectively, then the value of ( 2 p+3 q )
is
A . 16
B. 6
( c .14 )
( D )
12
482 ( A ) is a ( 3 times 3 ) matrix and ( B ) is its adjoint
matrix. If the determinant of ( B ) is 64
then the det ( A ) is
( A cdot 4 )
B. ±4
( c .pm 8 )
D. 8
12
483 ( mathrm{f}left|begin{array}{ccc}boldsymbol{x}^{2}+boldsymbol{x} & boldsymbol{x}+mathbf{1} & boldsymbol{x}-mathbf{2} \ mathbf{2} boldsymbol{x}^{2}+mathbf{3} boldsymbol{x}-mathbf{1} & mathbf{3} boldsymbol{x} & mathbf{3} boldsymbol{x}-mathbf{3} \ boldsymbol{x}^{mathbf{2}}+mathbf{2} boldsymbol{x}+mathbf{3} & mathbf{2} boldsymbol{x}-mathbf{1} & mathbf{2} boldsymbol{x}-mathbf{1}end{array}right|= )
( A x-12, ) then the value of ( A ) is
( A cdot 12 )
в. 24
( c .-12 )
( D .-24 )
12
484 Let ( boldsymbol{P}=left[boldsymbol{a}_{i j}right] ) be a ( boldsymbol{3} times boldsymbol{3} ) matrix and let
( Q=left(b_{i j}right) ) where ( b_{i j}=2^{i+j} a_{i j} ) for ( 1 leq )
( i, j leq 3 . ) If the determinant of ( P ) is 2 then
the determinant of the matrix ( Q ) is?
A ( cdot 2^{10} )
B . ( 2^{11} )
( c cdot 2^{12} )
D. ( 2^{13} )
12
485 A point ( boldsymbol{P}(mathbf{2},-mathbf{1}) ) is equidistant from
points ( (a, 7) ) and ( (-3, a) ). Find ( a )
12
486 The value of ( left|begin{array}{ccc}boldsymbol{a}-boldsymbol{b}-boldsymbol{c} & boldsymbol{2} boldsymbol{a} & boldsymbol{2} boldsymbol{a} \ boldsymbol{2} boldsymbol{b} & boldsymbol{b}-boldsymbol{c}-boldsymbol{a} & boldsymbol{2} boldsymbol{b} \ boldsymbol{2} boldsymbol{c} & boldsymbol{2} boldsymbol{c} & boldsymbol{c}-boldsymbol{a}-boldsymbol{b}end{array}right| ) will
n ( mathbf{2 C} )
A ( cdot(a+b+c)^{2} )
B ( cdot(a+b+c)^{3} )
c. ( (a-b-c)^{2} )
D ( cdot(a+b-c)^{2} )
12
487 ( mathbf{1 f} mathbf{A}=left[begin{array}{ccc}mathbf{1} & mathbf{5} & mathbf{- 6} \ mathbf{- 8} & mathbf{0} & mathbf{4} \ mathbf{3} & mathbf{- 7} & mathbf{2}end{array}right] ) then the
cofactors of the elements 3,-7,2 are
p,q,r respectively their ascending order
is
( mathbf{A} cdot mathbf{p}, mathbf{r}, mathbf{q} )
B. ( mathrm{q}, mathrm{r}, mathrm{p} )
( c cdot p, q, r )
( D cdot r, p, q )
12
488 Find the value of ( k ) if
( boldsymbol{A}(mathbf{4}, mathbf{1 1}), boldsymbol{B}(mathbf{2}, mathbf{5}), boldsymbol{C}(boldsymbol{6}, boldsymbol{k}) ) are collinear
points.
12
489 f ( x ) is a positive integer, then ( left|begin{array}{ccc}boldsymbol{x} ! & (boldsymbol{x}+mathbf{1}) ! & (boldsymbol{x}+mathbf{2}) ! \ (boldsymbol{x}+mathbf{1}) ! & (boldsymbol{x}+mathbf{2}) ! & (boldsymbol{x}+mathbf{3}) ! \ (boldsymbol{x}+mathbf{2}) ! & (boldsymbol{x}+mathbf{3}) ! & (boldsymbol{x}+mathbf{4}) !end{array}right| ) is equa
to
A ( cdot 2 x !(x+1) ! )
в. ( 2 x !(x+1) !(x+2) )
c. ( 2 x !(x+3) ! )
D. ( 2(x+1) !(x+2) !(x+3) ! )
12
490 ( mathbf{f} mathbf{Delta}=left|begin{array}{lll}boldsymbol{a} & mathbf{0} & mathbf{0} \ boldsymbol{b} & boldsymbol{c} & boldsymbol{a} \ boldsymbol{c} & boldsymbol{a} & boldsymbol{b}end{array}right| ) then ( left|begin{array}{lll}boldsymbol{p}^{2} boldsymbol{a} & boldsymbol{0} & boldsymbol{0} \ boldsymbol{p} boldsymbol{b} & boldsymbol{c} & boldsymbol{a} \ boldsymbol{p c} & boldsymbol{a} & boldsymbol{b}end{array}right| ) is
equal to
A ( . p Delta )
B . ( p^{2} Delta )
c. ( p^{3} Delta )
D. ( 2 p Delta )
12
491 Prove the following:
( left|begin{array}{ccc}boldsymbol{b} boldsymbol{c} & boldsymbol{b} boldsymbol{c}^{prime}+boldsymbol{b}^{prime} boldsymbol{c} & boldsymbol{b}^{prime} boldsymbol{c}^{prime} \ boldsymbol{c} boldsymbol{a} & boldsymbol{c} boldsymbol{a}^{prime}+boldsymbol{c}^{prime} boldsymbol{a} & boldsymbol{c}^{prime} boldsymbol{a}^{prime} \ boldsymbol{a b} & boldsymbol{a b}^{prime}+boldsymbol{d}^{prime} boldsymbol{b} & boldsymbol{d}^{prime} boldsymbol{b}^{prime}end{array}right|= )
[
left(b c^{prime}-b^{prime} cright)left(c a^{prime}-c^{prime} aright)left(a b^{prime}-d bright)
]
12
492 ( boldsymbol{B}_{1}+boldsymbol{B}_{2}+ldots ldots+boldsymbol{B}_{49} ) is equal to
A. ( B_{0} )
В. ( 7 B_{0} )
( mathbf{c} cdot 49 B_{0} )
D. 49
12
493 ( left|begin{array}{ccc}mathbf{2}^{3} & mathbf{3}^{3} & mathbf{3 . 2}^{mathbf{2}}+mathbf{3 . 2}+mathbf{1} \ mathbf{3}^{mathbf{3}} & mathbf{4}^{mathbf{3}} & mathbf{3 . 3}^{mathbf{2}}+mathbf{3 . 3}+mathbf{1} \ mathbf{4}^{mathbf{3}} & mathbf{5}^{mathbf{3}} & mathbf{3 . 4}^{mathbf{2}}+mathbf{3 . 4}+mathbf{1}end{array}right| ) is equal to
( A )
B.
( c cdot 2 )
D. 3
12
494 Suppose ( x, y, z ) are positive integers ( (neq )
1) if ( Delta=operatorname{det} ) of ( left[begin{array}{ccc}1 & log _{x} y & log _{x} z \ log _{y} x & 1 & log _{y} z \ sin (x+y) & -cos (x+y) & sin ^{2} zend{array}right] )
then ( Delta ) is 1 ) Independent of ( x ) II) Independent of ( y ) IIII Independent of ( z ) The which of the above statement is
are correct
A. only land II
D. all the three I,II,III
12
495 ( operatorname{et} Delta_{1}=left|begin{array}{ccc}x & y & x+y \ y & x+y & x \ x+y & x & yend{array}right| )
( boldsymbol{Delta}_{2}=left|begin{array}{ccc}mathbf{1} & boldsymbol{x} & boldsymbol{y} \ mathbf{1} & boldsymbol{x}+boldsymbol{y} & boldsymbol{y} \ mathbf{1} & boldsymbol{x} & boldsymbol{x}+boldsymbol{y}end{array}right| )
( boldsymbol{Delta}_{3}=left|begin{array}{ccc}boldsymbol{x} & boldsymbol{x}+boldsymbol{y} & boldsymbol{x}+boldsymbol{2} boldsymbol{y} \ -boldsymbol{x} & boldsymbol{x} & boldsymbol{0} \ boldsymbol{0} & -boldsymbol{x} & boldsymbol{x}end{array}right| )
( boldsymbol{Delta}_{4}=left|begin{array}{lll}1 & boldsymbol{x} & boldsymbol{x}^{2} \ mathbf{1} & boldsymbol{y} & boldsymbol{y}^{2} \ mathbf{1} & mathbf{1} & mathbf{1}end{array}right| )
then
12
496 Find the maximum value of ( left|begin{array}{ccc}mathbf{1} & mathbf{1} & mathbf{1} \ mathbf{1} & mathbf{1}+sin boldsymbol{theta} & mathbf{1} \ mathbf{1} & mathbf{1} & mathbf{1}+mathbf{c} mathbf{o} mathbf{s} boldsymbol{theta}end{array}right| ) 12
497 The minors and cofactors of -4 and 9 in
determinant ( left|begin{array}{ccc}-1 & -2 & 3 \ -4 & -5 & -6 \ -7 & 8 & 9end{array}right| ) are
respectively
A. 42,( 42 ; 3,3 )
В. -42,( 42 ;-3,-3 )
c. 42,( -42 ; 3,-3 )
D. 42,3: 42,3
12
498 The straight lines ( mathbf{x}+mathbf{2 y}-mathbf{9}=mathbf{0}, mathbf{3 x}+ )
( 5 y-5=0 ) and ( a x+b y-1=0 ) are
concurrent if the straight line ( 22 x- )
( 35 y-1=0 ) passes through the point
( A cdot(a, b) )
в. ( (b, a) )
c. ( (-a, b) )
D. (-a,-b)
12
499 The value of the determinant ( Delta= )
A ( a_{1} a_{2}+b_{1} b_{2} )
a
В ( cdotleft(a_{1} a_{2} a_{3}right)+left(b_{1} b_{2} b_{3}right) )
c. ( a_{1} a_{2} b_{1} b_{2}+a_{2} a_{3} b_{2} b_{3}+a_{3} a_{1} b_{3} b_{1} )
D. none of these
12
500 If ( boldsymbol{A}=left[begin{array}{cc}boldsymbol{a} & boldsymbol{p} \ boldsymbol{b} & boldsymbol{q} \ boldsymbol{c} & boldsymbol{r}end{array}right]_{mathbf{3} times mathbf{2}} ) then determinant
( left(A A^{T}right) ) is equal to
( mathbf{A} cdot mathbf{0} )
B. ( a^{2}+b^{2}+c^{2} )
c. ( p^{2}+q^{2}+r^{2} )
D. ( p^{2}+q^{2} )
12
501 Evaluate
( left|begin{array}{ccc}cos alpha cos beta & cos alpha sin beta & -sin alpha \ -sin beta & cos beta & 0 \ sin alpha cos beta & sin alpha sin beta & cos alphaend{array}right| )
12
502 Prove the following identities
( left|begin{array}{ccc}mathbf{1} & boldsymbol{a} & boldsymbol{a}^{2} \ boldsymbol{a}^{2} & mathbf{1} & boldsymbol{a} \ boldsymbol{a} & boldsymbol{a}^{2} & mathbf{1}end{array}right|=left(boldsymbol{a}^{3}-mathbf{1}right)^{2} )
12
503 If ( boldsymbol{a}=sin boldsymbol{theta}, boldsymbol{b}=sin (boldsymbol{theta}+boldsymbol{2} boldsymbol{pi} / mathbf{3}), boldsymbol{c}= )
( sin (theta+4 pi / 3), x=cos theta, y= )
( cos (theta+2 pi / 3), z=cos (theta+4 pi / 3) )
then value of
( boldsymbol{Delta}=left|begin{array}{ccc}boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{x} & boldsymbol{y} & boldsymbol{z} \ boldsymbol{b} boldsymbol{c} & boldsymbol{c} boldsymbol{a} & boldsymbol{a} boldsymbol{b}end{array}right| )
is
A . 1 ( overline{8} )
B. ( frac{3 sqrt{3}}{4} )
( c cdot frac{3 sqrt{3}}{8} )
D. ( frac{sqrt{3}}{4} )
12
504 The value of the determinant
( left|begin{array}{ccc}mathbf{1} & mathbf{1} & mathbf{1} \ m_{mathbf{C}} & m+mathbf{C}_{1} & m+mathbf{2} \ m & mathbf{C}_{1} \ m & boldsymbol{m}+mathbf{1} & boldsymbol{C}_{2}end{array}right| mathbf{~ i s ~ e q u a l ~ t o ~} )
( mathbf{A} cdot mathbf{1} )
B. –
( c cdot 0 )
D. None of these
12
505 ( left|begin{array}{ccc}x^{2}+x & x+1 & x+2 \ x^{2}+3 x-1 & 3 x & 3 x-3 \ x^{2}+2 x+3 & 2 x-1 & 2 x-1end{array}right|= )
( A x+B ) where ( A ) and ( B ) are
determinants of order ( 3 . ) Then ( A+2 B ) is
equal to
12
506 Find the values of ( x, ) if ( left|begin{array}{ll}mathbf{2} & mathbf{3} \ mathbf{4} & mathbf{5}end{array}right|=left|begin{array}{ll}boldsymbol{x} & mathbf{3} \ mathbf{2} boldsymbol{x} & mathbf{5}end{array}right| ) 12
507 If ( A ) is a ( 3 times 3 ) matrix and ( operatorname{det}(3 A)= )
( boldsymbol{k}{boldsymbol{d e t}(boldsymbol{A})}, boldsymbol{k}= )
( mathbf{A} cdot mathbf{9} )
B. 6
c. 1
D. 27
12
508 Which of the following are correct in
respect of the system of equations ( boldsymbol{x}+ )
( boldsymbol{y}+boldsymbol{z}=mathbf{8}, boldsymbol{x}-boldsymbol{y}+mathbf{2} boldsymbol{z}=boldsymbol{6} ) and ( boldsymbol{3} boldsymbol{x}- )
( boldsymbol{y}+mathbf{5} boldsymbol{z}=boldsymbol{k} ? )
1. They have no solution, if ( k=15 )
2. They have infinitely many solutions, if
( k=20 )
3. They have unique solution, if ( k=25 )
Select the correct answer using the code given below
( A cdot 1 ) and 2 only
B. 2 and 3 only
C. 1 and 3 only
D. 1,2 and 3
12
509 The graph of ( f(x) ) is shown above in the
( x y ) -plane. The points ( (0,3),(5 b, b) ) and
( (10 b,-b) ) are on the line described by
( f(x) . ) If ( b ) is a positive constant, find the
coordinates of point ( C )
( mathbf{A} cdot(5,1) )
B. (10,-1)
c. (15,-0.5)
(15, ( -0.5) )
D. (20,-2)
12
510 Applying ( boldsymbol{C}_{1} rightarrow boldsymbol{C}_{1}-^{8} boldsymbol{C}_{3}, ) we
getWithout expanding the determinant, prove that ( left|begin{array}{ccc}mathbf{4 1} & mathbf{1} & mathbf{5} \ mathbf{7 9} & mathbf{7} & mathbf{9} \ mathbf{2 9} & mathbf{5} & mathbf{3}end{array}right|=mathbf{0} )
12
511 If ( A ) is a singular matrix, then adj ( A ) is
A. non- singular
B. singular
c. symmetric
D. not defined
12
512 ( left|begin{array}{cc}mathbf{2} & -mathbf{4} \ mathbf{9} & boldsymbol{d}-mathbf{3}end{array}right|=mathbf{4} ) then ( boldsymbol{d}= )
A . 1
B. -11
c. 12
D. -13
12
513 s
I a-1 n 6
Let Aa=/(a 1)2 2n² 4n-2
(a – 1)3 3n² 3n² – 3n
Show that Aa = c , a constant.
a = 1
12
514 . If ( A, B, C ) are angles of angle and ( mid begin{array}{ccc}mathbf{1} & mathbf{1} \ mathbf{1}+boldsymbol{s} mathbf{i} boldsymbol{n} boldsymbol{A} & mathbf{1}+boldsymbol{operatorname { s i n }} boldsymbol{B} & mathbf{1}+ \ boldsymbol{s i n} boldsymbol{A}+boldsymbol{s i n}^{2} boldsymbol{A} & boldsymbol{s i n} boldsymbol{B}+boldsymbol{s i n}^{2} boldsymbol{B} & boldsymbol{s i n} boldsymbol{C}end{array} )
( =0 ) then triangle is isosceles
II. If ( boldsymbol{a}=mathbf{1}+mathbf{2}+mathbf{4}+— ) upto ( mathbf{n} )
terms ( boldsymbol{b}=mathbf{1}+mathbf{3}+mathbf{9}+— ) up to ( mathbf{n} )
terms ( c=1+5+25+—- ) up to ( n ) terms then ( Deltaleft|begin{array}{ccc}a & 2 b & 4 c \ 2 & 2 & 2 \ 2^{n} & 3^{n} & 5^{n}end{array}right|=0 )
A. I, II both are true
B. only lis true
c. only I I is true
D. neither of them are true
12
515 f ( P(x, y) ) is such that ( left|begin{array}{ccc}boldsymbol{x} & boldsymbol{y} & mathbf{1} \ boldsymbol{x}_{1} & boldsymbol{y}_{1} & mathbf{1} \ boldsymbol{x}_{2} & boldsymbol{y}_{2} & boldsymbol{1}end{array}right|+left|begin{array}{ccc}boldsymbol{x} & boldsymbol{y} & mathbf{1} \ boldsymbol{x}_{1} & boldsymbol{y}_{1} & boldsymbol{1} \ boldsymbol{x}_{3} & boldsymbol{y}_{3} & 1end{array}right|=mathbf{0} ) then the
line through ( A ) and ( P ) is
A. median of ( Delta A B C )
B. bisector of ( angle A )
c. altitude through vertex ( A )
D. perpendicular bisector of the side ( B C )
12
516 Solve for ( boldsymbol{x} )
( left|begin{array}{ccc}boldsymbol{x}-mathbf{2} & mathbf{2} boldsymbol{x}-mathbf{3} & mathbf{3} boldsymbol{x}-mathbf{4} \ boldsymbol{x}-mathbf{4} & mathbf{2} boldsymbol{x}-mathbf{9} & mathbf{3} boldsymbol{x}-mathbf{1} mathbf{6} \ boldsymbol{x}-mathbf{8} & mathbf{2} boldsymbol{x}-mathbf{2 7} & mathbf{3} boldsymbol{x}-mathbf{6 4}end{array}right|=mathbf{0} )
( A cdot frac{7}{4} )
B. ( frac{28}{3} )
( c cdot frac{28}{13} )
D. ( frac{14}{0} )
12
517 Prove that:
( left|begin{array}{ccc}-a^{2} & a b & a c \ a b & -b^{2} & b c \ a c & b c & -c^{2}end{array}right|=4 a^{2} b^{2} c^{2} )
12
518 11.
For all values of A, B, C and P, Q, R show that
(1994 – 4 Marks)
cos(A-P) cos(A-2) cos(A – R)
cos(B-P) cos(B-Q) cos(B – R) = 0
cos(C-P) cos(C-0) cos(C-R)
12
519 If the points ( (k, 2-2 k)(1-k, 2 k) ) and ( (-k ) ( -4,6-2 x) ) be collinear the possible values of k are
A ( cdot frac{1}{2} )
B. ( frac{1}{2} )
( c )
D. – –
12
520 Find the value of ( x ) if
( left|begin{array}{ccc}boldsymbol{x}-mathbf{2} & mathbf{2} boldsymbol{x}-mathbf{3} & mathbf{3} boldsymbol{x}-mathbf{4} \ boldsymbol{x}-mathbf{4} & mathbf{2} boldsymbol{x}-mathbf{9} & mathbf{3} boldsymbol{x}-mathbf{1 6} \ boldsymbol{x}-mathbf{8} & mathbf{2} boldsymbol{x}-mathbf{2 7} & mathbf{3} boldsymbol{x}-mathbf{6 4}end{array}right|=mathbf{0} ? )
12
521 Show that ( triangle A B C ) is an isosceles
triangle, if the determinant ( mid begin{array}{ccc}mathbf{1} & mathbf{1} \ mathbf{1}+cos boldsymbol{A} & mathbf{1}+cos boldsymbol{B} & mathbf{1} \ cos ^{2} boldsymbol{A}+cos boldsymbol{A} & cos ^{2} boldsymbol{B}+cos boldsymbol{B} & mathbf{c o s}^{2}end{array} )
( mathbf{D} )
12
522 Let A=(12)
– ( a
and B=1
0
,
(3
4
10 b) 0,6 EN. Then
[2006]
(a) there cannot exist any B such that AB = BA
(b) there exist more then one but finite number of B’s such
that AB=BA
(c) there exists exactly one B such that AB=BA
(d) there exist infinitely many B’s such that AB=BA
12
523 ( mid begin{array}{cccc}boldsymbol{a} & boldsymbol{b} & boldsymbol{c} & boldsymbol{d} \ boldsymbol{a} & boldsymbol{a}+boldsymbol{b} & boldsymbol{a}+boldsymbol{b}+boldsymbol{c} & boldsymbol{a}+boldsymbol{b}+boldsymbol{c} \ boldsymbol{a} & boldsymbol{2} boldsymbol{a}+boldsymbol{b} & boldsymbol{3} boldsymbol{a}+boldsymbol{2} boldsymbol{b}+boldsymbol{c} & boldsymbol{4} boldsymbol{a}+boldsymbol{3} boldsymbol{b}+boldsymbol{2} \ boldsymbol{a} & boldsymbol{3} boldsymbol{a}+boldsymbol{b} & boldsymbol{6} boldsymbol{a}+boldsymbol{3} boldsymbol{b}+boldsymbol{c} & boldsymbol{1} boldsymbol{0} boldsymbol{a}+boldsymbol{6} boldsymbol{b}+boldsymbol{3}end{array} ) 12
524 17. Let x e R and let
(
11 17 [2 x x
P=10 2 2 8 = 0 4 ol
and R= PQP-1
10 0 3 X x 6]
Then which of the following options is/are correct?
(JEE Adv. 2019)
[2 x x
(a) det R=det 0 4 0 +8, for all x ER
[x x 5
(b) For x = 1, there exists a unit vector ai+Bj+ył for
To
There exists a real number x such that PQ=QP
(C)
(d)
For x = 0, if
= a
[b]
a
. then a+b=5
[b]
12
525 Find the values of ( x, ) if ( left|begin{array}{cc}mathbf{2 x} & mathbf{5} \ mathbf{8} & boldsymbol{x}end{array}right|=left|begin{array}{ll}mathbf{6} & mathbf{5} \ mathbf{8} & mathbf{3}end{array}right| ) 12
526 If in the determinant ( Delta= )
( left|begin{array}{lll}boldsymbol{a}_{1} & boldsymbol{b}_{1} & boldsymbol{c}_{1} \ boldsymbol{a}_{2} & boldsymbol{b}_{2} & boldsymbol{c}_{2} \ boldsymbol{a}_{3} & boldsymbol{b}_{3} & boldsymbol{c}_{3}end{array}right|, boldsymbol{A}_{i}, boldsymbol{B}_{i}, boldsymbol{C}_{i} ) etc. be the co-
factors of ( a_{i}, b_{i}, c_{i} ) etc., then which of the
following relations is incorrect?
A ( cdot a_{1} A_{1}+b_{1} B_{1}+c_{1} C_{1}=Delta )
B . ( a_{2} A_{2}+b_{2} B_{2}+c_{2} C_{2}=Delta )
c. ( a_{3} A_{3}+b_{3} B_{3}+c_{3} C_{3}=Delta )
D. ( a_{1} A_{2}+b_{1} B_{2}+c_{1} C_{2}=Delta )
12
527 f ( a neq b neq c ) such that
( left|begin{array}{ccc}boldsymbol{a}^{3}-mathbf{1} & boldsymbol{b}^{mathbf{3}}-mathbf{1} & boldsymbol{c}^{mathbf{3}}-mathbf{1} \ boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{a}^{mathbf{2}} & boldsymbol{b}^{mathbf{2}} & boldsymbol{c}^{mathbf{2}}end{array}right|=mathbf{0} ) ther
A ( . a b+b c+c a=0 )
B . ( a+b+c=0 )
( c cdot a b c=1 )
D. ( a+b+c=1 )
12
528 Find the values of ( x, ) if ( left|begin{array}{ll}boldsymbol{x}+mathbf{1} & boldsymbol{x}-mathbf{1} \ boldsymbol{x}-mathbf{3} & boldsymbol{x}+mathbf{2}end{array}right|=left|begin{array}{cc}mathbf{4} & mathbf{- 1} \ mathbf{1} & mathbf{3}end{array}right| ) 12
529 ( left|begin{array}{ccc}mathbf{1} & mathbf{1} & mathbf{1} \ boldsymbol{p} & boldsymbol{q} & boldsymbol{r} \ boldsymbol{p} & boldsymbol{q} & boldsymbol{r}+mathbf{1}end{array}right| ) is equal to
A ( cdot q-p )
B. ( q+p )
( c cdot q )
( D )
12
530 Find the values of ( x ), if ( left|begin{array}{cc}boldsymbol{x}+mathbf{1} & boldsymbol{x}-mathbf{1} \ boldsymbol{x}-mathbf{3} & boldsymbol{x}+mathbf{2}end{array}right|=left|begin{array}{cc}mathbf{4} & mathbf{- 1} \ mathbf{1} & mathbf{3}end{array}right| ) 12
531 If ( boldsymbol{A}=left[begin{array}{lll}boldsymbol{a} & boldsymbol{0} & boldsymbol{0} \ boldsymbol{0} & boldsymbol{a} & boldsymbol{0} \ boldsymbol{0} & boldsymbol{0} & boldsymbol{a}end{array}right], ) then the value of
( |boldsymbol{A}||mathbf{A} mathbf{d} mathbf{j} boldsymbol{A}| )
A ( cdot a^{3} )
в. ( a^{6} )
( c cdot a^{9} )
( D cdot a^{2} )
12
532 Find the values of the following determinant, where ( boldsymbol{i}=sqrt{-1} ) ( left|begin{array}{cc}mathbf{2} boldsymbol{i} & -mathbf{3} i \ boldsymbol{i}^{mathbf{3}} & -mathbf{2} boldsymbol{i}^{5}end{array}right| ) 12
533 23. Let @ =-
23. Let o

+
, then the value of the det
.
, then the value of the det.
1
-1-0-
(2002 – 2 Marks
(a) 30
© 36?
(b) 30(0-1)
(d) 30(1-w)
12
534 The points ( (-a,-b),(0,0),(a, b) ) and ( left(a^{2},right. )
ab) are
A. collinear
B. vertices of a rectangle
c. vertices of a parallelogram
D. None of these
12
535 The value of
( begin{array}{|ccc|}1 & cos alpha-sin alpha & cos alpha+sin alpha \ 1 & cos beta-sin beta & cos beta+sin beta \ 1 & cos gamma-sin gamma & cos gamma+sin gammaend{array} mid )
A. ( left|begin{array}{ccc}1 & cos alpha & sin alpha \ 1 & cos beta & sin beta \ 1 & cos gamma & sin gammaend{array}right| )
B. ( left|begin{array}{ccc}1 & cos alpha & sin alpha \ 1 & cos beta & sin beta \ 1 & cos gamma & sin gammaend{array}right| )
( mathbf{c} cdotleft|begin{array}{lll}cos alpha & cos beta & 1 \ cos beta & cos gamma & 1 \ cos gamma & cos alpha & 1end{array}right| )
D. ( left|begin{array}{ccc}1 & cos alpha & sin alpha \ 1 & cos beta & sin beta \ 1 & cos gamma & sin gammaend{array}right| )
12
536 Match the entries of List – ( A ) and List ( -B ) 12
537 be a square matrix all of whose entries are integers.
[2008]
then A- exists but all its entries are not
18. Let A be a square matrix all
Then which one of the following is true?
(a) If det A=+1, then A-1 exists but all its entri
necessarily integers
(b) If det A++1, then A-1 exists and all its entries are non
integers
(©) If det A = + 1, then A-1 exists but all its entries are
integers
(d) If det A=+1, then A-1 need not exists
12
538 ( f f(x)=left|begin{array}{ccc}sin x & 1 & 0 \ 1 & 2 sin x & 1 \ 0 & 1 & 2 sin xend{array}right| ) then
( int_{-pi / 2}^{pi / 2} f(x) d x ) equals to
( A )
B.
( c )
D. ( frac{3 pi}{2} )
12
539 Find the value of the determinant:
[
left|begin{array}{ccc}
cos (theta+phi) & -sin (theta+phi) & cos 2 phi \
sin theta & cos theta & sin phi \
-cos theta & sin theta & cos phi
end{array}right|
]
12
540 If ( A ) is a square matrix of order 3 with
( |A|=4, ) then write the value of ( |-2 A| )
12
541 If the points ( (a, 0),(0, b) ) and (1,1) are collinear, then ( frac{1}{a}+frac{1}{b} ) equal to –
A . 1
B. 2
( c cdot 3 )
D. 4
12
542 Assertion
Points ( boldsymbol{P}(-sin (boldsymbol{beta}- )
( boldsymbol{alpha}),-cos beta), boldsymbol{Q}(cos (boldsymbol{beta}-boldsymbol{alpha}), sin beta) ) and
( boldsymbol{R}(cos (boldsymbol{beta}-boldsymbol{alpha}+boldsymbol{theta}), sin (boldsymbol{beta}-boldsymbol{theta})), ) where
( beta=frac{pi}{4}+frac{alpha}{2} ) are non-collinear.
Reason
Three given points are non-collinear if they form a triangle of non-zero area.
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 the Reason is correct
12
543 Let the three din
the three digit numbers A 28, 309, and 62 C, where A, B,
Care integers between 0 and 9, be divisible by a fixed
integer k Show that the determinant 8
12
9 C is divisible
B 2
(1990) – 4 Marks)
by ki
12
544 If ( D=left|begin{array}{ccc}a^{2}+1 & a b & a c \ b a & b^{2}+1 & b c \ c a & c b & c^{2}+1end{array}right| ) then
( D= )
12
545 ( boldsymbol{A}=left[begin{array}{lll}mathbf{4} & -mathbf{2} & mathbf{5}end{array}right], boldsymbol{B}=left[begin{array}{l}mathbf{2} \ mathbf{0} \ mathbf{3}end{array}right], ) then
( boldsymbol{A} boldsymbol{d} boldsymbol{j}(boldsymbol{B} boldsymbol{A})= )
( A cdotleft{begin{array}{lll}0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0end{array}right} )
( mathbf{B} cdotleft{begin{array}{lll}8 & -4 & 10 \ 0 & 0 & 0 \ 12 & -6 & 15end{array}right} )
( mathbf{c} cdotleft{begin{array}{ccc}8 & 0 & 12 \ -4 & 0 & -6 \ 10 & 0 & 5end{array}right} )
D. None of the above
12
546 If ( A ) and ( B ) are square matrices of order
3 such that ( |boldsymbol{A}|=-mathbf{1},|boldsymbol{B}|=mathbf{3}, ) then
( |mathbf{3} boldsymbol{A} boldsymbol{B}|= )
A . -9
B. -81
c. -27
D. 81
12
547 ( left|begin{array}{ccc}boldsymbol{b}+boldsymbol{c} & boldsymbol{a}-boldsymbol{c} & boldsymbol{a}-boldsymbol{b} \ boldsymbol{b}-boldsymbol{c} & boldsymbol{c}+boldsymbol{a} & boldsymbol{b}-boldsymbol{a} \ boldsymbol{c}-boldsymbol{b} & boldsymbol{c}-boldsymbol{a} & boldsymbol{a}+boldsymbol{b}end{array}right|= )
A. ( 4 a b c )
B. ( 6 a b c )
( c .8 a b c )
D. ( 2 a b c )
12
548 If ( Delta=left|begin{array}{lll}mathbf{5} & mathbf{3} & mathbf{8} \ mathbf{2} & mathbf{0} & mathbf{1} \ mathbf{1} & mathbf{2} & mathbf{3}end{array}right|, ) white the cofactor of
the element ( a_{32} )
12
549 If ( boldsymbol{x}^{boldsymbol{a}} boldsymbol{y}^{boldsymbol{b}}=boldsymbol{e}^{boldsymbol{m}}, boldsymbol{x}^{boldsymbol{c}} boldsymbol{y}^{boldsymbol{d}}=boldsymbol{e}^{boldsymbol{n}}, Delta_{mathbf{1}}= )
( left|begin{array}{ll}boldsymbol{m} & boldsymbol{b} \ boldsymbol{n} & boldsymbol{d}end{array}right|, triangle_{2}=left|begin{array}{ll}boldsymbol{a} & boldsymbol{m} \ boldsymbol{c} & boldsymbol{n}end{array}right| ) and ( Delta_{boldsymbol{3}}=left|begin{array}{ll}boldsymbol{a} & boldsymbol{b} \ boldsymbol{c} & boldsymbol{d}end{array}right| )
the value of ( x ) and ( y ) are respectively
A ( cdot frac{Delta_{1}}{Delta_{3}} ) and ( frac{Delta_{2}}{Delta_{3}} )
B. ( frac{Delta_{2}}{Delta_{1}} ) and ( frac{Delta_{3}}{Delta_{1}} )
( ^{mathbf{c}} cdot log left(frac{Delta_{1}}{Delta_{3}}right) a n d log left(frac{Delta_{2}}{Delta_{3}}right) )
( mathbf{D} cdot e^{Delta_{1} / Delta_{3}} ) and ( e^{Delta_{2} / Delta_{3}} )
12
550 If ( A ) is a square matrix of order 3 and ( |boldsymbol{A}|=4 . ) Find the value of ( |mathbf{2 A}| ) 12
551 |6i -3i 1
7. If 4 3i -1 = x+iy, then (1998 – 2 Marks)
en 20 3 il
(a) x=3, y=2
(b)x=1, y=3
(C) x=0, y=3
(d) x=0, y=0
12
552 ( fleft|begin{array}{ccc}mathbf{1} & boldsymbol{a} & boldsymbol{b} boldsymbol{c} \ mathbf{1} & boldsymbol{b} & boldsymbol{c a} \ mathbf{1} & boldsymbol{c} & boldsymbol{a b}end{array}right|=boldsymbol{lambda}left|begin{array}{ccc}boldsymbol{a}^{2} & boldsymbol{b}^{2} & boldsymbol{c}^{2} \ boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ mathbf{1} & boldsymbol{1} & boldsymbol{1}end{array}right|, ) then ( lambda ) is
equal to
( A cdot 1 )
B. -1
( c cdot 2 )
( D .-3 )
12
553 Show that the area of the triangle on the Argand diagram
formed by the complex numbers z, iz and z+iz is — Iz.
12
554 ( mathbf{f} mathbf{Delta}=left|begin{array}{lll}boldsymbol{a}_{11} & boldsymbol{a}_{12} & boldsymbol{a}_{13} \ boldsymbol{a}_{21} & boldsymbol{a}_{22} & boldsymbol{a}_{23} \ boldsymbol{a}_{31} & boldsymbol{a}_{32} & boldsymbol{a}_{33}end{array}right| ) and ( boldsymbol{c}_{i j}= )
( (-1)^{i+j} ) (determinant obtained by
deleting ith row and jth column), then ( left|begin{array}{lll}c_{11} & c_{12} & c_{13} \ c_{21} & c_{22} & c_{23} \ c_{31} & c_{32} & c_{33}end{array}right|=Delta^{2} )
( left|begin{array}{ccc}mathbf{1} & boldsymbol{x} & boldsymbol{x}^{2} \ boldsymbol{x} & boldsymbol{x}^{2} & mathbf{1} \ boldsymbol{x}^{2} & boldsymbol{1} & boldsymbol{x}end{array}right|=mathbf{7} ) and ( boldsymbol{Delta}= )
( left|begin{array}{ccc}x^{3}-1 & 0 & x-x^{4} \ 0 & x-x^{4} & x^{3}-1 \ x-x^{4} & x^{3}-1 & 0end{array}right|, ) then
A ( . Delta=7 )
B. ( Delta=343 )
c. ( Delta=-49 )
D. ( Delta=49 )
12
555 Prove that ( left[begin{array}{ccc}1 & a & a^{2} \ 1 & b & b^{2} \ 1 & c & c^{2}end{array}right]= )
( (a-b)(b-c)(c-a) )
12
556 Let a, b, c be positive and not all equal. Show that the value
а ь с
of the determinant
b ca is negative
с а ь
12
557 ( fleft|begin{array}{lll}boldsymbol{x} & boldsymbol{2} & boldsymbol{8} \ boldsymbol{2} & boldsymbol{8} & boldsymbol{x} \ boldsymbol{8} & boldsymbol{x} & boldsymbol{2}end{array}right|=left|begin{array}{lll}boldsymbol{3} & boldsymbol{x} & boldsymbol{7} \ boldsymbol{x} & boldsymbol{7} & boldsymbol{3} \ boldsymbol{7} & boldsymbol{3} & boldsymbol{x}end{array}right|= )
( left|begin{array}{ccc}mathbf{5} & mathbf{5} & boldsymbol{x} \ mathbf{5} & boldsymbol{x} & mathbf{5} \ boldsymbol{x} & mathbf{5} & mathbf{5}end{array}right|=mathbf{0} ) then ( boldsymbol{x} ) is equal to
( mathbf{A} cdot mathbf{0} )
B. -10
( c .3 )
D. None of these
12
558 Match the entries in column I with
column II
12
559 if ( a, b, c ) are unequal what is the condition that the value of following determinat is zero ( delta=left|begin{array}{lll}a & a^{2} & a^{3}+1 \ b & b^{2} & b^{3}+1 \ c & c^{2} & c^{3}+1end{array}right| )
A ( .1+a b c=0 )
B. ( a+b+c+1=0 )
c. ( (a-b)(b-c)(c-a)=0 )
D. None of these
12
560 Find the value of the determinant
without expansion ( left|begin{array}{ccc}b^{2}-a b & b-c & b c-a c \ a b-a^{2} & a-b & b^{2}-a b \ b c-a c & c-a & a b-a^{2}end{array}right| )
12
561 Show that the points ( boldsymbol{A}(-mathbf{3}, mathbf{3}), boldsymbol{B}(mathbf{0}, mathbf{0}) )
( C(3,-3) ) are collinear
12
562 ( left|begin{array}{lll}boldsymbol{a}+boldsymbol{b} & boldsymbol{b}+boldsymbol{c} & boldsymbol{c}+boldsymbol{a} \ boldsymbol{b}+boldsymbol{c} & boldsymbol{c}+boldsymbol{a} & boldsymbol{a}+boldsymbol{b} \ boldsymbol{c}+boldsymbol{a} & boldsymbol{a}+boldsymbol{b} & boldsymbol{b}+boldsymbol{c}end{array}right|=boldsymbol{K}left|begin{array}{lll}boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{b} & boldsymbol{c} & boldsymbol{a} \ boldsymbol{c} & boldsymbol{a} & boldsymbol{b}end{array}right| )
( operatorname{then} K= )
A . 1
B. 2
( c )
( D )
12
563 If ( a^{2}+b^{2}+c^{2}=-2 ) and ( f(x)= )
( left|begin{array}{ccc}mathbf{1}+boldsymbol{a}^{2} boldsymbol{x} & left(mathbf{1}+boldsymbol{b}^{2}right) boldsymbol{x} & left(mathbf{1}+boldsymbol{c}^{2}right) boldsymbol{x} \ left(mathbf{1}+boldsymbol{a}^{2}right) boldsymbol{x} & mathbf{1}+boldsymbol{b}^{2} boldsymbol{x} & left(mathbf{1}+boldsymbol{c}^{2}right) boldsymbol{x} \ left(mathbf{1}+boldsymbol{a}^{2}right) boldsymbol{x} & left(mathbf{1}+boldsymbol{b}^{2}right) boldsymbol{x} & mathbf{1}+boldsymbol{c}^{mathbf{2}} boldsymbol{x}end{array}right| ) then
( f(x) ) is a polynomial of degree
( A cdot )
B.
( c .3 )
( D )
12
564 If ( boldsymbol{A}=left[begin{array}{cc}mathbf{0} & mathbf{1} \ -mathbf{1} & mathbf{0}end{array}right] ) then determinant of ( [boldsymbol{A}] )
is
( mathbf{A} cdot mathbf{1} )
B. –
c. 0
D.
12
565 If the value of the determinant
( left|begin{array}{lll}boldsymbol{a} & boldsymbol{1} & boldsymbol{1} \ boldsymbol{1} & boldsymbol{b} & boldsymbol{1} \ boldsymbol{1} & boldsymbol{1} & boldsymbol{c}end{array}right| ) is positive, then
A ( . a b c>1 )
B . ( a b c>-8 )
c. ( a b c-2 )
12
566 f ( A+B+C=pi ), then
( left|begin{array}{ccc}sin (boldsymbol{A}+boldsymbol{B}+boldsymbol{C}) & sin boldsymbol{B} & cos boldsymbol{C} \ -sin boldsymbol{B} & boldsymbol{0} & boldsymbol{t a n} boldsymbol{A} \ cos (boldsymbol{A}+boldsymbol{B}) & -boldsymbol{t a n} boldsymbol{A} & boldsymbol{0}end{array}right| )
equals
A .
B. 2 ( sin B tan A cos C )
( c )
D. none of these
12
567 Find the value of ( K ) if the point ( A(2,3), B ) ( (4, K) ) and ( C(6,-3) ) are collinear? 12
568 For how many real values of ( ^{prime} m^{prime} ) the points ( boldsymbol{A}(boldsymbol{m}+mathbf{1}, mathbf{1}), boldsymbol{B}(boldsymbol{2 m}+mathbf{1}, boldsymbol{3}) ) and
( C(2 m+2,2 m) ) are collinear.
12
569 Let ( mathbf{P} ) and ( mathbf{Q} ) be ( mathbf{3} times mathbf{3} ) matrices with
( boldsymbol{P} neq boldsymbol{Q} . ) If ( boldsymbol{P}^{3}=boldsymbol{Q}^{3} ) and ( boldsymbol{P}^{2} boldsymbol{Q}=boldsymbol{Q}^{2} boldsymbol{P} )
then determinant of ( left(P^{2}+Q^{2}right) ) is equal
to:
A . -2
B.
( c cdot 0 )
D. –
12
570 ( left|begin{array}{ccc}boldsymbol{x}^{boldsymbol{n}} & boldsymbol{x}^{boldsymbol{n}+mathbf{2}} & boldsymbol{x}^{boldsymbol{n}+mathbf{3}} \ boldsymbol{y}^{boldsymbol{n}} & boldsymbol{y}^{boldsymbol{n}+mathbf{2}} & boldsymbol{y}^{boldsymbol{n}+boldsymbol{3}} \ boldsymbol{z}^{boldsymbol{n}} & boldsymbol{z}^{boldsymbol{n}+mathbf{2}} & boldsymbol{z}^{boldsymbol{n}+mathbf{3}}end{array}right|=(boldsymbol{x}-boldsymbol{y})(boldsymbol{y}- )
( (z-x)left(frac{1}{x}+frac{1}{y}+frac{1}{z}right), ) then the value
of ( n ) is
( A )
B. –
( c cdot 1 )
( D )
12
571 Find the integral value of ( x, ) if ( left|begin{array}{ccc}boldsymbol{x}^{2} & boldsymbol{x} & mathbf{1} \ mathbf{0} & boldsymbol{2} & mathbf{1} \ boldsymbol{3} & boldsymbol{1} & boldsymbol{4}end{array}right|=mathbf{2} mathbf{8} ) 12
572 The value of ( left|begin{array}{llll}boldsymbol{p} & boldsymbol{0} & boldsymbol{0} & boldsymbol{0} \ boldsymbol{a} & boldsymbol{q} & boldsymbol{0} & boldsymbol{0} \ boldsymbol{b} & boldsymbol{c} & boldsymbol{r} & boldsymbol{0} \ boldsymbol{d} & boldsymbol{e} & boldsymbol{f} & boldsymbol{s}end{array}right| ) is
( A cdot p+q+r+s )
B.
( c cdot a b+c d+e f )
D. pqrs
12
573 Prove that:
[
left|begin{array}{ccc}
boldsymbol{a}-boldsymbol{b}-boldsymbol{c} & boldsymbol{2} boldsymbol{a} & boldsymbol{2} boldsymbol{a} \
boldsymbol{2} boldsymbol{b} & boldsymbol{b}-boldsymbol{c}-boldsymbol{a} & boldsymbol{2} boldsymbol{b} \
boldsymbol{2} boldsymbol{c} & boldsymbol{2} boldsymbol{c} & boldsymbol{c}-boldsymbol{a}-boldsymbol{b}
end{array}right|=
]
( (a+b+c)^{3} )
12
574 The coefficient of ( x^{2} ) in the expansion of
the determinant
( left|begin{array}{ccc}x^{2} & x^{3}+1 & x^{5}+2 \ x^{3}+3 & x^{2}+x & x^{3}+x^{4} \ x+4 & x^{3}+x^{4} & 2^{3}end{array}right| ) is
A . -10
B. -8
( c .-2 )
D. – 6
E.
12
575 The characteristic equation of a matrix ( A ) is ( lambda^{3}-5 lambda^{2}-3 lambda+2 I=0 ) then
( |boldsymbol{a} boldsymbol{d} boldsymbol{j} boldsymbol{A}|= )
( mathbf{A} cdot mathbf{4} )
B . 25
c. 9
D. 30
12
576 If ( P=left[begin{array}{lll}1 & c & 3 \ 1 & 3 & 3 \ 2 & 4 & 4end{array}right] ) is the adjoint of a
( 3 times 3 ) matrix ( Q ) and ( operatorname{det} .(Q)=4, ) then ( c ) is
equal to.
( mathbf{A} cdot mathbf{0} )
B. 4
c. 5
D. 11
12
577 ( left|begin{array}{cc}sin ^{2} theta & cos ^{2} theta \ -cos ^{2} theta & sin ^{2} thetaend{array}right|= )
( mathbf{A} cdot cos 2 theta )
в. ( frac{1}{2}left(1+cos ^{2} 2 thetaright) )
c. ( frac{1}{2}left(1-sin ^{2} 2 thetaright) )
D. ( frac{1}{2} sin ^{2} 2 theta )
12
578 ( mathbf{f} mathbf{f}_{mathbf{3}}=left[begin{array}{ccc}mathbf{0} & mathbf{1} & mathbf{- 1} \ mathbf{2} & mathbf{1} & mathbf{3} \ mathbf{3} & mathbf{2} & mathbf{1}end{array}right], ) then
( left[boldsymbol{A}(boldsymbol{a} boldsymbol{d} boldsymbol{j} boldsymbol{A}) boldsymbol{A}^{-1}right] boldsymbol{A}= )
( mathbf{A} cdotleft[begin{array}{lll}6 & 0 & 0 \ 0 & 6 & 0 \ 0 & 0 & 6end{array}right] )
( mathbf{B} cdotleft[begin{array}{lll}4 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 4end{array}right] )
( mathbf{C} cdotleft[begin{array}{lll}2 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 2end{array}right] )
D. ( I )
12
579 Find the value of ( lambda ) if the following equations are consistent ( boldsymbol{x}+boldsymbol{y}-mathbf{3}=mathbf{0} )
( (1+lambda) x+(2+lambda) y-8=0 )
( boldsymbol{x}-(mathbf{1}+boldsymbol{lambda}) boldsymbol{y}+(boldsymbol{2}+boldsymbol{lambda})=mathbf{0} )
12
580 18.
sinx cos x cos x
sin x cos xl
The number of distinct real roots of COS X
cos x cos x sin x
VI
s
is
= 0 in the interval
(a) 0 (6) 2
(20015)
(d) 3
(
1
12
581 The value of ( left|begin{array}{ccc}-a^{2} & a b & a c \ a b & -b^{2} & b c \ a c & b c & -c^{2}end{array}right| ) is
A . a perfect cube
B. zero
c. a perfect square
D. negative
12
582 Evaluate ( left|begin{array}{lll}boldsymbol{x} & boldsymbol{x}^{2} & boldsymbol{x}^{2} \ boldsymbol{y} & boldsymbol{y}^{2} & boldsymbol{y}^{2} \ boldsymbol{z} & boldsymbol{z}^{2} & boldsymbol{z}^{3}end{array}right| ) 12
583 f ( p lambda^{4}+p lambda^{3}+p lambda^{2}+s lambda+t= )
( left|begin{array}{ccc}lambda^{2}+3 lambda & lambda+1 & lambda+3 \ lambda+1 & 2-lambda & lambda-4 \ lambda-3 & lambda+4 & 3 lambdaend{array}right|, ) then value
of t is
A . 16
B . 18
( c .17 )
D. 19
12
584 The value of determinant ( left|begin{array}{lll}1 / a & b c & a^{3} \ 1 / b & c a & b^{3} \ 1 / c & a b & c^{3}end{array}right| )
A ( cdot a^{2} b^{2} c^{2}(a-b)(b-c)(c-a) )
B. 0
c ( cdot(a-b)(b-c)(c-a) )
D. None of the above
12
585 Solve ( left.boldsymbol{D}=mid begin{array}{ccc}mathbf{1} & -mathbf{2} & mathbf{1} \ mathbf{2} & mathbf{1} & -mathbf{1} \ mathbf{1} & mathbf{3} & mathbf{1}end{array}right] ) 12
586 f ( triangle_{1}= )
( mid begin{array}{ccc}a_{1}^{2}+b_{1}+c_{1} & a_{1} a_{2}+b_{2}+c_{2} & a_{1} a_{3}+c \ b_{1} b_{2}+c_{1} & b_{2}^{2}+c_{2} & b_{2} b_{3} \ c_{3} c_{1} & c_{3} c_{2} & c_{3}^{2}end{array} )
and ( triangle_{2}=left|begin{array}{lll}a_{1} & b_{1} & c_{1} \ a_{2} & b_{2} & c_{2} \ a_{3} & b_{3} & c_{3}end{array}right|, ) then ( frac{triangle_{1}}{triangle_{2}} ) is
equal to
A ( cdot a_{1} b_{2} c_{3} )
B. ( a_{1} a_{2} a_{3} )
( c cdot a_{3} b_{2} c )
D. ( a_{1} b_{1} c_{1}+a_{2} b_{2} c_{2}+a_{3} b_{3} c_{3} )
12
587 In each of the following find the value of
( k, ) for which the points are collinear.
(ii) ( (8,1),(k,-4),(2,-5) )
12
588 ( (k, k),(2,3) ) and (4,-1) are collimear
So find the value of ( k )
12
589 ( mathbf{f}_{mathbf{1}}=left|begin{array}{ccc}mathbf{7} & boldsymbol{x} & mathbf{2} \ -mathbf{5} & boldsymbol{x}+mathbf{1} & mathbf{3} \ mathbf{4} & boldsymbol{x} & mathbf{7}end{array}right|, boldsymbol{Delta}_{2}= )
( left|begin{array}{ccc}boldsymbol{x} & boldsymbol{2} & boldsymbol{7} \ boldsymbol{x}+boldsymbol{1} & boldsymbol{3} & -boldsymbol{5} \ boldsymbol{x} & boldsymbol{7} & boldsymbol{4}end{array}right| ) then ( boldsymbol{Delta}_{1}-boldsymbol{Delta}_{2}=boldsymbol{0} ) for
( mathbf{A} cdot x=2 )
B. all real ( x )
c. ( x=0 )
D. none of these
12
590 Find ( left|begin{array}{ll}cos alpha & -sin alpha \ sin alpha & cos alphaend{array}right| ) 12
591 Ta 1 o7 [a 1 17 [f]
16. If A= 1 b d ,B= 0 d c ,U = 8 V = 0 ,x = y
1 b c f g h [h lol
and AX = U has infinitely many solutions, prove that
BX=V has no unique solution. Also show that if afd + 0,
then BX= V has no solution.
(2004 – 4 Marks)
12
592 ( f(x)=left|begin{array}{ccc}x & cos x & e^{x^{2}} \ sin x & x^{2} & sec x \ tan x & 1 & 2end{array}right| . ) Find
( boldsymbol{f}(mathbf{0}) ? )
( mathbf{A} cdot mathbf{0} )
( B )
( c .-r )
D. ( 2 pi )
12
593 Using properties of determinant, prove
the following:
[
mid begin{array}{ccc}
1+a^{2}-b^{2} & 2 a b & -2 b \
2 a b & 1-a^{2}+b^{2} & 2 a \
2 b & -2 a & 1-a^{2}-b^{2}
end{array}
]
( left(1+a^{2}+b^{2}right)^{3} )
12
594 Without expanding a determinant at any stage, show that
x²+x x+1 x-2
2×2 + 3x -1 3x 3x – 3) = xA+B , where A and B are
x2 + 2x + 3 2x-1 2x – 1
12
595 6.
Let A=
(1
2
(1
-1
1
1
1)
( 4
-3. and 10 B = -5
1
(1
2
0

2)
a . If B is
3)
1
the inverseof matrix A, then a is
(a) 5 (6) 1 (c) 2
[2004]
(d) 2
12
596 If ( a_{k}, b_{k}, c_{k} epsilon R ) for ( k=1,2,3 ) and
( left|begin{array}{lll}boldsymbol{a}_{1} & boldsymbol{b}_{1} & boldsymbol{c}_{1} \ boldsymbol{a}_{2} & boldsymbol{b}_{2} & boldsymbol{c}_{2} \ boldsymbol{a}_{3} & boldsymbol{b}_{3} & boldsymbol{c}_{3}end{array}right| )
then sum of the roots of the equation ( left|begin{array}{lll}a_{1}+i b_{1} x & i a_{1} x+b_{1} & c_{1} \ a_{2}+i b_{2} x & i a_{2} x+b_{2} & c_{2} \ a_{3}+i b_{3} x & i a_{3} x+b_{3} & c_{3}end{array}right| )
is
( mathbf{A} cdot a_{1}+a_{2}+a_{3} )
B. ( b_{1}+b_{2}+b_{3} )
( mathbf{c} cdot a_{1} b_{1}+a_{2} b_{2}+a_{3} b_{3} )
D. none of these
12
597 Solve the determinant ( left|begin{array}{ll}boldsymbol{x} & boldsymbol{y} \ -boldsymbol{y} & boldsymbol{x}end{array}right| ) 12
598 Write the minors and cofactors of each
element of the first column of the
following matrices ( boldsymbol{A}=left[begin{array}{lll}1 & a & b c \ 1 & b & c a \ 1 & c & a bend{array}right] )
12
599 ( mid begin{array}{ccc}mathbf{2} boldsymbol{a}_{1} boldsymbol{b}_{1} & boldsymbol{a}_{1} boldsymbol{b}_{1}+boldsymbol{a}_{2} boldsymbol{b}_{1} & boldsymbol{a}_{1} boldsymbol{b}_{3}+boldsymbol{a}_{3} boldsymbol{b}_{1} \ boldsymbol{a}_{1} boldsymbol{b}_{2}+boldsymbol{a}_{2} boldsymbol{b}_{1} & boldsymbol{2} boldsymbol{a}_{2} boldsymbol{b}_{2} & boldsymbol{a}_{2} boldsymbol{b}_{3}+boldsymbol{a}_{3} boldsymbol{b}_{2} \ boldsymbol{a}_{1} boldsymbol{b}_{3}+boldsymbol{a}_{3} boldsymbol{b}_{1} & boldsymbol{a}_{3} boldsymbol{b}_{2}+boldsymbol{b}_{3} boldsymbol{a}_{2} & boldsymbol{2} boldsymbol{a}_{3} boldsymbol{b}_{3}end{array} )
( A )
B.
( mathbf{c} cdot a_{1} b_{1} a_{2} b )
D. ( a_{1} b_{1}+a_{2} b_{2} )
12
600 If ( boldsymbol{A}=left|begin{array}{ll}mathbf{1 0} & mathbf{2} \ mathbf{3 0} & mathbf{6}end{array}right| )
then ( |boldsymbol{A}|= )
( A cdot O )
B. 10
( c cdot 12 )
D. 60
12
601 ( fleft(begin{array}{lll}1 & 0 & 1 \ 0 & 1 & 2 \ 0 & 0 & 4end{array}right], ) then show that
[
|mathbf{3} boldsymbol{A}|=mathbf{2 7}|boldsymbol{A}|
]
12
602 If every element of third order determinant of ( Delta ) is multiplied by 5 then value of new determinant equals
to,
A ( . Delta )
B. ( 5 Delta )
c. ( 25 Delta )
D. ( 125 Delta )
12
603 27. Let P and Q be 3 x 3 matrices P+Q. If P3= Q and P20 =
Q2P then determinant of (P2 +Q) is equal to: [2012]
(a) -2 (6) 1 (c) o (d) -1.
12
604 Expand: ( left|begin{array}{ccc}1 & -7 & 3 \ 5 & -6 & 0 \ 1 & 2 & -3end{array}right| ) 12
605 Evaluate :
( Delta=left|begin{array}{ccc}0 & sin alpha & -cos alpha \ -sin alpha & 0 & sin beta \ cos alpha & -sin beta & 0end{array}right| )
12
606 ( left|begin{array}{ccc}boldsymbol{a}+boldsymbol{b} & boldsymbol{a}+mathbf{2} boldsymbol{b} & boldsymbol{a}+boldsymbol{3} boldsymbol{b} \ boldsymbol{a}+mathbf{2} boldsymbol{b} & boldsymbol{a}+boldsymbol{3} boldsymbol{b} & boldsymbol{a}+boldsymbol{4} boldsymbol{b} \ boldsymbol{a}+boldsymbol{4} boldsymbol{b} & boldsymbol{a}+boldsymbol{5} boldsymbol{b} & boldsymbol{a}+boldsymbol{6} boldsymbol{b}end{array}right|= )
A ( cdot a^{2}+b^{2}+c^{2}-3 a b )
B. ( 3 a b c )
( c cdot 3 a+5 b )
( D )
12
607 ( mathrm{f}left|begin{array}{ccc}boldsymbol{x} & boldsymbol{x}+boldsymbol{y} & boldsymbol{x}+boldsymbol{y}+boldsymbol{z} \ mathbf{2} boldsymbol{x} & mathbf{3} boldsymbol{x}+mathbf{2} boldsymbol{y} & boldsymbol{4} boldsymbol{x}+boldsymbol{3} boldsymbol{y}+boldsymbol{2} boldsymbol{z} \ boldsymbol{3} boldsymbol{x} & boldsymbol{6} boldsymbol{x}+boldsymbol{3} boldsymbol{y} & boldsymbol{1} mathbf{0} boldsymbol{x}+boldsymbol{6} boldsymbol{y}+boldsymbol{3} boldsymbol{z}end{array}right|=mathbf{6} boldsymbol{4} )
then find ( x )
12
608 If ( [mathbf{x}] ) stands greatest integer ( leq mathbf{x} ) then
the value of ( left|begin{array}{ccc}{[boldsymbol{e}]} & {[boldsymbol{pi}]} & {left[boldsymbol{pi}^{2}-boldsymbol{6}right]} \ {[boldsymbol{pi}]} & boldsymbol{pi}^{2}-boldsymbol{6} & {[boldsymbol{e}]} \ {left[boldsymbol{pi}^{2}-boldsymbol{6}right]} & {[boldsymbol{e}]} & {[boldsymbol{pi}]}end{array}right| ) equals
( A cdot-8 )
B. 8
( c cdot-1 )
( D )
12
609 Calculate the values of the
determinants:
( left|begin{array}{cccc}mathbf{0} & mathbf{1} & mathbf{1} & mathbf{1} \ mathbf{1} & boldsymbol{b}+boldsymbol{c} & boldsymbol{a} & boldsymbol{a} \ mathbf{1} & boldsymbol{b} & boldsymbol{c}+boldsymbol{a} & boldsymbol{b} \ mathbf{1} & boldsymbol{c} & boldsymbol{c} & boldsymbol{a}+boldsymbol{b}end{array}right| )
12
610 Let ( Delta= )
( left|begin{array}{ccc}sin theta cos phi & sin theta sin phi & cos theta \ cos theta cos phi & cos theta sin phi & -sin theta \ -sin theta sin phi & sin theta cos phi & 0end{array}right|, ) then
A. ( Delta ) is independent of ( theta )
B. ( Delta ) is independent of ( phi )
( mathrm{c} cdot Delta ) is a constant
D. none of these
12
611 37. 14-
37. IfA=
), then adj(x2 + 124) is equal to
, then adj (3A2 + 12A) is equal to :
(JEE M 2017]
[ 72 -847
L-63 51
a)
-84
51
a [51 84]
() [84 72
12
612 If ( boldsymbol{A}=left[begin{array}{cc}-mathbf{5} & mathbf{2} \ mathbf{1} & -mathbf{3}end{array}right], ) then adj ( mathbf{A} ) is equal to
A. ( left[begin{array}{ll}-3 & -2 \ -1 & -5end{array}right] )
(年) 0
В. ( left[begin{array}{cc}3 & -2 \ -1 & 5end{array}right] )
c. ( left[begin{array}{ll}5 & 1 \ 2 & 3end{array}right] )
D. ( left[begin{array}{ll}3 & 2 \ 1 & 5end{array}right] )
12
613 Let ( boldsymbol{A}=left[begin{array}{cc}mathbf{5} & mathbf{8} \ mathbf{8} & mathbf{1 3}end{array}right] ) then show that ( boldsymbol{A} )
satisfies the equation ( x^{2}-18 x+1= )
( mathbf{0} )
12
614 ( operatorname{Let} boldsymbol{D}_{1}=left|begin{array}{lll}boldsymbol{a} & boldsymbol{b} & boldsymbol{a}+boldsymbol{b} \ boldsymbol{c} & boldsymbol{d} & boldsymbol{c}+boldsymbol{d} \ boldsymbol{a} & boldsymbol{b} & boldsymbol{a}-boldsymbol{b}end{array}right|, quad boldsymbol{D}_{2}= )
( left|begin{array}{ccc}boldsymbol{a} & boldsymbol{c} & boldsymbol{a}+boldsymbol{c} \ boldsymbol{b} & boldsymbol{d} & boldsymbol{b}+boldsymbol{d} \ boldsymbol{a} & boldsymbol{c} & boldsymbol{a}+boldsymbol{b}+boldsymbol{c}end{array}right|, ) then the value of
( left|frac{boldsymbol{D}_{1}}{boldsymbol{D}_{2}}right|, ) where ( boldsymbol{b} neq mathbf{0} ) and ( boldsymbol{a} boldsymbol{d} neq boldsymbol{b} boldsymbol{c}, ) is
12
615 f ( a, b, c ) are ( p ) th ( , q ) thand ( r ) th terms of a
( mathrm{GP}, ) then ( left|begin{array}{lll}log boldsymbol{a} & boldsymbol{p} & mathbf{1} \ log boldsymbol{b} & boldsymbol{q} & mathbf{1} \ log boldsymbol{c} & boldsymbol{r} & mathbf{1}end{array}right| ) is equal to
( mathbf{A} cdot mathbf{0} )
B.
c. ( log a b c )
D. none of these
12
616 ( f(a, b, c text { are in } A P ) then Prove that ( left|begin{array}{ccc}boldsymbol{x}+mathbf{2} & boldsymbol{x}+mathbf{3} & boldsymbol{x}+mathbf{2} boldsymbol{a} \ boldsymbol{x}+mathbf{3} & boldsymbol{x}+mathbf{4} & boldsymbol{x}+mathbf{2} boldsymbol{b} \ boldsymbol{x}+mathbf{4} & boldsymbol{x}+mathbf{5} & boldsymbol{x}+mathbf{2} boldsymbol{c}end{array}right|=mathbf{0} ) 12
617 If ( boldsymbol{A}=left[begin{array}{ll}boldsymbol{alpha} & boldsymbol{2} \ boldsymbol{2} & boldsymbol{alpha}end{array}right] ) and ( left|boldsymbol{A}^{3}right|=mathbf{1 2 5}, ) then the
value of ( alpha ) is
( A cdot pm 1 )
( B cdot pm 2 )
( c cdot pm 3 )
( D .pm 5 )
12
618 Find ( (5 sqrt{3}+3 sqrt{2}) Delta ) where
( Delta=left|begin{array}{ccc}sqrt{13}+sqrt{3} & sqrt{5} & 2 sqrt{5} \ sqrt{15}+sqrt{26} & sqrt{10} & 5 \ 3+sqrt{65} & 5 & sqrt{15}end{array}right| )
12
619 ( f )
( a, b, c ) are all different and if ( left|begin{array}{lll}boldsymbol{a} & boldsymbol{a}^{2} & mathbf{1}+boldsymbol{a}^{3} \ boldsymbol{b} & boldsymbol{b}^{2} & mathbf{1}+boldsymbol{b}^{3} \ boldsymbol{c} & boldsymbol{c}^{2} & boldsymbol{1}+boldsymbol{c}^{3}end{array}right|=mathbf{0} ) then ( -boldsymbol{a} boldsymbol{b} boldsymbol{c}= )
12
620 If the entries in a ( 3 times 3 ) determinant are
either 0 or 1 , then the greatest value of their determinats is:
A
B. 2
( c cdot 3 )
D.
12
621 Evaluate the following determinants
without expansion as far as possible. ( left|begin{array}{ccc}1 & b c & b c(b+c) \ 1 & c a & c a(c+a) \ 1 & a b & a b(a+b)end{array}right| )
12
622 If ( left|begin{array}{lll}a & a^{3} & a^{4}-1 \ b & b^{3} & b^{4}-1 \ c & c^{3} & c^{4}-1end{array}right|=0 ) and ( a, b, c ) are all
distinct then ( a b c(a b+b c+c a) ) is equal
to
A ( . a+b+c )
в. ( a b c )
c. 0
D. none of these
12
623 If ( A ) is a square matrix of order ( n ), then
( |mathbf{A} mathbf{d} mathbf{j} A| ) is
( mathbf{A} cdot|A|^{2} )
B cdot ( |A|^{n} )
C ( cdot|A|^{n-1} )
D・ |A|
12
624 The value of determinant ( left|begin{array}{lll}mathbf{1} & boldsymbol{x} & boldsymbol{y}+boldsymbol{z} \ mathbf{1} & boldsymbol{y} & boldsymbol{z}+boldsymbol{x} \ mathbf{1} & boldsymbol{z} & boldsymbol{x}+boldsymbol{y}end{array}right| )
( mathbf{A} cdot mathbf{0} )
B. ( x+y+z )
c. ( 1+x+y+z )
D. ( (x-y)(y-z)(z-x) )
12
625 Match the statements in Column I with
statements in column II
12
626 Find the value of the determinant:
[
left|begin{array}{ccc}
cos (theta+phi) & -sin (theta+phi) & cos 2 phi \
sin theta & cos theta & sin phi \
-cos theta & sin theta & cos phi
end{array}right|
]
12
627 If ( m ) and ( p ) are positive ( (m geq p) ) and
( boldsymbol{Delta}(boldsymbol{m}, boldsymbol{p})= )
( left|begin{array}{ccc}m & C_{p} & m \ m+1 & C_{p} \ m+2 & m+1 & C_{p+1} & m+1 \ m_{p} & m+2 & m_{p+2} \ & & m+2end{array}right| ) and
( m_{boldsymbol{p}}=mathbf{0} ) if ( mathbf{m}<mathbf{p}, ) then
This question has multiple correct options
A ( . Delta(2,1) / Delta(1,0)=4 )
B. ( Delta(4,3) / Delta(3,2)=2 )
( mathbf{c} cdot Delta(4,3) / Delta(2,1)=5 )
D. ( Delta(4,3) / Delta(1,0)=10 )
12
628 Evaluate the following:
( left|begin{array}{ccc}1 & a & b c \ 1 & b & c a \ 1 & c & a bend{array}right| )
12
629 Find the values of ( x, ) if
(i) ( left|begin{array}{ll}2 & 4 \ 5 & 1end{array}right|=left|begin{array}{ll}2 x & 4 \ 6 & xend{array}right| )
(ii) ( left|begin{array}{ll}2 & 3 \ 4 & 5end{array}right|= )
( left|begin{array}{ll}boldsymbol{x} & boldsymbol{3} \ boldsymbol{2} boldsymbol{x} & boldsymbol{5}end{array}right| )
12
630 The value of
( A cdot O )
B. ( 30^{text {th }} )
( c cdot 30^{-x} )
D. None of these
12
631 Assertion
Let ( p<0 ) and ( alpha_{1}, alpha_{2}, dots, alpha_{9} ) be the nine
roots of ( boldsymbol{x}^{mathbf{9}}=boldsymbol{p}, ) then
( boldsymbol{Delta}=left|begin{array}{lll}boldsymbol{alpha}_{1} & boldsymbol{alpha}_{2} & boldsymbol{alpha}_{3} \ boldsymbol{alpha}_{4} & boldsymbol{alpha}_{5} & boldsymbol{alpha}_{6} \ boldsymbol{alpha}_{4} & boldsymbol{alpha}_{8} & boldsymbol{alpha}_{9}end{array}right|=0 )
Reason
If two rows of a determinant are
identical, then determinant equals zero
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
632 ut
7
Let a, b, c be any real numbers. Suppose that there are real
numbers x, y, z not all zero such that x=cy + bz, y=az + cx,
and z=bx + ay. Then a2 + b2 + c2 + 2abc is equal to
[2008]
(a) 2 (b) 1 (c) 0 (d) 1
12
633 x – 4 2x 2x
38. If 2X X-4 2x
28 2X X-4
=(A+Bx)(x- A)2 , then the ordered
pair (A, B) is equal to :
(a) (-4, 3) (b) (-4,5)
[JEE M 2018]
(d) (-4,-5)
(c) (4,5)
12
634 Let ( Delta_{mathrm{o}}=left[begin{array}{lll}a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33}end{array}right] ) and let ( Delta_{1} )
denote the determinant formed by the
cofactors of elements of ( Delta_{0} ) and ( Delta_{2} )
denote the determinant formed by the
cofactor of ( Delta_{1}, ) similarly ( Delta_{n} ) denotes the
determinant formed by the cofactors of
( Delta_{n-1} ) then the determinant value of ( Delta_{n} )
is
( mathbf{A} cdot Delta_{0}^{2 n} )
B . ( Delta_{0}^{2^{2}} )
( mathbf{C} cdot Delta_{0}^{n^{2}} )
D. ( Delta^{2}_{0} )
12
635 Find the value of determinant, ( left|begin{array}{ccc}1 & a & a^{2}-b c \ 1 & b & b^{2}-c a \ 1 & c & c^{2}-a bend{array}right| ) 12
636 14.
[1 o o [10 of
A= 0 1 1 and 1 = 0 1 0 and
To 2 4. Lo o 1
dI), then the value of c and d are
(20055)
(a) (6,-11) (b) (6,11) (C) (-6,11) (d) (6,-11)
12
637 ( mathbf{y}=sin mathbf{x}, boldsymbol{y}_{n}=frac{boldsymbol{d}^{n}(sin boldsymbol{x})}{boldsymbol{d} boldsymbol{x}^{n}} )
( operatorname{then}left|begin{array}{lll}boldsymbol{y} & boldsymbol{y}_{1} & boldsymbol{y}_{2} \ boldsymbol{y}_{3} & boldsymbol{y}_{4} & boldsymbol{y}_{5} \ boldsymbol{y}_{6} & boldsymbol{y}_{7} & boldsymbol{y}_{8}end{array}right|=? )
( A cdot-sin x )
B.
( c cdot sin x )
D. ( cos )
12
638 If ( triangle(r)=left|begin{array}{cc}r & r^{3} \ 1 & n(n+1)end{array}right|, ) then ( sum_{r=1}^{n} triangle(r) )
is equal to
( ^{A} cdot sum_{i=1}^{n} r )
B. ( sum_{n=1}^{n} r )
( c cdot sum^{n} )
D. ( sum_{n=1}^{n} r^{4} )
12
639 ( begin{array}{c}text { If } boldsymbol{A}+boldsymbol{B}+boldsymbol{C}=boldsymbol{pi}, text { then } \ left|begin{array}{ccc}sin (boldsymbol{A}+boldsymbol{B}+boldsymbol{C}) & sin boldsymbol{B} & cos C \ -sin boldsymbol{0} & boldsymbol{operatorname { t a n }} boldsymbol{A}end{array}right|= \ cos (boldsymbol{A}+boldsymbol{B}) quad-boldsymbol{operatorname { t a n } boldsymbol { A }} & boldsymbol{0}end{array} mid ) 12
640 If ( left|begin{array}{lll}a & a^{3} & a^{4}-1 \ b & b^{3} & b^{4}-1 \ c & c^{3} & c^{4}-1end{array}right|=0 ) and ( a, b, c ) are all
distinct, then ( a b c(a b+b c+c a) ) is
equal to
A. ( a+b+c )
B. abc
( c cdot 0 )
D. none of these
12
641 ( mathbf{f} mathbf{Delta}=left|begin{array}{cc}mathbf{f}(boldsymbol{x}) & boldsymbol{f}left(frac{mathbf{1}}{boldsymbol{x}}right)+boldsymbol{f}(boldsymbol{x}) \ mathbf{1} & boldsymbol{f}left(frac{mathbf{1}}{boldsymbol{x}}right)end{array}right|=mathbf{0} )
where;
( boldsymbol{f}(boldsymbol{x})=boldsymbol{a}+boldsymbol{b} boldsymbol{x}^{n} ) and ( boldsymbol{f}(boldsymbol{2})=mathbf{1 7}, ) then
( boldsymbol{f}(mathbf{5}) ) is:
A . 126
в. 326
( c .428 )
D. 626
12
642 ( fleft|begin{array}{ccc}boldsymbol{x}+mathbf{1} & mathbf{3} & mathbf{5} \ mathbf{2} & boldsymbol{x}+mathbf{2} & mathbf{5} \ mathbf{2} & mathbf{3} & boldsymbol{x}+mathbf{4}end{array}right|=mathbf{0}, ) then ( boldsymbol{x}=? )
A . 1,9
В. -1,9
c. -1,-9
D. 1,-9
12
643 If ( A=left[begin{array}{ll}1 & 1 \ 0 & 1end{array}right], ) then ( operatorname{det}left(A+A^{2}+right. )
( left.boldsymbol{A}^{3}+boldsymbol{A}^{4}+boldsymbol{A}^{5}right) ) is
( mathbf{A} cdot mathbf{1} )
B. 32
c. 25
D. 30
12
644 If the vectors ( vec{a}, vec{b}, vec{c} ) are coplanar, then the value of ( left|begin{array}{ccc}overrightarrow{boldsymbol{a}} & overrightarrow{boldsymbol{b}} & overrightarrow{boldsymbol{c}} \ overrightarrow{boldsymbol{a}} cdot overrightarrow{boldsymbol{a}} & overrightarrow{boldsymbol{a}} cdot overrightarrow{boldsymbol{b}} & overrightarrow{boldsymbol{a}} cdot overrightarrow{boldsymbol{c}} \ overrightarrow{boldsymbol{b}} cdot overrightarrow{boldsymbol{a}} & overrightarrow{boldsymbol{b}} cdot overrightarrow{boldsymbol{b}} & overrightarrow{boldsymbol{b}} cdot overrightarrow{boldsymbol{c}}end{array}right|= )
( mathbf{A} cdot mathbf{1} )
B.
( c cdot-1 )
D. ( vec{a}+vec{b}+vec{c} )
12
645 What is the value of ( y ) if ( (y, 3),(-5,6) )
and (-8,8) are collinear?
A . -1
B . 2
c. ( frac{1}{2} )
D. ( -frac{1}{2} )
12
646 ( fleft(a_{1}, a_{2}, a_{3}, dots, a_{n}, dots a r e text { in } G P ) then the right.
determinant ( Delta= )
( left|begin{array}{ccc}log a_{n} & log a_{n+1} & log a_{n+2} \ log a_{n+3} & log a_{n+4} & log a_{n+5} \ log a_{n+6} & log a_{n+7} & log a_{n+8}end{array}right| ) is equal
to
A . 0
B. 1
( c cdot 2 )
D.
12
647 If ( A ) is an invertible matrix of order ( n )
then the determinant of adj ( A ) is equal
to:
( mathbf{A} cdot|A|^{n} )
В ( cdot|A|^{n+1} )
c. ( |A|^{n-1} )
D. ( |A|^{n+2} )
12
648 The number of distinct real roots of the
[
text { guation, }left|begin{array}{ccc}
cos x & sin x & sin x \
sin x & cos x & sin x \
sin x & sin x & cos x
end{array}right|=0
]
in the interval ( left[-frac{pi}{4}, frac{pi}{4}right] ) is/are
( A )
B . 2
( c cdot 1 )
( D )
12
649 Suppose ( f(x) ) is a function satisfying the following conditions.
( (a) f(0)=2, f(1)=1 )
(b) ( f(x) ) has a maximum at ( x=5 / 2, ) and
(c) for all ( x in R )
( boldsymbol{f}^{prime}(boldsymbol{x})= )
( mid begin{array}{ccc}2 a x & 2 a x-1 & 2 a x+b+ \ b & b+1 & -1 \ 2(a x+b) & 2 a x+2 b+1 & 2 a x+bend{array} )
where ( a, b ) are constants, then
12
650 The value of the determinant
( left|begin{array}{ccc}b^{2}-a b & b-c & b c-a c \ a b-a^{2} & a-b & b^{2}-a b \ b c-a c & c-a & a b-a^{2}end{array}right| )
A ( . a b c )
B. ( a+b+c )
( c )
( mathbf{D} cdot a b+b c+c a )
12
651 ( operatorname{Det}left{begin{array}{ccc}1^{2} & 2^{2} & 3^{2} \ 2^{2} & 3^{2} & 4^{2} \ 3^{2} & 4^{2} & 5^{2}end{array}right}=dots )
A . -8
B. -7
( c .-6 )
D. ( -21 / 4 )
12
652 24.
The number of values of k for which the linear equations
4x + ky + 2z=0, kx + 4y+z=0 and 2x +2y+z= 0 possess
a non-zero solution is
[2011]
(a) 2 (b) 1 (C) zero (d) 3
12
653 Find the value of ( x ) if ( left|begin{array}{ccc}3 & 5 & x \ 2 & 4 & 1 \ -1 & 2 & 3end{array}right|=0 ) 12
654 ( left|begin{array}{ccc}frac{1}{a} & a^{2} & b c \ frac{1}{b} & b^{2} & c a \ frac{1}{c} & c^{2} & a bend{array}right|= )
A ( cdot ) abc
B. ( a+b+c )
( c cdot 0 )
D. ( 4 a b c )
12
655 When the determinant ( left|begin{array}{lll}cos 2 x & sin ^{2} x & cos 4 x \ sin ^{2} x & cos 2 x & cos ^{2} x \ cos 4 x & cos ^{2} x & cos 2 xend{array}right| ) is expanded in
powers of ( sin x, ) then the constant term in that expression is
( A )
B.
( c cdot-1 )
D.
12
656 ( f e^{i theta}=cos theta+i sin theta, ) find the value of
( -left|begin{array}{ccc}mathbf{1} & boldsymbol{e}^{i boldsymbol{pi} / mathbf{3}} & boldsymbol{e}^{i boldsymbol{pi} / mathbf{4}} \ boldsymbol{e}^{-i boldsymbol{pi} / mathbf{3}} & mathbf{1} & boldsymbol{e}^{boldsymbol{i} mathbf{2} boldsymbol{pi} / mathbf{3}} \ boldsymbol{e}^{-boldsymbol{i} boldsymbol{pi} / mathbf{4}} & boldsymbol{e}^{-boldsymbol{i} mathbf{2} boldsymbol{pi} / mathbf{3}} & mathbf{1}end{array}right|-mathbf{2}^{frac{1}{2}} )
12
657 ( f(a+b+c=0, ) then one root of ( left|begin{array}{ccc}boldsymbol{a}-boldsymbol{x} & boldsymbol{c} & boldsymbol{b} \ boldsymbol{c} & boldsymbol{b}-boldsymbol{x} & boldsymbol{a} \ boldsymbol{b} & boldsymbol{a} & boldsymbol{c}-boldsymbol{x}end{array}right|=0 ) is
( A cdot a+b )
B.
( c cdot b+c )
D. atca
12
658 If ( D_{P}=left|begin{array}{ccc}boldsymbol{P} & mathbf{1 5} & mathbf{8} \ boldsymbol{P}^{2} & mathbf{3 5} & mathbf{9} \ boldsymbol{P}^{3} & mathbf{2 5} & mathbf{1 0}end{array}right|, ) then ( boldsymbol{D}_{1}+ )
( D_{2}+D_{3}+D_{4}+D_{5} ) is equal to
A. -29000
в. -25000
c. 25000
D. none of these
12
659 19. Let A be a 2 x 2 matrix
Statement-1: adj (adj A)=A
[2009]
Statement-2: (adj A F|A|
(a) Statement-1 is true, Statement-2 is true.
Statement-2 is not a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is false.
(C) Statement -1 is false, Statement-2 is true.
(d) Statement-1 is true, Statement -2 is true.
Statement-2 is a correct explanation for Statement-1.
12
660 Find the value of the following determinants. ( left|begin{array}{cc}mathbf{5} & mathbf{3} \ -mathbf{7} & mathbf{0}end{array}right| ) 12
661 The value of ( triangle= )
( begin{array}{cc}mathbf{2} & boldsymbol{a}+boldsymbol{r}+mathbf{2} \ boldsymbol{a}+boldsymbol{r}+mathbf{2} & mathbf{2}(boldsymbol{a}+mathbf{1})(boldsymbol{r}+mathbf{1}) \ boldsymbol{a}+boldsymbol{r} & boldsymbol{a}(boldsymbol{r}+mathbf{1})+boldsymbol{r}(boldsymbol{a}+mathbf{1})end{array} )
( A )
в. ( -2 a(r+1) )
c. ( a(a r+r+a) )
D. -1
12
662 Find equation of line joining (1,2) and
(3,6) using determinants.
12
663 ( mathbf{f} boldsymbol{A}=left|begin{array}{ccc}mathbf{1} & mathbf{- 1} & mathbf{1} \ mathbf{0} & mathbf{2} & -mathbf{3} \ mathbf{2} & mathbf{1} & mathbf{0}end{array}right| ) and ( boldsymbol{B}=(boldsymbol{a} boldsymbol{d} boldsymbol{j} boldsymbol{A}) )
and ( C=5 A, ) then ( frac{|a d j B|}{|C|} ) is?
( mathbf{A} cdot mathbf{5} )
B . 25
c. -1
D.
12
664 , then prove that
Illustration 3.84 If x+y+z=
sin x sin y sin z
cosx cos y cos z = 0.
[cos x cos y cos z
12
665 4.
[1 4 4
If the adjoint of a 3 x 3 matrix P is
2 1 7
, then the
[
113]
(2012)
possible value(s) of the determinant of Pis (are)
(a) 2 (6) -1 (c) 1 (d) 2
12
666 Find the value of the determinant
( left|begin{array}{cc}mathbf{2} boldsymbol{i} & -mathbf{3} boldsymbol{i} \ boldsymbol{i}^{3} & -mathbf{2} boldsymbol{i}end{array}right| ) where ( boldsymbol{i}=sqrt{-boldsymbol{i}} )
12
667 Evaluate ( mid begin{array}{ccc}cos left(x+x^{2}right) & sin left(x+x^{2}right) & -cos (x+1) \ sin left(x-x^{2}right) & cos left(x-x^{2}right) & sin (x-x) \ sin 2 x & 0 & sin 2 x^{2}end{array} )
A ( cdot sin left(2 x+2 x^{2}right) )
B. ( -sin left(2 x+2 x^{2}right) )
( c cdot cos left(2 x+2 x^{2}right) )
D. ( -cos left(2 x+2 x^{2}right) )
12
668 If the system of linear equations
( 2 x+2 a y+a z=0 )
( 2 x+3 b y+b z=0 )
( 2 x+4 c y+c z=0 )
where ( a, b, c epsilon R ) are non-zero and
distinct; has a non-zero solution, then:
A ( cdot frac{1}{a}, frac{1}{b}, frac{1}{c} ) are in A.P
B . ( a+b+c=0 )
c. ( a, b, c ) are in A.P
( mathbf{D} cdot a, b, c ) are in G.P
12
669 If ( A ) is an ( n times n ) non-singular matrix,
then ( |boldsymbol{A} boldsymbol{d} boldsymbol{j} boldsymbol{A}| ) is:
( mathbf{A} cdot|A|^{n} )
B ( cdot|A|^{n+1} )
c. ( |A|^{n-1} )
D. ( |A|^{n-2} )
12
670 ( mathbf{f} mathbf{a}^{2}+boldsymbol{b}^{2}+boldsymbol{c}^{2}=-mathbf{2} ) and ( boldsymbol{f}(boldsymbol{x})= )
( left|begin{array}{ccc}mathbf{1}+boldsymbol{a}^{2} boldsymbol{x} & left(mathbf{1}+boldsymbol{b}^{2}right) boldsymbol{x} & left(mathbf{1}+boldsymbol{c}^{2}right) boldsymbol{x} \ left(mathbf{1}+boldsymbol{a}^{2}right) boldsymbol{x} & mathbf{1}+boldsymbol{b}^{2} boldsymbol{x} & left(mathbf{1}+boldsymbol{c}^{2}right) boldsymbol{x} \ left(mathbf{1}+boldsymbol{a}^{2}right) boldsymbol{x} & left(mathbf{1}+boldsymbol{b}^{2}right) boldsymbol{x} & mathbf{1}+boldsymbol{c}^{2} boldsymbol{x}end{array}right| )
Then, ( f(x) ) is a polynomial of degree
A .2
B. 3
( c .4 )
( D )
12
671 If 3 points ( A(1, a, b), B(a, 2, b), C(a, b, 3) ) are
collinear, then find ( a+b=? )
12
672 Consider the following statements:
1. Determinant is a square matrix.
2. Determinant is a number associated
with a square matrix. Which of the above statements is/are
correct?
A . 1 only
B. 2 only
c. Both 1 and 2
D. Neither 1 nor 2
12
673 Solve:
( left|begin{array}{ccc}mathbf{0} & -mathbf{3} & boldsymbol{x} \ boldsymbol{x}+mathbf{1} & mathbf{3} & mathbf{1} \ mathbf{4} & mathbf{1} & mathbf{5}end{array}right|=mathbf{0} )
12
674 If the determinant ( left|begin{array}{lll}boldsymbol{a} & boldsymbol{b} & boldsymbol{a} boldsymbol{t}-boldsymbol{b} \ boldsymbol{b} & boldsymbol{c} & boldsymbol{b} boldsymbol{t}-boldsymbol{c} \ boldsymbol{2} & boldsymbol{1} & boldsymbol{0}end{array}right|=0 )
if ( a, b, c ) are in
A. ( A . P )
в. G.Р.
c. ( H . P )
D. ( k=1 / 2 )
12
675 ( operatorname{Show}left|begin{array}{ccc}boldsymbol{a x} & boldsymbol{b y} & boldsymbol{c z} \ boldsymbol{x}^{2} & boldsymbol{y}^{2} & boldsymbol{z}^{2} \ mathbf{1} & mathbf{1} & mathbf{1}end{array}right|=left|begin{array}{ccc}boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{x} & boldsymbol{y} & boldsymbol{z} \ boldsymbol{y} boldsymbol{z} & boldsymbol{z} boldsymbol{x} & boldsymbol{x} boldsymbol{y}end{array}right| ) 12
676 ( left(begin{array}{lll}mathbf{7} & mathbf{1} & mathbf{5} \ mathbf{8} & mathbf{0} & mathbf{0}end{array}right)left(begin{array}{l}mathbf{2} \ mathbf{3} \ mathbf{1}end{array}right)+mathbf{5}left(begin{array}{l}mathbf{1} \ mathbf{0}end{array}right) ) is equal to
A ( cdotleft(begin{array}{l}16 \ 27end{array}right. )
B. ( left(begin{array}{l}27 \ 16end{array}right. )
c. ( left(begin{array}{l}15 \ 16end{array}right. )
D. ( left(begin{array}{l}16 \ 15end{array}right. )
12
677 Calculate the values of the
determinants:
( left|begin{array}{ccc}boldsymbol{b}+boldsymbol{c} & boldsymbol{a} & boldsymbol{a} \ boldsymbol{b} & boldsymbol{c}+boldsymbol{a} & boldsymbol{b} \ boldsymbol{c} & boldsymbol{c} & boldsymbol{a}+boldsymbol{b}end{array}right| )
12
678 Evaluate the determinant :
( left|begin{array}{ccc}mathbf{1}+boldsymbol{a} & mathbf{0} & mathbf{0} \ mathbf{0} & mathbf{1}+boldsymbol{a} & mathbf{0} \ mathbf{0} & mathbf{0} & mathbf{1}+boldsymbol{a}end{array}right| )
12
679 ( left|begin{array}{ccc}(boldsymbol{b}+boldsymbol{c})^{2} & boldsymbol{a}^{2} & boldsymbol{a}^{2} \ boldsymbol{b}^{2} & (boldsymbol{c}+boldsymbol{a})^{2} & boldsymbol{b}^{2} \ boldsymbol{c}^{2} & boldsymbol{c}^{2} & (boldsymbol{a}+boldsymbol{b})^{2}end{array}right| ) is equal
to
( mathbf{A} cdot a b c(a+b+c)^{2} )
B ( cdot 2 a b c(a+b+c)^{2} )
( mathbf{c} cdot 2 a b c(a+b+c)^{3} )
( mathbf{D} cdot 2 a b c(a+b+c) )
12
680 Prove the following identities :
[
begin{array}{ccc}
boldsymbol{a}+boldsymbol{b}+mathbf{2 c} & boldsymbol{a} & boldsymbol{b} \
boldsymbol{c} & boldsymbol{b}+boldsymbol{c}+mathbf{2} boldsymbol{a} & boldsymbol{b} \
boldsymbol{c} & boldsymbol{a} & boldsymbol{c}+boldsymbol{a}+boldsymbol{2} boldsymbol{b}
end{array} mid=
]
( mathbf{2}(boldsymbol{a}+boldsymbol{b}+boldsymbol{c})^{3} )
12
681 Prove that: ( left|begin{array}{ccc}boldsymbol{b}+boldsymbol{c} & boldsymbol{a} & boldsymbol{b} \ boldsymbol{c}+boldsymbol{a} & boldsymbol{c} & boldsymbol{a} \ boldsymbol{a}+boldsymbol{b} & boldsymbol{b} & boldsymbol{c}end{array}right|=(boldsymbol{a}+boldsymbol{b}+ )
( c)(a-c)^{2} )
12
682 Suppose ( a, b, c epsilon R ) and ( a b c=1 . ) If ( A= ) ( left[begin{array}{ccc}2 a & b & c \ b & 2 c & a \ c & a & 2 bend{array}right] ) is such that ( |A|left|A^{prime}right|=64 )
and ( |A|>0, ) then find the value of
( left(a^{3}+b^{3}+c^{3}right)^{4} )
( left(A^{prime} text { denotes transpose of a matrix } A .right) )
A . 1
B. 2
( c .3 )
( D )
12
683 Find the integral value of ( x, ) if ( left|begin{array}{ccc}boldsymbol{x}^{2} & boldsymbol{x} & mathbf{1} \ mathbf{0} & boldsymbol{2} & mathbf{1} \ boldsymbol{3} & boldsymbol{1} & boldsymbol{4}end{array}right|=mathbf{2} mathbf{8} ) 12
684 ( mathbf{f}left|begin{array}{lll}boldsymbol{x}^{boldsymbol{k}} & boldsymbol{x}^{boldsymbol{k}+boldsymbol{2}} & boldsymbol{x}^{boldsymbol{k}+boldsymbol{3}} \ boldsymbol{y}^{boldsymbol{k}} & boldsymbol{y}^{boldsymbol{k}+boldsymbol{2}} & boldsymbol{y}^{boldsymbol{k}+boldsymbol{3}} \ boldsymbol{z}^{boldsymbol{k}} & boldsymbol{z}^{boldsymbol{k}+boldsymbol{2}} & boldsymbol{z}^{boldsymbol{k}+boldsymbol{3}}end{array}right| )
三室 ( (x-y)(y-z)(z-x)left(frac{1}{x}+frac{1}{y}+frac{1}{z}right) )
then
A. ( k=-3 )
B. ( k=-1 )
( mathbf{c} cdot k=1 )
D. ( k=3 )
12
685 For positive numbers ( boldsymbol{x}, boldsymbol{y}, boldsymbol{z} ) the
numerical value of the determinant ( left|begin{array}{ccc}mathbf{1} & log _{x} boldsymbol{y} & log _{boldsymbol{x}} boldsymbol{z} \ log _{boldsymbol{y}} boldsymbol{x} & boldsymbol{3} & log _{boldsymbol{y}} boldsymbol{z} \ log _{boldsymbol{z}} boldsymbol{x} & log _{boldsymbol{z}} boldsymbol{y} & boldsymbol{5}end{array}right| ) is
( mathbf{A} cdot mathbf{0} )
B. ( log x log y log z )
( c .1 )
D. 8
12
686 If ( I ) is the unit matrix of order ( n, ) where
( k neq 0 ) is a constant then ( operatorname{adj}(k I)= )
A ( cdot k^{n}(operatorname{adj} I) )
B. ( k(operatorname{adj} I) )
c. ( k^{2}(operatorname{adj} I) )
D. ( k^{n-1}(text { adj } I) )
12
687 Find the value of: ( left|begin{array}{ccc}mathbf{1} & mathbf{1} & mathbf{1} \ boldsymbol{x}^{2} & boldsymbol{y}^{2} & boldsymbol{z}^{2} \ boldsymbol{x}^{3} & boldsymbol{y}^{3} & boldsymbol{z}^{3}end{array}right| )
( mathbf{A} cdot mathbf{0} )
B. ( (x-y)(y-z)(z-x)(x y+y z+z x) )
C. ( (x+y x+z x)(x+y)(y+z)(z+x) )
D. ( (x+y+z) )
12
688 If the area of the triangle formed by ( (0,0),(a, 0) ) and ( left(frac{1}{2}, aright) ) is equal to ( frac{1}{2} s q ) unit, then the values of ( a ) are :
( A ldots pm 2 )
B. ±3
( c .pm 1 )
D. ±4
( mathrm{E} cdot pm 5 )
12
689 If ( boldsymbol{A}=left[begin{array}{rrr}3 & -3 & 4 \ 2 & -3 & 4 \ 0 & -1 & 1end{array}right] ) then find
( boldsymbol{A} boldsymbol{d} boldsymbol{j}(boldsymbol{A} boldsymbol{d} boldsymbol{j} boldsymbol{A}) )
A. ( left[begin{array}{lll}3 & -3 & 4 \ 2 & -3 & 4 \ 0 & -1 & 1end{array}right] )
В. ( left[begin{array}{ccc}3 & 3 & 4 \ 2 & -3 & -4 \ 0 & -1 & 1end{array}right] )
( begin{array}{llll}text { C. } & {left[begin{array}{ccc}3 & 3 & 4 \ 2 & -3 & 4 \ 0 & 1 & 1end{array}right]}end{array} )
D. ( left[begin{array}{ccc}3 & -3 & 4 \ 2 & -3 & -4 \ 0 & 1 & 1end{array}right] )
12
690 A set of points which do not lie on the same line are called as
A. collinear
B. non-collinear
c. concurrent
D. square
12
691 If ( boldsymbol{A}(boldsymbol{x})=left|begin{array}{ccc}boldsymbol{x}+mathbf{1} & mathbf{2} boldsymbol{x}+mathbf{1} & mathbf{3} boldsymbol{x}+mathbf{1} \ mathbf{2} boldsymbol{x}+mathbf{1} & mathbf{3} boldsymbol{x}+mathbf{1} & boldsymbol{x}+mathbf{1} \ boldsymbol{3} boldsymbol{x}+mathbf{1} & boldsymbol{x}+mathbf{1} & boldsymbol{2} boldsymbol{x}+mathbf{1}end{array}right| )
( operatorname{then} int_{0}^{1} boldsymbol{A}(boldsymbol{x}) boldsymbol{d} boldsymbol{x}= )
A . -15
( B cdot frac{-1}{2} )
( c .-30 )
( D )
12
692 ( fleft|begin{array}{ll}boldsymbol{x} & boldsymbol{y} \ boldsymbol{4} & boldsymbol{2}end{array}right|=mathbf{7} ) and ( left|begin{array}{ll}boldsymbol{2} & boldsymbol{3} \ boldsymbol{y} & boldsymbol{x}end{array}right|=boldsymbol{4} ) then
A ( cdot x=-3, y=-frac{5}{2} )
B. ( x=-frac{5}{2}, y=-3 )
c. ( _{x}=-3, y=frac{5}{2} )
D. ( x=-frac{5}{2}, y=3 )
12
693 5.
The parameter, on which the value of the determinant
1
a
cos(p-d)x cos px
sin(p-d)x sin px
cos(p+d)x
sin(p+.d)x
does not depend
upon is
(a) a
(b) p
(c) d
(1997 – 2 Marks)
(d) x
12
694 Evaluate the determinant :
( left|begin{array}{cc}cos theta & -sin theta \ sin theta & cos thetaend{array}right| )
12
695 Prove the following:
[
left|begin{array}{ccc}
boldsymbol{a}+boldsymbol{b}+mathbf{2} boldsymbol{c} & boldsymbol{a} & boldsymbol{b} \
boldsymbol{c} & boldsymbol{b}+boldsymbol{c}+mathbf{2} boldsymbol{a} & boldsymbol{b} \
boldsymbol{c} & boldsymbol{a} & boldsymbol{c}+boldsymbol{a}+mathbf{2} boldsymbol{b}
end{array}right|=
]
( mathbf{2}(boldsymbol{a}+boldsymbol{b}+boldsymbol{c})^{3} )
12
696 f ( a, b, c ) are real numbers such that ( left|begin{array}{ccc}boldsymbol{b}+boldsymbol{c} & boldsymbol{c}+boldsymbol{a} & boldsymbol{a}+boldsymbol{b} \ boldsymbol{c}+boldsymbol{a} & boldsymbol{a}+boldsymbol{b} & boldsymbol{b}+boldsymbol{c} \ boldsymbol{a}+boldsymbol{b} & boldsymbol{b}+boldsymbol{c} & boldsymbol{c}+boldsymbol{a}end{array}right|=0, ) then show
that either ( boldsymbol{a}+boldsymbol{b}+boldsymbol{c}=boldsymbol{0} ) or, ( boldsymbol{a}=boldsymbol{b}=boldsymbol{c} )
12
697 Let ( omega=-frac{1}{2}+i frac{sqrt{3}}{2}, ) then the value of the
determinant
( left|begin{array}{ccc}1 & 1 & 1 \ 1 & -1-omega^{2} & omega^{2} \ 1 & omega^{2} & omega^{4}end{array}right|, ) is
This question has multiple correct options
( A .3 omega )
В. ( 3 omega(omega-1) )
( c cdot 3 omega^{2} )
D. ( 3(-2 omega-1) )
12
698 Adj ( left(A d jleft[begin{array}{cc}2 & -3 \ 4 & 6end{array}right]right)= )
A ( cdotleft[begin{array}{cc}2 & -3 \ 4 & 6end{array}right] )
в. ( left[begin{array}{cc}6 & 3 \ -4 & 2end{array}right] )
с. ( left[begin{array}{cc}-6 & 3 \ -4 & -2end{array}right] )
D ( cdotleft[begin{array}{cc}-6 & -3 \ 4 & -2end{array}right] )
12
699 If ( boldsymbol{A}=left|begin{array}{ll}mathbf{0} & mathbf{0} \ mathbf{1} & mathbf{1}end{array}right| ) then the value of ( boldsymbol{A}+ )
( boldsymbol{A}^{2}+boldsymbol{A}^{3}+ldots+boldsymbol{A}^{n}=? )
( A cdot A )
B. nA
c. ( (n+1) A )
( D )
12
700 Prove the following:
( left|begin{array}{llll}boldsymbol{x} & boldsymbol{a} & boldsymbol{a} & boldsymbol{a} \ boldsymbol{a} & boldsymbol{x} & boldsymbol{a} & boldsymbol{a} \ boldsymbol{a} & boldsymbol{a} & boldsymbol{x} & boldsymbol{a} \ boldsymbol{a} & boldsymbol{a} & boldsymbol{a} & boldsymbol{x}end{array}right|=(boldsymbol{x}+boldsymbol{3} boldsymbol{a})(boldsymbol{x}-boldsymbol{a})^{3} )
12
701 2. The values of a lying between 0 = 0 and 0 = 7/2 and
satisfying the equation
1+sine cose 4sin 40
sin 1+cos e 4 sin 40 = 0 are
sin’e cosạe 1+4 sin 40
a. 77/24
& b. 57/24 08
c. 117/24
d. Tt/24 (IIT-JEE 1988)
12
702 ( left|begin{array}{ccc}cos C & tan A & 0 \ sin B & 0 & -tan A \ 0 & sin B & cos Cend{array}right| ) has the value
( mathbf{A} cdot mathbf{0} )
B.
( c cdot sin A sin B cos B )
D. none of these
12
703 ( mathbf{f} boldsymbol{x}, boldsymbol{y}, boldsymbol{z} in boldsymbol{R} & boldsymbol{Delta}= )
( left|begin{array}{ccc}boldsymbol{x} & boldsymbol{x}+boldsymbol{y} & boldsymbol{x}+boldsymbol{y}+boldsymbol{z} \ mathbf{2} boldsymbol{x} & mathbf{5} boldsymbol{x}+mathbf{2} boldsymbol{y} & mathbf{7} boldsymbol{x}+mathbf{5} boldsymbol{y}+mathbf{2} z \ mathbf{3} boldsymbol{x} & mathbf{7} boldsymbol{x}+mathbf{3} boldsymbol{y} & mathbf{9} boldsymbol{x}+mathbf{7} boldsymbol{y}+mathbf{3} boldsymbol{z}end{array}right|=-mathbf{1} mathbf{6} )
then value of ( x ) is
A . -2
B. -3
( c cdot 2 )
( D )
12
704 If ( A ) is a ( 3 * 3 ) singular matrix then
( boldsymbol{A}(boldsymbol{A} boldsymbol{d} boldsymbol{j} boldsymbol{A})= )
( A cdot ) Det ( A )
B.
( c cdot c )
D. ±1
12
705 If ( A, B, C ) are the angles of a triangle,
then the value of determinant
( left|begin{array}{ccc}-1+cos B & cos C+cos B & cos B \ cos C+cos A & -1+cos A & cos A \ -1+cos B & -1+cos A & -1end{array}right| )
( mathbf{A} cdot mathbf{0} )
B.
( c .-1 )
( D )
12
706 20.
Tf Pisa 3 x 3 matrix such that pl = 2P+1, where PT is the
transpose of P and I is the 3 x 3 identity matrix, then there
x] rol
exists a column matrix X = y 20 such that
z] [0]
(2012)
(b) PX=X
(a) PX=0
0
(c) PX= 2X
(d) PX=-X
12
707 Minor ( m_{33} ) of the determinant
( left|begin{array}{ccc}2 & 3 & 5 \ 2 & -1 & 8 \ 1 & 2 & 4end{array}right| ) is
12
708 2.
l, m, n are the pth, qth and th term of a G. P. all positive,
log 1 p 1
then log m q 1 equals
[2002]
log nr 1
(a) -1 (b) 2 (C) 1 (d) 0
12
709 ( mathbf{A}=left[begin{array}{ccc}-boldsymbol{q} boldsymbol{r} & boldsymbol{p}(boldsymbol{q}+boldsymbol{r}) & boldsymbol{p} boldsymbol{r}+boldsymbol{p} boldsymbol{q} \ boldsymbol{p} boldsymbol{q}+boldsymbol{q} boldsymbol{r} & -boldsymbol{p} boldsymbol{r} & boldsymbol{p} boldsymbol{q}+boldsymbol{q} boldsymbol{r} \ boldsymbol{q} boldsymbol{r}+boldsymbol{p} boldsymbol{r} & boldsymbol{q} boldsymbol{r}+boldsymbol{p} boldsymbol{r} & boldsymbol{-} boldsymbol{p} boldsymbol{q}end{array}right] )
then ( |boldsymbol{A}| ) equals:
( ^{A} cdotleft(sum p qright)^{2} )
( ^{mathrm{B}}left(sum p^{2} q^{2}right)^{2} )
( c cdotleft[sum(q r)^{3}right. )
( ^{mathrm{D}} cdotleft(sum p qright)^{frac{3}{2}} )
12
710 Let ( a, b, c ) be real numbers with ( a^{2}+ )
( b^{2}+c^{2}=1 )
show that the equation ( mid begin{array}{ccc}boldsymbol{a} boldsymbol{x}-boldsymbol{b} boldsymbol{y}-boldsymbol{c} & boldsymbol{b} boldsymbol{x}+boldsymbol{a} boldsymbol{y} & boldsymbol{c} boldsymbol{x}+ \ boldsymbol{b} boldsymbol{x}+boldsymbol{a} boldsymbol{y} & boldsymbol{-} boldsymbol{a} boldsymbol{x}+boldsymbol{b} boldsymbol{y}-boldsymbol{c} & boldsymbol{c} boldsymbol{y}+ \ boldsymbol{c} boldsymbol{x}+boldsymbol{a} & boldsymbol{c} boldsymbol{y}+boldsymbol{b} & boldsymbol{-} boldsymbol{a} boldsymbol{x}-boldsymbol{b}end{array} )
represents a straight line.
12
711 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{3} & mathbf{2} \ mathbf{1} & mathbf{4}end{array}right], ) then ( boldsymbol{A}(boldsymbol{A} boldsymbol{d} boldsymbol{j} cdot boldsymbol{A}) ) equals-
A. ( left[begin{array}{cc}10 & 0 \ 0 & 10end{array}right] )
В. ( left[begin{array}{cc}0 & 10 \ 10 & 0end{array}right] )
c. ( left[begin{array}{ll}10 & 1 \ 1 & 10end{array}right] )
D. none of these
12
712 ( left|begin{array}{ccc}boldsymbol{x}+boldsymbol{alpha} & boldsymbol{beta} & gamma \ boldsymbol{alpha} & boldsymbol{x}+boldsymbol{beta} & gamma \ boldsymbol{alpha} & boldsymbol{beta} & boldsymbol{x}+boldsymbol{gamma}end{array}right|=mathbf{0}, ) then ( boldsymbol{x} ) is
equal to
( mathbf{A} cdot 0,-(alpha+beta+gamma) )
B. ( 0, alpha+beta+gamma )
c. ( 1, alpha+beta+gamma )
D. ( 0, alpha^{2}+beta^{2}+gamma^{2} )
12
713 Find ( left|begin{array}{lll}mathbf{4} & mathbf{6} & mathbf{2} \ mathbf{3} & mathbf{0} & mathbf{9} \ mathbf{7} & mathbf{1} & mathbf{5}end{array}right| ) 12
714 ( mathbf{f} mathbf{A}=left[begin{array}{lll}mathbf{1} & mathbf{5} & mathbf{- 6} \ mathbf{- 8} & mathbf{0} & mathbf{4} \ mathbf{3} & mathbf{- 7} & mathbf{2}end{array}right], ) then the
cofactor of ( -7= )
A . 44
B. 43
( c cdot 40 )
D. 39
12
715 Let
( f(x)=left|begin{array}{ccc}sec ^{2} x & 1 & 1 \ cos ^{2} x & cos ^{2} x & operatorname{cosec}^{2} x \ 1 & cos ^{2} x & cot ^{2} xend{array}right| )
then
12
716 Solve the following determinant :
( begin{array}{|ccc|}15 & 11 & 7 \ 11 & 17 & 14 \ 10 & 16 & 13end{array} )
12
717 (2000 – TV US)
25. Let a, b, c be real numbers with a2 + b2 + c2= 1. Show that
the equation
Jax – by-c
bx + ay
cxta
b6+ay
ax + by-c
cytb
axta
cy+b 1=0
– ax – bytc.
represents a straight line
(20016 Marts)
12
718 A matrix A of order ( 3 times 3 ) has
determinant 6. What is the value of ( |mathbf{3} boldsymbol{A}| )
( ? )
12
719 If ( a neq b neq c, ) the value of ( x ) which
satisfies the question ( left|begin{array}{ccc}mathbf{0} & boldsymbol{x}-boldsymbol{a} & boldsymbol{x}-boldsymbol{b} \ boldsymbol{x}+boldsymbol{a} & mathbf{0} & boldsymbol{x}-boldsymbol{c} \ boldsymbol{x}+boldsymbol{b} & boldsymbol{x}+boldsymbol{c} & mathbf{0}end{array}right|=mathbf{0} ) is
( mathbf{A} cdot x=0 )
B. ( x=a )
( mathbf{c} cdot x=b )
D. ( x=c )
12
720 Find the value of the following determinant:
( left|begin{array}{cc}-4 & frac{-6}{35} \ 7 & frac{35}{5} \ 5 & 5end{array}right| )
A ( cdot frac{15}{34} )
в. ( frac{32}{45} )
c. ( frac{25}{33} )
D. ( frac{38}{35} )
12
721 Eliminating ( a, b, c ) from ( x=frac{a}{b-c}, y= )
( frac{b}{c-a}, z=frac{c}{a-b} ) we get
( mathbf{A} cdotleft|begin{array}{lll}1 & -x & x \ 1 & -y & y \ 1 & -z & zend{array}right|=0 )
B. ( left|begin{array}{ccc}1 & -x & x \ 1 & 1 & -y \ 1 & z & 1end{array}right|=0 )
c. ( left|begin{array}{ccc}1 & -x & x \ y & 1 & -y \ -z & z & 1end{array}right|=0 )
D. none of these
12
722 Using the factor theorem it is found
that ( b+c, c+a ) and ( a+b ) are three
factors of the determinant ( left|begin{array}{ccc}-2 a & a+b & a+c \ b+a & -2 b & b+c \ c+a & c+b & -2 cend{array}right| . ) The other factor
in the value of the determinant is
A . 4
B. 2
( mathbf{c} cdot a+b+c )
D. none of these
12
723 ( left|begin{array}{lll}mathbf{1} & boldsymbol{omega} & boldsymbol{omega}^{2} \ boldsymbol{omega} & boldsymbol{omega}^{2} & mathbf{1} \ boldsymbol{omega}^{2} & mathbf{1} & boldsymbol{omega}end{array}right|=ldots )
( A )
B.
( c cdot 2 )
D. -1
12
724 Numbers of ways in which 75600 can be
resolved as product of two divisors which are relatively prime?
A .44
B. 8
( c .9 )
D. 16
12
725 If
( a, b, c ) all are non-zero and unequal and ( left|begin{array}{ccc}mathbf{1}+boldsymbol{a} & mathbf{1} & mathbf{1} \ mathbf{1} & mathbf{1}+boldsymbol{b} & mathbf{1} \ mathbf{1} & mathbf{1} & mathbf{1}+boldsymbol{c}end{array}right|=mathbf{0}, ) then
( 1+frac{1}{a}+frac{1}{b}+frac{1}{c} ) is equal to?
12
726 Prove that ( left|begin{array}{ccc}mathbf{1}+boldsymbol{a} & mathbf{1} & mathbf{1} \ mathbf{1} & mathbf{1}+boldsymbol{b} & mathbf{1} \ mathbf{1} & mathbf{1} & mathbf{1}+boldsymbol{c}end{array}right|= )
( boldsymbol{a b c}left(mathbf{1}+frac{mathbf{1}}{boldsymbol{a}}+frac{mathbf{1}}{boldsymbol{b}}+frac{mathbf{1}}{boldsymbol{c}}right) )
12
727 f ( boldsymbol{A}+boldsymbol{B}+boldsymbol{C}=boldsymbol{pi}, ) then
( left|begin{array}{ccc}sin (boldsymbol{A}+boldsymbol{B}+boldsymbol{C}) & sin boldsymbol{B} & cos boldsymbol{C} \ -sin boldsymbol{B} & boldsymbol{0} & tan boldsymbol{A} \ cos (boldsymbol{A}+boldsymbol{B}) & -tan boldsymbol{A} & 0end{array}right| )
equal to
( mathbf{A} cdot mathbf{0} )
B. ( 2 sin B tan A cos C )
( c .1 )
D. None of these
12
728 Evaluate the following determinant:
[
left|begin{array}{ccc}
102 & 18 & 36 \
1 & 3 & 4 \
17 & 3 & 6
end{array}right|
]
12
729 Evaluate the following:
( left|begin{array}{ccc}boldsymbol{x}+boldsymbol{lambda} & boldsymbol{x} & boldsymbol{x} \ boldsymbol{x} & boldsymbol{x}+boldsymbol{lambda} & boldsymbol{x} \ boldsymbol{x} & boldsymbol{x} & boldsymbol{x}+boldsymbol{lambda}end{array}right| )
12
730 Evaluate the following determinant:
( left|begin{array}{ccc}1 & -3 & 2 \ 4 & -1 & 2 \ 3 & 5 & 2end{array}right| )
12
731 f the matrix ( boldsymbol{A}=left[begin{array}{ccc}mathbf{6} & boldsymbol{x} & mathbf{2} \ mathbf{2} & mathbf{- 1} & mathbf{2} \ mathbf{- 1 0} & mathbf{5} & mathbf{2}end{array}right] ) is
singular matrix. Find the value of ( x )
12
732 How that the points ( boldsymbol{P}(-boldsymbol{2}, boldsymbol{3}, boldsymbol{5}), boldsymbol{Q}(boldsymbol{1}, boldsymbol{2}, boldsymbol{3}) ) and ( boldsymbol{R}(boldsymbol{7}, boldsymbol{0},-boldsymbol{1}) )
are collinear.
12
733 Evaluate ( left[begin{array}{cc}sqrt{mathbf{3}} & sqrt{mathbf{5}} \ -sqrt{mathbf{5}} & mathbf{3} sqrt{mathbf{3}}end{array}right] ) 12
734 ( boldsymbol{I f} mathbf{A}=left[begin{array}{ll}mathbf{1} & mathbf{3} \ mathbf{2} & mathbf{1}end{array}right], ) then the determinant
( mathbf{A}^{2}-2 mathbf{A}: )
A . 5
B . 25
( c .-5 )
D. -25
12
735 Using properties of determinants, prove
the following:
( left|begin{array}{ccc}a^{2} & b c & a c+c^{2} \ a^{2}+a b & b^{2} & a c \ a b & b^{2}+b c & c^{2}end{array}right|=4 a^{2} b^{2} c^{2} )
12
736 rove: ( left|begin{array}{ccc}-cos alpha & sin beta & 0 \ 0 & -sin alpha & cos alpha \ sin alpha & 0 & -sin betaend{array}right|=0 ) 12
737 Evaluate the determinants ( left|begin{array}{cc}2 & 4 \ -5 & -1end{array}right| ) 12
738 Evaluate the following:
( left|begin{array}{ccc}boldsymbol{x} & mathbf{1} & mathbf{1} \ mathbf{1} & boldsymbol{x} & mathbf{1} \ mathbf{1} & mathbf{1} & boldsymbol{x}end{array}right| )
12
739 Using the properties of determinants,
find the value of
( left|begin{array}{ccc}mathbf{0} & boldsymbol{a} & -boldsymbol{b} \ -boldsymbol{a} & boldsymbol{0} & -boldsymbol{c} \ boldsymbol{b} & boldsymbol{c} & boldsymbol{0}end{array}right| )
12
740 If a + p, b #9, C # r and
TP
a
a
b cl
q c = 0. Then find the
b rl
P
value of
(1991 – 4 Marks)
p-aq-br-c
12
741 If ( boldsymbol{A}=left[begin{array}{cc}mathbf{2} & -mathbf{3} \ mathbf{- 4} & mathbf{1}end{array}right], ) then adj ( left(mathbf{3} boldsymbol{A}^{2}+right. )
( 12 A) ) is equal to.
( mathbf{A} cdotleft[begin{array}{cc}72 & -84 \ -63 & 51end{array}right] )
В. ( left[begin{array}{ll}51 & 63 \ 84 & 72end{array}right] )
c. ( left[begin{array}{ll}51 & 84 \ 63 & 72end{array}right] )
D. ( left[begin{array}{cc}72 & -63 \ -84 & 51end{array}right] )
12
742 Maximum value of a second order
determinant whose every element is either 0,1 or 2 only is:
A.
B.
( c cdot 2 )
D. 4
12
743 Find the determinant:
( left|begin{array}{ccc}1 & x & x^{2} \ 1 & y & y^{2} \ 1 & z & z^{2}end{array}right| )
12
744 Write the value of the determinant
[
left[begin{array}{cc}
boldsymbol{p} & boldsymbol{p}+mathbf{1} \
boldsymbol{p}-mathbf{1} & boldsymbol{p}
end{array}right]
]
when ( p=1342 )
12
745 ( mathbf{a}=left|begin{array}{ccc}boldsymbol{a} & mathbf{5}-boldsymbol{i} & mathbf{7}+boldsymbol{i} \ mathbf{5}+boldsymbol{i} & boldsymbol{b} & mathbf{3}+boldsymbol{i} \ mathbf{7}-boldsymbol{i} & boldsymbol{3}-boldsymbol{i} & boldsymbol{c}end{array}right|, ) then ( boldsymbol{Delta} ) is
always
A . real
B. imaginary
( c cdot 0 )
D. None of these
12
746 Evaluate the following determinant:
( left|begin{array}{ccc}1 & 3 & 5 \ 2 & 6 & 10 \ 31 & 11 & 38end{array}right| )
12
747 If ( boldsymbol{A}=left[boldsymbol{a}_{i j}right] ) is a ( mathbf{4} times mathbf{4} ) matrix and ( boldsymbol{c}_{i j} ) is
the co-factor of the element ( boldsymbol{a}_{boldsymbol{i} j} ) in ( |boldsymbol{A}| )
then the expression ( a_{11} c_{11}+a_{12} c_{12}+ )
( boldsymbol{a}_{13} boldsymbol{c}_{13}+boldsymbol{a}_{14} boldsymbol{c}_{14} ) equals
( mathbf{A} cdot mathbf{0} )
B. –
c. 1
D. ( |A| )
12
748 toppr
Q Type your question
List I
The value of the determinant
A. ( quadleft|begin{array}{lll}x+2 & x+3 & x+5 \ x+4 & x+6 & x+9 \ x+8 & x+11 & x+15end{array}right| ) is
If one of the roots of the
equation
B. ( left|begin{array}{ccc}7 & 6 & x^{2}-13 \ 2 & x^{2}-13 & 2 \ x^{2}-13 & 3 & 7end{array}right|=0 )
is ( x+2 ), then the sum of all other
five roots is
The value of
C. ( quadleft|begin{array}{ccc}sqrt{6} & 2 i & 3+sqrt{6} \ sqrt{12} & sqrt{3}+sqrt{8} i & 3 sqrt{2}+sqrt{6} i \ sqrt{18} & sqrt{2}+sqrt{12} i & sqrt{27}+2 iend{array}right| )
If ( f(theta)= )
D. ( quadleft|begin{array}{ccc}cos ^{2} theta & cos theta sin theta & -sin theta \ cos theta sin theta & sin ^{2} theta & cos theta \ sin theta & -cos theta & 0end{array}right| )
then ( f(pi / 3) )
( mathbf{A} cdot A-i v, B-i i i, C-i i, D-i )
B . ( A-i, B-i i i, C-i i, D-i v )
c. ( A-i i, B-i i i, C-i v, D-i )
D. ( A-i i i, B-i v, C-i i, D-i )
12
749 If ( boldsymbol{a} boldsymbol{x}_{1}^{2}+boldsymbol{b} boldsymbol{y}_{1}^{2}+boldsymbol{c} boldsymbol{z}_{1}^{2}=boldsymbol{a} boldsymbol{x}_{2}^{2}+boldsymbol{b} boldsymbol{y}_{2}^{2}+ )
( boldsymbol{c} boldsymbol{z}_{2}^{2}=boldsymbol{a} boldsymbol{x}_{3}^{2}+boldsymbol{b} boldsymbol{y}_{3}^{2}+boldsymbol{c} boldsymbol{z}_{3}^{2}=boldsymbol{d} ) and
( boldsymbol{a} boldsymbol{x}_{2} boldsymbol{x}_{3}+boldsymbol{b} boldsymbol{y}_{2} boldsymbol{y}_{3}+boldsymbol{c} boldsymbol{z}_{2} boldsymbol{z}_{3}=boldsymbol{a} boldsymbol{x}_{3} boldsymbol{x}_{1}+ )
( boldsymbol{b} boldsymbol{y}_{3} boldsymbol{y}_{1}+boldsymbol{c} boldsymbol{z}_{3} boldsymbol{z}_{1}=boldsymbol{a} boldsymbol{x}_{1} boldsymbol{x}_{2}+boldsymbol{b} boldsymbol{y}_{1} boldsymbol{y}_{2}+ )
( boldsymbol{c} boldsymbol{z}_{1} boldsymbol{z}_{2}=boldsymbol{f}, ) then prove that
( left|begin{array}{lll}boldsymbol{x}_{1} & boldsymbol{y}_{1} & boldsymbol{z}_{1} \ boldsymbol{x}_{2} & boldsymbol{y}_{2} & boldsymbol{z}_{2} \ boldsymbol{x}_{3} & boldsymbol{y}_{3} & boldsymbol{z}_{3}end{array}right|=(boldsymbol{d}- )
( f)left[frac{d+2 f}{a b c}right]^{1 / 2}(boldsymbol{a}, boldsymbol{b}, boldsymbol{c} neq mathbf{0}) )
12
750 Prove that ( left|begin{array}{ccc}boldsymbol{a}^{2}+mathbf{1} & boldsymbol{a} boldsymbol{b} & boldsymbol{a} boldsymbol{c} \ boldsymbol{a} boldsymbol{b} & boldsymbol{b}^{2}+mathbf{1} & boldsymbol{b} boldsymbol{c} \ boldsymbol{c} boldsymbol{a} & boldsymbol{c b} & boldsymbol{c}^{2}+1end{array}right|= )
( mathbf{1}+boldsymbol{a}^{2}+boldsymbol{b}^{2}+boldsymbol{c}^{2} )
12
751 Assertion
( begin{array}{ccc}mathbf{Delta}= & & \ & sin pi & cos (boldsymbol{x}+boldsymbol{pi} / mathbf{4}) & tan (boldsymbol{x}- \ sin (boldsymbol{x}-boldsymbol{pi} / mathbf{4}) & -cos (boldsymbol{pi} / mathbf{2}) & log (boldsymbol{x} \ cot (boldsymbol{pi} / mathbf{4}+boldsymbol{x}) & log (boldsymbol{y} / boldsymbol{x}) & tan end{array} )
0
Reason
A skew symmetric determinant of odd
order equals 0
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
752 The system of equations
a x + y +z= a – 1
x + ay+z= a -1
x+y+ az= a – 1
has infinite solutions, if a is
(a) -2
(c) not-2
2
(b) either – 2 or 1
(d) 1
12
753 If ( n ) is a positive integer, then the value
of the determinant ( left|begin{array}{ccc}boldsymbol{a}^{n}-boldsymbol{x} & boldsymbol{a}^{boldsymbol{n}+1}-boldsymbol{x} & boldsymbol{a}^{boldsymbol{n}+mathbf{2}}-boldsymbol{x} \ boldsymbol{a}^{boldsymbol{n}+mathbf{3}}-boldsymbol{x} & boldsymbol{a}^{boldsymbol{n}+mathbf{4}}-boldsymbol{x} & boldsymbol{a}^{boldsymbol{n}+mathbf{5}}-boldsymbol{x} \ boldsymbol{a}^{boldsymbol{n}+mathbf{6}}-boldsymbol{x} & boldsymbol{a}^{boldsymbol{n}+mathbf{7}}-boldsymbol{x} & boldsymbol{a}^{boldsymbol{n}+boldsymbol{8}}-boldsymbol{x}end{array}right|= )
( mathbf{A} cdot mathbf{1} )
B.
( c .-1 )
D. None of these
12
754 f ( x, y ) and ( z ) are all distinct and ( left|begin{array}{lll}boldsymbol{x} & boldsymbol{x}^{2} & boldsymbol{1}+boldsymbol{x}^{3} \ boldsymbol{y} & boldsymbol{y}^{2} & boldsymbol{1}+boldsymbol{y}^{3} \ boldsymbol{z} & boldsymbol{z}^{2} & boldsymbol{1}+boldsymbol{z}^{3}end{array}right|=mathbf{0}, ) then the value of
( x y z ) is.
A . -4
B. –
( c .-2 )
D. – –
12
755 Assertion ( operatorname{Let} boldsymbol{A}=left[begin{array}{ccc}mathbf{0} & boldsymbol{c} & -boldsymbol{b} \ -boldsymbol{c} & boldsymbol{0} & boldsymbol{a} \ boldsymbol{b} & boldsymbol{-} boldsymbol{a} & boldsymbol{0}end{array}right] ) and ( boldsymbol{t} in boldsymbol{C} )
( boldsymbol{a} boldsymbol{d} boldsymbol{j}(boldsymbol{t} boldsymbol{I}-boldsymbol{A})=boldsymbol{t}^{2} boldsymbol{I}+boldsymbol{t} boldsymbol{A}+boldsymbol{A}^{2}+left(boldsymbol{a}^{2}+right. )
( left.b^{2}+c^{2}right) I )
Reason
( boldsymbol{a} boldsymbol{d} boldsymbol{j} boldsymbol{A}=boldsymbol{A}^{2}+left(boldsymbol{a}^{2}+boldsymbol{b}^{2}+boldsymbol{c}^{2}right) boldsymbol{I} )
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
756 Evaluate ( left|begin{array}{cc}sin 60^{circ} & cos 60^{circ} \ -sin 30^{circ} & cos 30^{circ}end{array}right| ) 12
757 For what value of ( x, ) will the points ( (-1, x) ) (-3,2) and (-4,4) lie on a line?
( A cdot-3 )
B. 3
( c cdot-2 )
D.
12
758 If ( boldsymbol{A} ) is matrix of order ( boldsymbol{3}, ) then ( boldsymbol{d} boldsymbol{e} boldsymbol{t}(boldsymbol{k} boldsymbol{A}) )
is:
A ( cdot k^{3} d e t(A) )
B . ( k^{2} d e t(A) )
c. ( k d e t(A) )
D. ( operatorname{det}(A) )
12
759 If ( left|begin{array}{lll}boldsymbol{a} & boldsymbol{a}^{2} & mathbf{1}+boldsymbol{a}^{3} \ boldsymbol{b} & boldsymbol{b}^{2} & mathbf{1}+boldsymbol{b}^{3} \ boldsymbol{c} & boldsymbol{c}^{2} & boldsymbol{1}+boldsymbol{c}^{3}end{array}right|=mathbf{0} ) and vectors
( left(1, a, a^{2}right),left(1, b, b^{2}right) ) and ( left(1, c, c^{2}right) ) are non
coplanar then ( a b c ) equals
A . -1
B. 1
( c cdot 0 )
D.
12
760 8.
Let @=-
XV
. Then the value of the determinant
-1-02
(2002)
(a) 30
(b) 3ola -1) (c) 302
(d) 3011–@)
Ti
12
761 Prove that ( left|begin{array}{lll}boldsymbol{a} & boldsymbol{a} & boldsymbol{a} \ boldsymbol{a} & boldsymbol{b} & boldsymbol{b} \ boldsymbol{a} & boldsymbol{b} & boldsymbol{c}end{array}right|=boldsymbol{a}(boldsymbol{b}-boldsymbol{c})(boldsymbol{a}-boldsymbol{b}) )
Hence find the value of ( left|begin{array}{lll}mathbf{3} & mathbf{3} & mathbf{3} \ mathbf{3} & mathbf{5} & mathbf{5} \ mathbf{3} & mathbf{5} & mathbf{7}end{array}right| )
12
762 Let ( n ) and ( r ) be two positive integers such that ( n geq r+2 . ) Suppose ( Delta(n, r)= )
( left|begin{array}{ccc}n & C_{r} & ^{n} C_{r+1} & ^{n} C_{r+2} \ ^{n+1} C_{r} & ^{n+1} C_{r+1} & ^{n+1} C_{r+2} \ ^{n+2} C_{r} & ^{n+2} C_{r+1} & ^{n+2} C_{r+2}end{array}right| ) Show that
( Delta(n, r)=frac{n+2}{n+2} C_{3} Delta(n-2, r-1) ) Hence
or otherwise,
A ( cdot frac{left(^{n+2} C_{3}right)left(^{n+1} C_{3}right) ldots . .left(^{n-r+3} C_{3}right)}{left.left(r+2 C_{3}right)left(r+1 C_{3}right) ldots . .^{3} C_{3}right)} )
B. ( -frac{left(^{n+2} C_{3}right)left(^{n+1} C_{3}right) ldots . .left(^{n-r+3} C_{3}right)}{left(r+2 C_{3}right)left(^{r+1} C_{3}right) ldotsleft(^{3} C_{3}right)} )
c. ( frac{(n+3}{(r+3)}(sqrt[n+2]{(r+2)}) ldots . .left(^{n-r+3} C_{3}right) )
D. ( -frac{left.left(^{n+3} C_{3}right)(n+2)_{3}right) ldots . .left(^{n-r+3} C_{3}right)}{left(r+3 C_{3}right)left(r+2 C_{3}right) ldotsleft(^{3} C_{3}right)} )
12
763 Show that ( left|begin{array}{ccc}mathbf{1} & mathbf{1} & mathbf{1} \ boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{a}^{2} & boldsymbol{b}^{2} & boldsymbol{c}^{2}end{array}right|=(boldsymbol{a}-boldsymbol{b})(boldsymbol{b}- )
( c)(c-a) )
12
764 Calculate the values of the
determinants:
( left|begin{array}{cccc}mathbf{7} & mathbf{1 3} & mathbf{1 0} & mathbf{6} \ mathbf{5} & mathbf{9} & mathbf{7} & mathbf{4} \ mathbf{8} & mathbf{1 2} & mathbf{1 1} & mathbf{7} \ mathbf{4} & mathbf{1 0} & mathbf{6} & mathbf{3}end{array}right| )
12
765 Consider the matrix ( A, B, C, D ) with
order ( 2 times 3,3 times 4,4 times 4,4 times 2 )
respectively. Let ( boldsymbol{x}=left(boldsymbol{alpha} boldsymbol{A} boldsymbol{B} boldsymbol{gamma} boldsymbol{C}^{2} boldsymbol{D}right)^{3} )
where ( alpha ) and ( gamma ) are scalars. Let ( |x|= )
( kleft|A B C^{2} Dright|^{3} . ) Then the value of ( k )
A ( cdot alpha^{6} gamma^{6} )
в. ( alpha gamma )
c. ( alpha^{3} gamma^{3} )
D. ( alpha^{6} gamma^{12} )
12
766 Which of these are correct?
(a) If any two rows or columns of a determinant are identical, then the
value of the determinant is zero.
(b) If the corresponding rows and columns of a determinant are
interchanged, then the value of the determinant does not change.
(c) If any two rows (or column) of a determinant are interchanged, then the value of the determinant changes in
( operatorname{sign} )
A ( cdot(a) ) and ( (b) )
B. ( (b) ) and ( (c) )
c. ( (a) ) and ( (c) )
D. ( (a),(b) ) and ( (c) )
12
767 ( begin{array}{ccc}sin ^{2} x & cos ^{2} x & 1 \ cos ^{2} x & sin ^{2} x & 1 \ -10 & 12 & 2end{array} mid= )
( mathbf{A} cdot mathbf{0} )
B. ( 12 cos ^{2} x-10 sin ^{2} x )
c. ( 12 sin ^{2} x-10 cos ^{2} x-2 )
D. ( 10 sin 2 x )
12
768 Using properties of determinants, prove
that
[
boldsymbol{omega}left|begin{array}{ccc}
boldsymbol{x} & boldsymbol{y} & boldsymbol{z} \
boldsymbol{x}^{2} & boldsymbol{y}^{2} & boldsymbol{z}^{2} \
boldsymbol{y}+boldsymbol{z} & boldsymbol{z}+boldsymbol{x} & boldsymbol{x}+boldsymbol{y}
end{array}right|=(boldsymbol{x}-
]
( boldsymbol{y})(boldsymbol{y}-boldsymbol{z})(boldsymbol{z}-boldsymbol{x})(boldsymbol{x}+boldsymbol{y}+boldsymbol{z}) )
12
769 Using properties of determinants, prove
[
operatorname{that}left[begin{array}{ccc}
boldsymbol{b}+boldsymbol{c} & boldsymbol{q}+boldsymbol{r} & boldsymbol{y}+boldsymbol{z} \
boldsymbol{c}+boldsymbol{a} & boldsymbol{r}+boldsymbol{p} & boldsymbol{z}+boldsymbol{x} \
boldsymbol{a}+boldsymbol{b} & boldsymbol{p}+boldsymbol{q} & boldsymbol{x}+boldsymbol{y}
end{array}right]=
]
( mathbf{2}left[begin{array}{lll}boldsymbol{a} & boldsymbol{p} & boldsymbol{x} \ boldsymbol{b} & boldsymbol{q} & boldsymbol{y} \ boldsymbol{c} & boldsymbol{r} & boldsymbol{z}end{array}right] )
12
770 If area of a triangle is 35 sq units with
vertices (2,-6),(5,4) and ( (k, 4), ) then find the value of ( k )
12
771 ( left|begin{array}{ccc}a^{2} & b^{2} & c^{2} \ (a+1)^{2} & (b+1)^{2} & (c+1)^{2} \ (a-1)^{2} & (b-1)^{2} & (c-1)^{2}end{array}right|= )
( k(a-b)(b-c)(c-a), ) then find the
value of ( -boldsymbol{k} )
12
772 Evaluate
[
left|begin{array}{ccc}
2 & 7 & 3 \
-4 & 3 & -1 \
0 & -3 & 7
end{array}right|
]
12
773 Prove that:
[
begin{array}{ccc}
boldsymbol{a}+boldsymbol{b}+mathbf{2 c} & boldsymbol{a} & boldsymbol{b} \
boldsymbol{c} & boldsymbol{b}+boldsymbol{c}+mathbf{2} boldsymbol{a} & boldsymbol{b} \
boldsymbol{c} & boldsymbol{a} & boldsymbol{c}+boldsymbol{a}+boldsymbol{2} boldsymbol{b}
end{array} mid=
]
( mathbf{2}(boldsymbol{a}+boldsymbol{b}+boldsymbol{c})^{3} )
12
774 If ( boldsymbol{A}=left|begin{array}{ll}mathbf{3} & mathbf{4} \ mathbf{1} & mathbf{2}end{array}right|, ) find the value of ( mathbf{3}|boldsymbol{A}| ) 12
775 Let the three digit numbers ( A 28,3 B 9 )
and ( 62 C, ) where ( A, B, C ) are integers
between 0 and 9 , be divisible by a fixed integer ( k, ) Show that the determinant ( left|begin{array}{lll}boldsymbol{A} & boldsymbol{3} & boldsymbol{6} \ boldsymbol{8} & boldsymbol{9} & boldsymbol{C} \ boldsymbol{2} & boldsymbol{B} & boldsymbol{2}end{array}right| ) is also divisible by the
same integer ( boldsymbol{k} )
12
776 ( P, Q, R ) are three collinear points. The coordinates of ( mathrm{P} ) and ( mathrm{R} ) are (3,4) and (11
10) respectively and PQ is equal to 2.5 units. Coordinates of ( Q ) are-
A ( cdot(5,11 / 2) )
B. (11,5/2)
c. ( (5,-11 / 2) )
D. ( (-5,11 / 2) )
12
777 ( fleft(begin{array}{cc}x+y & x-y \ 2 x+z & x+zend{array}right)=left(begin{array}{cc}0 & 0 \ 1 & 1end{array}right), ) then
the values of ( x, y ) and ( z ) are respectively
( mathbf{A} cdot 0,0,1 )
в. 1,1,0
( mathrm{c} cdot-1,0,0 )
( mathbf{D} cdot 0,0,0 )
12
778 If the system of equations
x-ky – z= 0, kx – y – z=0, x + y – z = 0 has a non-zero
solution, then the possible values of k are (20005)
(a) -1,2 (6) 1,2 (c) 0,1 (d) -1, 1
12
779 ( mathbf{f} boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{- 1} & mathbf{2} \ mathbf{3} & mathbf{0} & -mathbf{2} \ mathbf{1} & mathbf{0} & mathbf{3}end{array}right] ) verify that
( boldsymbol{A}(boldsymbol{a} boldsymbol{d} boldsymbol{j} boldsymbol{A})=|boldsymbol{A}| boldsymbol{I} )
12
780 The value of determinant is
[
left|begin{array}{ccc}
boldsymbol{a}+boldsymbol{b} & boldsymbol{a}+mathbf{2 b} & boldsymbol{a}+mathbf{3} boldsymbol{b} \
boldsymbol{a}+mathbf{2} boldsymbol{b} & boldsymbol{a}+mathbf{3} boldsymbol{b} & boldsymbol{a}+mathbf{4} boldsymbol{b} \
boldsymbol{a}+mathbf{4} boldsymbol{b} & boldsymbol{a}+mathbf{5} boldsymbol{b} & boldsymbol{a}+boldsymbol{6 b}
end{array}right|
]
12
781 Using properties of determinants, prove
the following:
( left(begin{array}{ccc}mathbf{1} & boldsymbol{x} & boldsymbol{x} \ mathbf{2} boldsymbol{x} & boldsymbol{x}(boldsymbol{x}-mathbf{1}) & boldsymbol{x}(boldsymbol{x} boldsymbol{x} \ mathbf{3} boldsymbol{x}(mathbf{1}-boldsymbol{x}) & boldsymbol{x}(boldsymbol{x}-mathbf{1})(boldsymbol{x}-mathbf{2}) & boldsymbol{x}(boldsymbol{x}+mathbf{1})end{array}right. )
( 6 x^{2}left(1-x^{2}right) )
12
782 A square matrix ( mathbf{B} ) of order ( mathbf{3}, ) has ( |boldsymbol{B}|= )
( mathbf{7}, ) find ( mid boldsymbol{B} ) adj ( mathbf{B} mid )
12
783 Find the maximum value of ( left|begin{array}{ccc}mathbf{1} & mathbf{1} & mathbf{1} \ mathbf{1} & mathbf{1}+sin boldsymbol{theta} & mathbf{1} \ mathbf{1} & mathbf{1} & mathbf{1}+cos boldsymbol{pi}end{array}right| ) 12
784 Assertion
consider the determinant ( Delta= ) ( left|begin{array}{lll}a_{1}+b_{1} x^{2} & a_{1} x^{2}+b_{1} & c_{1} \ a_{2}+b_{2} x^{2} & a_{2} x^{2}+b_{2} & c_{2} \ a_{3}+b_{3} x^{2} & a_{3} x^{2}+b_{3} & c_{3}end{array}right|=0, ) where
( boldsymbol{x}_{i}, boldsymbol{y}_{i}, boldsymbol{z}_{i} in boldsymbol{R} quad(boldsymbol{i}=mathbf{1}, boldsymbol{2}, boldsymbol{3}), quad boldsymbol{x} in boldsymbol{R} )
The values of ( x ) satisfying ( Delta=0 ) are
( boldsymbol{x}=mathbf{1},-mathbf{1} )
Reason
( left|begin{array}{lll}boldsymbol{a}_{1} & boldsymbol{b}_{1} & boldsymbol{c}_{1} \ boldsymbol{a}_{2} & boldsymbol{b}_{2} & boldsymbol{c}_{2} \ boldsymbol{a}_{3} & boldsymbol{b}_{3} & boldsymbol{c}_{3}end{array}right|=mathbf{0}, ) then ( boldsymbol{Delta}=mathbf{0} )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
785 If ( boldsymbol{A}=left[begin{array}{ll}2 & 0 \ 0 & 3end{array}right], ) then show that ( |3 A|= )
( mathbf{9}|boldsymbol{A}| )
12
786 f ( f(x)= ) ( left|begin{array}{ccc}mathbf{1} & mathbf{1} & mathbf{1} \ mathbf{2} boldsymbol{x} & boldsymbol{x}-mathbf{1} & boldsymbol{x} \ mathbf{3} boldsymbol{x}(boldsymbol{x}-mathbf{1}) & (boldsymbol{x}-mathbf{1})(boldsymbol{x}-mathbf{2}) & boldsymbol{x}(boldsymbol{x}-mathbf{1})end{array}right| )
then ( boldsymbol{f}(mathbf{5 0})= )
4.0
B. 2
( c_{1} )
( D )
( E )
12
787 If ( a neq b neq c, ) the value of ( x ) which
satisfies the question ( left|begin{array}{ccc}mathbf{0} & boldsymbol{x}-boldsymbol{a} & boldsymbol{x}-boldsymbol{b} \ boldsymbol{x}+boldsymbol{a} & mathbf{0} & boldsymbol{x}-boldsymbol{c} \ boldsymbol{x}+boldsymbol{b} & boldsymbol{x}+boldsymbol{c} & mathbf{0}end{array}right|=mathbf{0} ) is
( mathbf{A} cdot x=0 )
B. ( x=a )
( mathbf{c} cdot x=b )
D. ( x=c )
12
788 uet (Padj ())=2 (u) uute
12. Let a, 2. uR. Consider the system
ax +2y=a
3x – 2y = u
T the system of linear equations
ment(s) is (are) correct?
(JEE Adv. 2016)
tem has a unique solution for all
1= -3, then the system has infinitely many solutio
for all values of 2 and u.
(b) Ifa -3, then the system has a unique solution
values of 2 andu.
© If 2 +u = 0, then the system has infinitely many
solutions for a = -3.
(d) If a+u = 0, then the system has no solution for
a=-3.
12
789 ( mathrm{f}left|begin{array}{ccc}boldsymbol{x} & boldsymbol{2} & boldsymbol{x} \ boldsymbol{x}^{2} & boldsymbol{x} & boldsymbol{6} \ boldsymbol{x} & boldsymbol{x} & boldsymbol{6}end{array}right|=boldsymbol{a} boldsymbol{x}^{4}+boldsymbol{b} boldsymbol{x}^{3}+boldsymbol{c} boldsymbol{x}^{2}+ )
( boldsymbol{d} boldsymbol{x}+boldsymbol{e}, ) then ( boldsymbol{5} boldsymbol{a}+boldsymbol{4} boldsymbol{b}+boldsymbol{3} boldsymbol{c}+boldsymbol{2} boldsymbol{d}+boldsymbol{e} ) is
equal to
A. 11
B. -11
c. 12
D. -12
E . 13
12
790 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{4} & mathbf{2} \ mathbf{3} & mathbf{3}end{array}right], ) then adj ( (operatorname{adj} boldsymbol{A}) ) is equal
to
( mathbf{A} cdotleft[begin{array}{cc}3 & -2 \ -3 & 4end{array}right] )
в. ( left[begin{array}{ll}4 & 2 \ 3 & 3end{array}right] )
D. None of these
12
791 If ( boldsymbol{A}=left[begin{array}{cc}mathbf{2} & -mathbf{3} \ mathbf{4} & mathbf{1}end{array}right], ) then adjoint of matrix
( A ) is
A. ( left[begin{array}{cc}1 & 3 \ -4 & 2end{array}right] )
В. ( left[begin{array}{cc}1 & -3 \ -4 & 2end{array}right] )
c. ( left[begin{array}{cc}1 & 3 \ 4 & -2end{array}right] )
D. ( left[begin{array}{cc}-1 & -3 \ -4 & 2end{array}right] )
12
792 Show that ( left|begin{array}{lll}boldsymbol{b} boldsymbol{c} & boldsymbol{b}+boldsymbol{c} & mathbf{1} \ boldsymbol{c} boldsymbol{a} & boldsymbol{c}+boldsymbol{a} & mathbf{1} \ boldsymbol{a} boldsymbol{b} & boldsymbol{a}+boldsymbol{b} & mathbf{1}end{array}right|= )
( (boldsymbol{a}-boldsymbol{b})(boldsymbol{b}-boldsymbol{c})(boldsymbol{c}-boldsymbol{a}) )
12
793 If ( |boldsymbol{A}|=2, ) where ( boldsymbol{A} ) is a ( mathbf{3} times mathbf{3} ) matrix.
Then find ( |boldsymbol{3} boldsymbol{A}| )
12
794 Find cofactors of the elements of the ( operatorname{matrix} boldsymbol{A}=left[begin{array}{ll}-1 & 2 \ -3 & 4end{array}right] ) 12
795 Prove that ( left|begin{array}{ccc}a^{2} & b c & a c+c^{2} \ a^{2}+a b & b^{2} & a c \ a b & b^{2}+b c & c^{2}end{array}right| )
( =4 a^{2} b^{2} c^{2} )
12
796 If ( d ) is the determinant of a square
matrix A of order ( n ), then the
determinant of its adjoint is:
A . ( d^{n} )
B. ( d^{n-1} )
( c cdot d^{n-2} )
D. ( d )
12
797 In a third order determinant, each
element of the first column consists of
sum of two terms, each element of the second column consists of sum of three
terms and each element of the
third column consists of sum of four
terms. Then it can be decomposed into n determinants, where n has the value
( mathbf{A} cdot mathbf{1} )
B. 9
c. 16
D. 24
12
798 Using properties of determinants prove
that :
( left|begin{array}{lll}boldsymbol{x} & boldsymbol{a} & boldsymbol{a} \ boldsymbol{a} & boldsymbol{x} & boldsymbol{a} \ boldsymbol{a} & boldsymbol{a} & boldsymbol{x}end{array}right|=(boldsymbol{x}+mathbf{2} boldsymbol{a})(boldsymbol{x}-boldsymbol{a})^{2} )
12
799 Let ( boldsymbol{A}=left[boldsymbol{a}_{i j}right] ) and ( boldsymbol{B}=left[boldsymbol{b}_{boldsymbol{i} j}right] ) be two ( boldsymbol{3} times boldsymbol{3} )
real matrices such that ( b_{i j}= )
( (3)^{(i+j-2)} a_{i j}, ) where ( i, j=1,2,3 . ) If the
determinant of ( B ) is 81 , then the
determinant of A is :
A. ( 1 / 3 )
B. 3
c. ( 1 / 81 )
D. ( 1 / 9 )
12
800 Evaluate the following determinant:
( left|begin{array}{lll}boldsymbol{a} & boldsymbol{h} & boldsymbol{g} \ boldsymbol{h} & boldsymbol{b} & boldsymbol{f} \ boldsymbol{g} & boldsymbol{f} & boldsymbol{c}end{array}right| )
12
801 30. If A is a 3 x 3 non-singular matrix such that AA’ = A’A and
B=A-1 A’, then BB’ equals:
[JEE M 2014
(a) B-1
(b) (B-1)’ (©) I+B
(d) I
12
802 ( left|begin{array}{ccc}1 & a & a^{2}-b c \ 1 & b & b^{2}-c a \ 1 & c & c^{2}-a bend{array}right| ) is equal to
( A )
B . ( sum a^{2}(b-c) )
c. ( 2 sum a^{2}(b-c) )
D. ( -2 sum a b(a-b) )
12
803 43.
Let a and ß be the roots of the equation x2 + x + 1 = 0. Then
for y#0 in R,
y+1 a B1
a y+B 1
is equal to: [JEEM 2019-9 April (M)
IB 1 yta
(a) y(y2 – 1)
(b) y(y – 3)
C) y
(d) y – 1
12
804 ( f x neq 0 ) and ( left|begin{array}{ccc}1 & x & 2 x \ 1 & 3 x & 5 x \ 1 & 3 & 4end{array}right|=0, ) then ( x= )
( A )
B. –
( c cdot 2 )
( D ldots-2 )
12
805 Evaluate the following:
( left|begin{array}{ccc}boldsymbol{x}+boldsymbol{lambda} & boldsymbol{x} & boldsymbol{x} \ boldsymbol{x} & boldsymbol{x}+boldsymbol{lambda} & boldsymbol{x} \ boldsymbol{x} & boldsymbol{x} & boldsymbol{x}+boldsymbol{lambda}end{array}right| )
12
806 If ( boldsymbol{x}+boldsymbol{a}+boldsymbol{b}+boldsymbol{c}=boldsymbol{0}, ) then what is the
value of ( left|begin{array}{ccc}boldsymbol{x}+boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ boldsymbol{a} & boldsymbol{x}+boldsymbol{b} & boldsymbol{c} \ boldsymbol{a} & boldsymbol{b} & boldsymbol{x}+boldsymbol{c}end{array}right| ? )
( mathbf{A} cdot mathbf{0} )
B ( cdot(A+B+C)^{2} )
c. ( A^{2}+B^{2}+C^{2} )
D. ( A+B+C-2 )
12
807 Evaluate the following:
( left|begin{array}{ccc}boldsymbol{x}+boldsymbol{lambda} & boldsymbol{x} & boldsymbol{x} \ boldsymbol{x} & boldsymbol{x}+boldsymbol{lambda} & boldsymbol{x} \ boldsymbol{x} & boldsymbol{x} & boldsymbol{x}+boldsymbol{lambda}end{array}right| )
12
808 Calculate the values of the
determinants:
( left|begin{array}{cccc}mathbf{1} & mathbf{1} & mathbf{1} & mathbf{1} \ mathbf{1} & mathbf{2} & mathbf{3} & mathbf{4} \ mathbf{1} & mathbf{3} & mathbf{6} & mathbf{1 0} \ mathbf{1} & mathbf{4} & mathbf{1 0} & mathbf{2 0}end{array}right| )
12
809 ( operatorname{Det}left{begin{array}{ccc}-2 a & a+b & c+a \ b+a & -2 b & b+c \ c+a & c+b & -2 cend{array}right}= )
( A cdot(a+b)(b+c)(c+a) )
В . ( (a-b)(b-c)(c-a) )
( c cdot 4(a+b)(b+c)(c+a) )
D. ( 4(a-b)(b-c)(c-a) )
12
810 [
begin{array}{c}
operatorname{Let}left|begin{array}{ccc}
a^{2}+1 & a b & a c \
a b & b^{2}+1 & b c \
a c & b c & c^{2}+1
end{array}right|=k+ \
a^{2}+b^{2}+c^{2}
end{array}
]
then ( 4 k ) is
12
811 Evaluate ( left|begin{array}{cc}sqrt{mathbf{6}} & sqrt{mathbf{5}} \ sqrt{mathbf{2 0}} & sqrt{mathbf{2 4}}end{array}right| ) 12
812 Find the value of ( x ) if
( left|begin{array}{ccc}mathbf{3} & mathbf{4} & mathbf{1} \ mathbf{0} & boldsymbol{x} & mathbf{8} \ mathbf{3} & mathbf{- 1} & mathbf{4}end{array}right|=mathbf{0} )
12
813 ( boldsymbol{A}=left[begin{array}{ccc}mathbf{5} & mathbf{5} boldsymbol{alpha} & boldsymbol{alpha} \ mathbf{0} & boldsymbol{alpha} & mathbf{5} boldsymbol{alpha} \ mathbf{0} & mathbf{0} & mathbf{5}end{array}right] ; ) If ( left|boldsymbol{A}^{2}right|=mathbf{2 5}, ) then
( |boldsymbol{alpha}|= )
A . 5
B ( .5^{2} )
( c cdot 1 )
( D )
( therefore )
12
814 Solve for ( mathbf{y}:left|begin{array}{ccc}boldsymbol{x}+boldsymbol{y} & boldsymbol{x} & boldsymbol{x} \ boldsymbol{x} & boldsymbol{x}+boldsymbol{y} & boldsymbol{x} \ boldsymbol{x} & boldsymbol{x} & boldsymbol{x}+boldsymbol{y}end{array}right|= )
( mathbf{1 6}(boldsymbol{3} boldsymbol{x}+boldsymbol{4}) )
12
815 Show that:
( left|begin{array}{ccc}boldsymbol{a}^{2} & boldsymbol{b}^{2} & boldsymbol{c}^{2} \ boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \ mathbf{1} & mathbf{1} & mathbf{1}end{array}right|=-(boldsymbol{a}-boldsymbol{b})(boldsymbol{b}-boldsymbol{c})(boldsymbol{c}- )
( boldsymbol{a}) )
12
816 ( boldsymbol{A}=left|begin{array}{ccc}mathbf{1} & mathbf{1} & mathbf{1} \ mathbf{1} & mathbf{1}+boldsymbol{x} & mathbf{1} \ mathbf{1} & mathbf{1} & mathbf{1}+boldsymbol{y}end{array}right|= )
A . ( x y )
в. ( x+y )
c ( . x-y )
D. ( x^{2} y^{2} )
12
817 The value of which of the following
determinants can be non-zero?
( mathbf{A} cdotleft|begin{array}{lll}a_{1}+a_{2} & a_{2} & a_{1} \ a_{4}+a_{5} & a_{5} & a_{4} \ a_{7}+a_{8} & a_{8} & a_{7}end{array}right| )
( mathbf{B} cdotleft|begin{array}{lll}a_{1}+2 a_{2} & a_{2} & a_{3} \ a_{4}+2 a_{5} & a_{5} & a_{6} \ a_{7}+2 a_{8} & a_{8} & a_{9}end{array}right| )
( mathbf{C} cdotleft|begin{array}{lll}k a_{4} & k a_{5} & k a_{6} \ a_{4} & a_{5} & a_{6} \ a_{7} & a_{8} & a_{9}end{array}right| )
D. None of these
12
818 Solve
( left|begin{array}{ccc}mathbf{1} & boldsymbol{b} boldsymbol{c} & boldsymbol{a}(boldsymbol{b}+boldsymbol{c}) \ mathbf{1} & boldsymbol{c} boldsymbol{a} & boldsymbol{b}(boldsymbol{c}+boldsymbol{a}) \ mathbf{1} & boldsymbol{a} boldsymbol{b} & boldsymbol{c}(boldsymbol{a}+boldsymbol{b})end{array}right| )
( mathbf{A} cdot mathbf{0} )
B.
( c cdot a b c )
D. ( a b+b c+c a )
12
819 ( mathbf{f} boldsymbol{z}=left|begin{array}{ccc}mathbf{3}+mathbf{3} boldsymbol{i} & mathbf{5}-boldsymbol{i} & mathbf{7}-mathbf{3} i \ boldsymbol{i} & mathbf{2} boldsymbol{i} & -mathbf{3} i \ boldsymbol{3}-mathbf{2} boldsymbol{i} & mathbf{5}+boldsymbol{i} & mathbf{7}+mathbf{3} boldsymbol{i}end{array}right| ), then
A. z is purely real
B. z is purely imaginary
( c cdot 0 )
D. none of these
12
820 Using properties of determinants, prove
[
text { the following: }left|begin{array}{ccc}
mathbf{1} & mathbf{1} & mathbf{1} \
boldsymbol{a} & boldsymbol{b} & boldsymbol{c} \
boldsymbol{a}^{mathbf{3}} & boldsymbol{b}^{mathbf{3}} & boldsymbol{c}^{3}
end{array}right|=(boldsymbol{a}-
]
( b)(b-c)(c-a)(a+b+c) )
12
821 2.
If (71) is a cube root of unity, then
1 1+i+02 02
(1995
-i
-i +0-1
-1
(a) 0
(6) 1
(c)
i
(d) w
12
822 ( operatorname{Let} boldsymbol{A}=left[begin{array}{ll}boldsymbol{a} & boldsymbol{b} \ boldsymbol{c} & boldsymbol{d}end{array}right] ) and ( boldsymbol{f}(boldsymbol{lambda})=operatorname{det}(boldsymbol{A}- )
( boldsymbol{lambda} boldsymbol{I}), ) then
This question has multiple correct options
A ( cdot f(lambda)=lambda^{2}-(a+d) lambda+a d-b c )
в. ( f(A)=O )
c. ( A^{2}=O ) implies ( A^{r}=O forall r geq 2 )
D. none of these
12
823 7. The number of distinct real roots of
sin x
cos x
cos x
VI
s
cos x cos x
sin x cos x = 0 in the interval
cos x sin x
b. 2
d. 3
(IIT-JEE 2001)
a. 0
c. 1
12
824 Consider the determinant ( Delta= ) ( left|begin{array}{lll}a_{1} & a_{2} & a_{3} \ b_{1} & b_{2} & b_{3} \ c_{1} & c_{2} & c_{3}end{array}right| )
( M_{i j}= ) Minor of the element of ( i^{t h} ) row ( & ) ( j^{t h} ) column ( C_{i j}= ) Cofactor of element of ( i^{t h} ) row ( & ) ( j^{t h} ) column
( a_{2} cdot C_{12}+b_{2} cdot C_{22}+c_{2} cdot C_{32} ) is equal to
A.
B. ( Delta )
( c cdot 2 Delta )
D. ( Delta^{2} )
12
825 ( operatorname{Let} boldsymbol{D}_{boldsymbol{r}}=left|begin{array}{ccc}mathbf{2}^{boldsymbol{r}-mathbf{1}} & mathbf{2} mathbf{.} boldsymbol{3}^{boldsymbol{r}-mathbf{1}} & boldsymbol{4} mathbf{.} mathbf{5}^{boldsymbol{r}-mathbf{1}} \ boldsymbol{alpha} & boldsymbol{beta} & boldsymbol{gamma} \ mathbf{2}^{boldsymbol{n}}-mathbf{1} & boldsymbol{3}^{boldsymbol{n}}-mathbf{1} & mathbf{5}^{boldsymbol{n}}-mathbf{1}end{array}right| )
Then, the value of ( sum_{r=1}^{n} D_{r} ) is
( mathbf{A} cdot alpha beta mathbf{gamma} )
B . ( 2^{n} alpha+3^{n} beta+4^{n} gamma )
( c cdot 2 alpha+3 beta+4 gamma )
D. None of these
12
826 ( f(x)=left|begin{array}{lll}a^{-x} & e^{x ln a} & x^{2} \ a^{-3 x} & e^{3 x ln a} & x^{4} \ a^{-5 x} & e^{5 x ln a} & 1end{array}right|, ) then
A. ( f(x) cdot f(-x)=0 )
B. ( f(x)-f(-x)=0 )
c. ( f(x)+f(-x)=0 )
D. None of these
12
827 Prove that
[
left|begin{array}{ccc}
boldsymbol{x}+boldsymbol{y}+mathbf{2} boldsymbol{z} & boldsymbol{x} & boldsymbol{y} \
boldsymbol{z} & boldsymbol{y}+boldsymbol{z}+mathbf{2} boldsymbol{x} & boldsymbol{y} \
boldsymbol{z} & boldsymbol{x} & boldsymbol{z}+boldsymbol{x}+boldsymbol{2} boldsymbol{y}
end{array}right|
]
( 2(x+y+z)^{3} )
12
828 ( left|begin{array}{ccc}1 ! & 2 ! & 3 ! \ 2 ! & 3 ! & 4 ! \ 3 ! & 4 ! & 5 !end{array}right|=2016 K )
then value of ( boldsymbol{K} ) is
A . 24
B. 84
c. ( frac{1}{24} )
D. ( frac{1}{84} )
12
829 If the lines ( 3 x+2 y-5=0,2 x-5 y+ )
( mathbf{3}=mathbf{0}, mathbf{5 x}+mathbf{b y}+mathbf{c}=mathbf{0} ) are concurrent
then ( mathbf{b}+mathbf{c}= )
A. 7
B. – –
( c cdot 6 )
D.
12
830 Find the values of the following
determinants
( left|begin{array}{cc}mathbf{2} boldsymbol{i} & -mathbf{3} i \ boldsymbol{i}^{3} & -mathbf{2} boldsymbol{i}^{5}end{array}right| ) where ( boldsymbol{i}=sqrt{-mathbf{1}} )
12
831 Evaluate the following:
( left|begin{array}{ccc}1 & a & b c \ 1 & b & c a \ 1 & c & a bend{array}right| )
12
832 [
mathbf{f} boldsymbol{a}_{mathbf{1}} boldsymbol{f}_{mathbf{1}}(boldsymbol{x})+boldsymbol{a}_{mathbf{2}} boldsymbol{f}_{mathbf{2}}(boldsymbol{x})+boldsymbol{a}_{mathbf{3}} boldsymbol{f}_{mathbf{3}}(boldsymbol{x})=mathbf{0}
]
where ( a_{1}, a_{2}, a_{3} ) are constants (not all
zero) and ( f_{1}, f_{2}, f_{3} ) are twice
differentiable functions.Then ( D= )
( left|begin{array}{ccc}boldsymbol{f}_{1}(boldsymbol{x}) & boldsymbol{f}_{2}(boldsymbol{x}) & boldsymbol{f}_{3}(boldsymbol{x}) \ boldsymbol{D} boldsymbol{f}_{1}(boldsymbol{x}) & boldsymbol{D} boldsymbol{f}_{2}(boldsymbol{x}) & boldsymbol{D} boldsymbol{f}_{3}(boldsymbol{x}) \ boldsymbol{D}^{2} boldsymbol{f}_{1}(boldsymbol{x}) & boldsymbol{D}^{2} boldsymbol{f}_{2}(boldsymbol{x}) & boldsymbol{D}^{2} boldsymbol{f}_{3}(boldsymbol{x})end{array}right| ) qual to ( D f_{1}(x)=frac{d}{d x} f_{1} )
12
833 The value of ( frac{1}{x-y}left|begin{array}{ccc}1 & 0 & 0 \ 3 & x^{3} & 1 \ 5 & y^{3} & 1end{array}right| ) is
( mathbf{A} cdot x+y )
B . ( x^{2}-x y+y^{2} )
c. ( x^{2}+x y+y^{2} )
D. ( x^{3}-y^{3} )
12
834 ( fleft(a_{1}, a_{2}, a_{3}, dots, a_{n}, dots ) are in GP, then right.
the value of the determinant ( begin{array}{|ccc|}log a_{n} & log a_{n+1} & log a_{n+2} \ log a_{n+3} & log a_{n+4} & log a_{n+5} \ log a_{n+6} & log a_{n+7} & log a_{n+8}end{array} mid ), is
A . 0
B. 1
( c cdot 2 )
( D .-2 )
12
835 If ( boldsymbol{A}=left[begin{array}{ll}2 & 5 \ 2 & 1end{array}right] ) and ( B=left[begin{array}{cc}4 & -3 \ 2 & 5end{array}right], ) verify
that ( |boldsymbol{A B}|=|boldsymbol{A}||boldsymbol{B}| )
12
836 Find the value of ( lambda ) for which the points
( (6,-1,2),(8,-7, lambda) ) and (5,2,4) are
collinear.
12
837 ( left|begin{array}{ccc}mathbf{1}+boldsymbol{i} & mathbf{1}-boldsymbol{i} & mathbf{1} \ mathbf{1}-boldsymbol{i} & boldsymbol{i} & mathbf{1}+boldsymbol{i} \ boldsymbol{i} & mathbf{1}+boldsymbol{i} & mathbf{1}-boldsymbol{i}end{array}right| ) is a
A. real number
B. irrational number
c. complex member
D. Purely imaginary
12
838 If ( A B C ) is a triangle, then the vectors
( (-1, cos C, cos B),(cos C,-1, cos A) )
and ( (cos B, cos C,-1) ) are
A. linearly independent for all triangles
B. linearly dependent for all triangles
c. linearly independent for all isosceles triangles
D. none of these
12
839 Using properties of determinant, prove
[
text { that }left|begin{array}{lll}
boldsymbol{b}+boldsymbol{c} & boldsymbol{a}-boldsymbol{b} & boldsymbol{a} \
boldsymbol{c}+boldsymbol{a} & boldsymbol{b}-boldsymbol{c} & boldsymbol{b} \
boldsymbol{a}+boldsymbol{b} & boldsymbol{c}-boldsymbol{a} & boldsymbol{c}
end{array}right|=mathbf{3} boldsymbol{a} boldsymbol{b} boldsymbol{c}-boldsymbol{a}^{3}-
]
( b^{3}-c^{3} )
12
840 ( mathbf{f}left|begin{array}{ccc}boldsymbol{a} & boldsymbol{b} & boldsymbol{a} boldsymbol{alpha}+boldsymbol{b} \ boldsymbol{b} & boldsymbol{c} & boldsymbol{b} boldsymbol{alpha}+boldsymbol{c} \ boldsymbol{a} boldsymbol{alpha}+boldsymbol{b} & boldsymbol{b} boldsymbol{alpha}+boldsymbol{c} & boldsymbol{0}end{array}right|=mathbf{0} . ) Prove
that ( a, b, c ) are in G.P. or ( alpha ) is a root of
( boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{b} boldsymbol{x}+boldsymbol{c}=mathbf{0} )
12
841 Find the value of following determinant. ( left|begin{array}{cc}-1 & 7 \ 2 & 4end{array}right| ) 12
842 Prove that: ( 2left|begin{array}{cc}8 & -5 \ -2 & 6end{array}right|=left|begin{array}{cc}14 & -2 \ -4 & 6end{array}right| ) 12
843 If ( boldsymbol{A}=left|begin{array}{lll}boldsymbol{a} & boldsymbol{0} & boldsymbol{0} \ boldsymbol{0} & boldsymbol{a} & boldsymbol{0} \ boldsymbol{0} & boldsymbol{0} & boldsymbol{a}end{array}right|, ) then the value of
( |boldsymbol{A}||boldsymbol{a} boldsymbol{d} boldsymbol{j}(boldsymbol{A})| ) is
A ( cdot a^{3} )
в. ( a^{6} )
( c cdot a^{9} )
D. ( a^{2} )
12
844 The points (2,-3),(4,3) and ( (5, k / 2) ) are on the same straight line. The value(s) of k is (are):
A . 12
B. -12
( c .pm 12 )
D. 12 or 6
12
845 ( f(x)=left|begin{array}{ccc}a^{-x} & e^{x log _{e} a} & x^{2} \ a^{-3 x} & e^{3 x log _{e} a} & x^{4} \ a^{-5 x} & e^{5 x log _{e} a} & 1end{array}right|, ) then
A ( cdot g(x)+g(-x)=0 )
B . ( g(x)-g(-x)=0 )
C. ( g(x) times g(-x)=0 )
D. none of these
12
846 If ( A ) is a square matrix such that ( A^{2}= )
( A, ) then ( |A| ) equals
( mathbf{A} cdot 0 ) or 1
B. – 2 or 2
( c .-3 ) or 3
D. None of these
12
847 h non-zero entries and let A2=I,
(d) less wall
Let A be a 2 x 2 matrix with non-zero entries
where I is 2 x 2 identity matrix. Define
TT(A) = sum of diagonal elements of A and
A=determinant of matrix A.
Statement-1: Tr(A)=0.
Statement-2: A=1.
(a) Statement -1 is true, Statement -2 is true,
is not a correct explanation for Statement -1.
(b) Statement -1 is true, Statement -2 is false.
(C) Statement -1 is false, Statement -2 is true.
(d) Statement – 1 is true, Statement 2 is true; Statement-
is a correct explanation for Statement-1.
[2010]
ment-2 is true ; Stctement-2
12
848 If ( A ) is an idempotent matrix satisfying, ( (I-0.4 A)^{-1}=I-alpha A, ) where ( I ) is the
unit matrix of the same order as that of
( A, ) then the value of ( |9 alpha| ) is equal to
12
849 ( f x neq 0 ) and ( left|begin{array}{ccc}1 & x & 2 x \ 1 & 3 x & 5 x \ 1 & 3 & 4end{array}right|=0, ) then ( x= )
( A )
B. –
( c cdot 2 )
( D ldots-2 )
12
850 Find the value of the determinant ( left|begin{array}{ccc}1 & 0 & 0 \ 2 & cos x & sin x \ 3 & sin x & cos xend{array}right| )
( mathbf{A} cdot cos 2 x )
B.
( c cdot 0 )
D. ( sin 2 x )
12
851 ( mathbf{A}=left[begin{array}{ccc}mathbf{1}^{2} & mathbf{2}^{mathbf{2}} & mathbf{3}^{2} \ mathbf{2}^{mathbf{2}} & mathbf{3}^{2} & mathbf{4}^{2} \ mathbf{3}^{mathbf{2}} & mathbf{4}^{mathbf{2}} & mathbf{5}^{mathbf{2}}end{array}right], ) then ( |boldsymbol{A} boldsymbol{d} boldsymbol{j} boldsymbol{A}|= )
( A )
B . 16
( c cdot 64 )
D. 128
12
852 The adjoint of the matric ( boldsymbol{A}= ) ( left[begin{array}{lll}1 & 0 & 2 \ 2 & 1 & 0 \ 0 & 3 & 1end{array}right] ) is
A. ( left[begin{array}{ccc}-1 & 6 & 2 \ -2 & 1 & -4 \ 6 & 3 & 1end{array}right] )
B.
[
left[begin{array}{ccc}
1 & 6 & -2 \
-2 & 1 & 4 \
6 & -3 & 1
end{array}right]
]
( c )
[
left[begin{array}{ccc}
6 & 1 & 2 \
4 & -1 & 2 \
6 & 3 & -1
end{array}right]
]
D.
[
left[begin{array}{ccc}
-6 & 2 & 1 \
4 & -2 & 1 \
3 & 1 & -6
end{array}right]
]
12
853 The straight lines ( imath_{1}, imath_{2} ) and ( imath_{3} ) are parallel and lie in the same plane. A
total of ( mathrm{m} ) points are taken on the line ( imath_{1} )
n points on ( imath_{2}, ) and ( mathrm{k} ) points on ( imath_{3} . ) How many triangles are there whose vertices are at these points?
12

Hope you will like above questions on determinants and follow us on social network to get more knowledge with us. If you have any question or answer on above determinants questions, comments us in comment box.

Stay in touch. Ask Questions.
Lean on us for help, strategies and expertise.

Leave a Reply

Your email address will not be published. Required fields are marked *