BANSALCLASSES TARGET IIT JEE 2007
MATHEMATICS
NUCLEUS
QUESTION BANK ON
DETERMINANT & MATRICES
Time Limit : 4 Sitting Each of 75 Minutes duration approx.
Question bank on Determinant & Matrices There are 102 questions in this question bank. Select the correct alternative : (Only one is correct) a2
Q.1
a
The The value value of the determ determin inant ant cos (nx) cos ( n 1) x cos (n 2) x is independent of : si sin (n 1) x
sin (nx )
(A) n Q.2
(B) a
A
1
(C)
2 1 a
1
1
1 b
1
1
(B) a1 b1 c1
(C)
A
1 1 = 0 , then the value of a 1 + b1 + c1 is 1 c
a b c
If
, &
If A =
1
cos ( ) cos ( ) =
1 cos ( )
1
(B) cos cos cos (D) zero
Lcos sin O –1 M sin cos P , A is given by N Q
(A) –A
(B) AT
(C) –AT
(D) A
If the the syst system em of equa equati tion onss ax+ y + z = 0 , x + by+ z = 0 & x + y+ cz = 0 (a, (a, b, c 1) has a non-trivial solution, then the value of (A)
Q.8
cos ( )
are real numbers numbers , then D = cos ( )
(A) 1 (C) cos + cos + cos
Q.7
(D)
If A and B are symmetric symmetric matrices, matrices, then ABA is (A) symmetric matrix (B) skew symmetric (C) diagonal matrix (D) scalar matrix
cos ( )
Q.6
(D) A2
2
1
Q.5
(D) a , n and x
0 1 1 A 4 3 4 then the inverse of will be 2 3 3 4
If a, b, c are are all differ different ent from from zero zero & (A) abc
Q.4
(B)
sin (n 2) x
(C ) x
A is an involut involutary ary matrix matrix given given by A =
(A) 2A
Q.3
1
1
1 1 a
1 1 b
(B) 0
Consid Consider er the matric matrices es A =
1 1 c
is : (C ) 1
4 6 1 2 4 3 0 2 , B = 0 1 , C = 1 2 5 1 2
(i) (AB)TC (ii) CTC(AB)T (A) exactly one is defined (C) exactly three are defined
Bansal C lasses
(D) none of these
3 1 . Out of the given matrix products 2
(iii) CTAB and (iv) ATABBTC (B) exactly two are defined (D) all four are defined
Q. B. on Determinant & Matrices
[2]
Q.9
The value of of a for which the system system of equations equations ; a 3x+(a+1)3 y + (a+ 2)3 z = 0 , ax + (a + 1) y + (a + 2) z = 0 & x + y + z = 0 has a nonnon-zer zero o solutio solution n is : (A) 1 (B) 0 (C ) 1 (D) none of these
Q.10 Q.10
If A =
F 1 G0 H
F 1 (A) G H 0
Q.11 Q.11
Q.12 Q.12
J , then An (where n N) equals
1K na I
J
1 K
F 1 (B) G H 0
1 cos 2 x
4 sin 2x
sin 2 x
cos 2 x
1 4 sin 2x
(A) 2
(B) 4
If A =
L3 4 O L 2 M1 6P and B = M 6 N Q N 3O
0PQ
naI
J
0 K
5O 1PQ
F n (D) G H 0
naI
J
n K
, then the maximum value of f (x) =
(C ) 6
L3 (B) M N 1
If px4 + qx 3 + rx 2 + sx + t
F 1 (C) G H 0
sin 2 x
(D) 8
then X such that A + 2X = B equals 5O 0PQ
L5 (C) M N 1
2
0P
x 2 3x
x 1
x3
x 1
2x
x 3 then t =
x3
x4
(B) 0
(D) none of these
3x
(C) 21
(D) none
If A and B are are invertible invertible matrices, matrices, which one of the following following statements statements is not not correct correct (A) Adj. A = |A| A –1 (B) det (A –1) = |det (A)| –1 (C) (A + B) –1 = B –1 + A –1 (D) (AB) –1 = B –1A –1
If D = ba ca
ab
ac
b2 1
bc
cb
c2 1
(A) (A) 1 + a2 + b2 + c2
Q.16 Q.16
J K
4 sin 2x
a2 1
Q.15 Q.15
1
cos 2 x
(A) 33 Q.14
n 2 aI
1 sin 2 x
Let f (x) (x) =
L2 (A) M N 1
Q.13 Q.13
a I
If A =
F a Gc H
then D =
(B) a2 + b2 + c2
(C) (C) (a + b + c)2
(D) none
bI
J satisfies the equation x2 – (a + d)x + k = 0, then
dK
(A) k = bc
Bansal C lasses
(B) k = ad
(C) k = a 2 + b2 + c2 + d2
Q. B. on Determinant & Matrices
(D) ad–bc
[3]
a If a, b, c > 0 & x, y, z R , then the determi determinant nant b c
x
Q.17
y
(B) ax bycz
(A) ax bycz Q.18
The The determ determina inant nt
sin
cos
sin ( ) cos sin
a 2
b y
z
c z
(C) a2x b2yc2z
2
2
(D) zero
co c os 2 sin
is :
cos
ar
Let A =
(B) GP
Lx M x M MN x
(C) HP
(D) none
O P x x P , then A –1 exists if x x PQ x
x
(A) x 0 (C) 3x + 0, 0
(B) 0 (D) x 0, 0 1
For positive numbers x, y & z the numerical value of the determinant determinant logy x log z x
(A) 0 Q.24 Q.24
1
ap
(A) AP
Q.23
1 =
If a 2 a 3 a q = 0 , then p, q, r are in : a 3 a 4
Q.22 Q.22
2
y
1
If A and B are non singular Matrices of of same order order then Adj. (AB) is (A) Adj. (A) (Adj. B) (B) (Adj. B) (Adj. A) (C) Adj. A + Adj. B (D) none of these a 1
Q.21 Q.21
z
2
a x
(B) independent of (D) independent of & both
(A) 0 (C) independent of Q.20
y
2
x
Identify the incorrect incorrect statement in respect of two square square matrices A and B conformable conformable for sum and product. product. (A) tr (A + B) = t r (A) + tr (B) (B) tr (A) = tr (A), R T (C) tr (A ) = t r (A) (D) tr (AB) tr (BA) cos ( )
Q.19
z
2
a b b c c a x
(B) 1
If K R 0 then det. {adj (KIn)} is equal to (A) K n – 1 (B) K n(n – 1)
Bansal C lasses
log x y 1 log z y
(C ) 3
(D) none
(C) K n
(D) K
Q. B. on Determinant & Matrices
logx z logy z is 1
[4]
Q.25
Q.26 Q.2 6
b1 c1
c1 a1
a 1 b1
The The determ determina inant nt b 2 c2
c2 a 2
a2
b 3 c 3
c3 a 3
b2 = a 3 b3
a1
b1
c1
a1
b1
c1
a1
b1
c1
a1
b1
c1
(A) a 2
b2
c2
(B) 2 a 2
b2
c2
(C) 3 a 2
b2
c2
(D) 4 a 2
b2
c2
a3
b3
c3
a3
b3
c3
a3
b3
c3
a3
b3
c3
Which of the following is an orthogonal orthogonal matrix
6 / 7 2 / 7 3 / 7 6/7 (A) 2 / 7 3 / 7 3 / 7 6 / 7 2 / 7 (C)
Q.27
(B)
6 / 7 2 / 7 3 / 7 2/ 7 3/ 7 6/ 7 3 / 7 6 / 7 2 / 7
6/ 7 2 / 7 3/ 7 2 / 7 3 / 7 (D) 2 / 7 6 / 7 2 / 7 3 / 7
1 a x
a y
az
b x
1 b y
b z
c x
cy
1 c z
The The determ determina inant nt
6 / 7 2 / 7 3 / 7 2 / 7 3 / 7 6 / 7 3 / 7 6 / 7 2 / 7
=
(A) (1 + a + b + c) (1+ x + y+ z) 3 (ax + by+ cz) (B) (B) a(x + y) + b (y+ z) + c(z + x) (xy+ (xy+ yz + zx) zx) (C) x(a + b) + y(b+ c) + z(c + a) (ab (ab + bc + ca) (D) none of these Q.28
Which of the following following statements is incorrect for a square square matrix A. ( | A | (A) If A is a diagonal matrix, A –1 will also be a diagonal matrix (B) If A is a symmetric matrix, matr ix, A –1 will also be a symmetric matrix (C) If A –1 = A A is an idempotent matrix (D) If A –1 = A A is an involutary matrix x
Q.29
The The deter determin minan antt
(A) (C) Q.30
1 3 1 12
C1
x
C2
x
y
C3
C1
y
C2
y
C 3 =
z
C1
z
C2
z
C3 1
0)
xyz xyz (x+ y z) (y+ z x)
xyz (x + y) y) (y + z) z) (z + x) x)
(B)
xyz (x y) (y z) (z x)
(D) none
4
Which Which of of the the follow following ing is a nilpoten nilpotentt matrix matrix (A)
1 0
0
1
Bansal C lasses
(B)
cos sin sin cos
(C)
0 1
0
0
Q. B. on Determinant & Matrices
(D)
1 1
1
1
[5]
a3
a
Q.31
If a, b, c are all different different and b b 3 c3
c
a 4 1
then : b 4 1 = 0 , then c4 1
(A) abc (a ( ab + bc + ca) = a + b + c (C) abc (a + b + c) = ab + bc + ca Q.32
(B) (a + b + c) (ab + bc + ca) = abc (D) none of these
Give the correct correct order order of initials initials T or F for following statements. Use T if statement is true and F if it is false. Statement-1 : If A is an invertible 3 × 3 matrix and B is a 3 × 4 matrix, then A –1B is defined Statement-2 : It is never true that A + B, A – B, and AB are all defined. Statement-3 : Every matrix none of whose entries are zero is invertible. Statement-4 : Every invertible matrix is square and has no two rows the same. (A) TFFF (B) TTFF (C) TFFT (D) TTTF
1
Q.33 Q.33
If
is one of the
imaginary cube roots of of unity unity,, then the value of the determinant 3
2 (A) 1 Q.34
(B) 2
(D) none
Identi Identify fy the correct correct statemen statementt : (A) If system of n simultaneous linear equations has a unique solution, then coefficient matrix is singular (B) If system of of n simultaneous linear equations has a unique solution, then coefficient matrix is non singular (C) If A –1 exists , (adjA) –1 may or may not exist
(D) F(x) =
cos x sin x 0 sin x cos x 0 , 0 0 0 ap
Q.35
(C ) 3
3 2 = 1 1
1 x
If the determ determina inant nt b q m y c r
n z
then F(x) . F(y) F(y) = F(x – y) y)
uf
element of v g splits into exactly K determinants of order 3, each element w h
which contains only one term, then the value of K, is (A) 6 (B) 8 (C ) 9
(D) 12
Q.36
A and B are two given matrices such that the order of A is 3×4 , if A B and BA are both defined then (A) order of B is 3 × 4 (B) order of BA is 4 × 4 (C) order of B A is 3 × 3 (D) BA is undefined
Q.37 Q.37
If the the syst system em of equa equati tion onss x + 2y+ 3z = 4 , x + py+ 2z = 3 , x + 4y+ z = 3 has an infinite number of solutions, solutions , then : (A) p = 2 , = 3 (B) p = 2 , = 4 (C) 3 p = 2 (D) none of these
Bansal C lasses
Q. B. on Determinant & Matrices
[6]
Q.38
If A =
cos 2 sin cos
sin cos sin 2
;
B =
cos 2 sin cos
sin cos sin 2
are such that, AB is a null matrix, then which of the following should necessarily be an odd integral multiple of (A)
Q.39 Q.39
2
. (B)
a b Let D1 = c d a b
a b a c c d and D2 = b d a b a c
ad bc, is (A) – 2
Q.40
(B) 0
cos sin sin cos
2
1 a x 2 If a2 + b2 + c2 = – 2 and f (x) = (1 a ) x (1 a 2 ) x
Q.42 Q.42
Matri Matrix x A=
(A)
Q.43
(B) 1
x 1 2
3 y 2
64 0 0 0 64 0 0 0 64
D1 D2
where b
0 and
(D) 2b
, n I 2 (D) (D) A is a skew skew symm symmet etri ric, c, for for = n ; n I 2
(1 b ) x 1 b 2 x (1 b 2 ) x
2
(1 c ) x (1 c 2 ) x then f (x) is a polynomial of degree 1 c2x
(C ) 2
(D) 3
2 4 , if x y z = 60 and 8x + 4y + 3z = 20 , then A (adj A) is equal to z
(B)
88 0 0 0 88 0 0 0 88
(C)
68 0 0 0 68 0 0 0 68
(D)
34 0 0 0 34 0 0 0 34
The The valu values es of for which the following equations sinx – cos y + (+1)z = 0; cosx + siny – z = 0; x +( + 1)y + cos z = 0 have non trivial solution, is (A) = n, R – {0} (B) = 2n, is any rational number (C) = (2n + 1),
Q.44
a c b d then the value of a b c
(B) A is symmetric, for = (2n + 1)
R
(A) 0
(D) +
which of the following statement holds good?
(C) A is an orthogonal matrix for R
Q.41 Q.41
(C) – 2b
For a given given matrix matrix A = (A) A = A –1
(C) –
R +, n I
(D) = (2n + 1) , 2
R, n I
If A is matrix matrix such that A2 + A + 2I = O, then which which of the th e following is INCORRECT ? (A)A is non-si non-sing ngula ularr
(B) A O
(C) (C) A is symm symmet etri ricc
1 (D) (D) A –1 = – (A + I) 2
(Where I is unit matrix of order 2 and O is null matrix of order 2 )
Bansal C lasses
Q. B. on Determinant & Matrices
[7]
Q.45
Q.46
The system system of equatio equations ns : 2 2x cos + y sin2 – 2sin = 0 x sin2 + 2y sin2 = – 2 cos x sin – y cos = 0 , for all values of , can (A) (A) have have a uniq unique ue non non - trivi trivial al solu soluti tion on (C) have infinite solutions
1 2
The number number of solution solution of the matrix matrix equatio equation n X 2 = (A) more than 2
Q.47
Q.48 Q.48
(B) (B) not not have have a solu solutio tion n (D) have a trivial solution
(B) 2
Let Let A + 2B =
1 2 0 6 3 3 5 3 1
(D) 0
For a non - zero, zero, real a, b and and c
(C ) 2
a
c b 2 c 2
b (A) – 4
1 Given Given A = 2 (A)
(B) 0 3 1 ; I = 2 0
2 1 5 2 1 6 0 1 2
and 2A – B =
c
Q.51
is
If x, y, y, z are not all all simultaneously simultaneously equal to zero, zero, satisfying satisfying the system system of equation equationss (sin3 ) x y + z = 0 (cos (cos 2 ) x + 4 y + 3 z = 0 2 x + 7 y + 7z = 0 then the number of principal values of is (A) 2 (B) 4 (C ) 5 (D) 6
a 2 b 2
Q.50
3
(C ) 1
then Tr Tr (A) – Tr(B) Tr (B) has the value equal to (A) 0 (B) 1
Q.49
1
a b
(D) none
c a
= abc, then the values of is
c2 a 2 b
(C ) 2
(D) 4
0 1 . If A – I is a singular matrix then
(B) 2 – 3 – 4 = 0
(C) 2 + 3 + 4 = 0
(D) 2 – 3 – 6 = 0
If the system system of equations, equations, a2 x ay = 1 a & bx + (3 2b)y 2b) y = 3 + a possess a unique solution x = 1, y = 1 then : (A) a = 1 ; b = 1 (B) a = 1 , b = 1 (C ) a = 0 , b = 0 (D) none
Bansal C lasses
Q. B. on Determinant & Matrices
[8]
Q.52 Q.52
Q.53
Q.54 Q.54
sin 1 1 1 sin , where 0 < 2, then Let Let A = sin 1 sin 1 (A) Det (A) = 0 (B) Det A (0, ) (C) Det (A) [2, [2, 4]
(D) (D) Det Det A [2, )
Number of value of 'a' for which the system of equations, a2 x + (2 a) y = 4 + a2 a x + (2a 1) y = a5 2 possess no solution is (A) 0 (B) 1 (C ) 2
(D) infinite
If A =
0 1 2 1 2 3 , 3 a 1
(A) (A) a = 1, c = – 1
–1
A
1 / 2 1 / 2 1 / 2 3 c , then = 4 5 / 2 3 / 2 1 / 2
(B) (B) a = 2, c = –
1 2
(C) a = – 1, c = 1
(D) a =
1 2
1
,c=
2
Q.55
Number Number of of triplets triplets of a, b & c for which which the system system of equations, equations, ax by = 2a b and (c + 1) x + cy = 10 10 a + 3 b has infinitely many solutions solutions and x = 1, y = 3 is one of the solutions, solutions, is : (A) exactly one (B) exactly two (C) exactly three (D) infinitely many
Q.56
D is a 3 x 3 diagonal diagonal matrix. Which of of the following following statements statements is not not true? (A) D = D (B) AD = DA for every matrix A of order 3 x 3 –1 (C) D if exists is a scalar matrix (D) none of these
Q.57
The following system of equations 3x – 7y + 5z = 3; 3x + y + 5z = 7 and 2x + 3y + 5z = 5 are (A) (A) cons consis iste tent nt with with triv trivia iall solu soluti tion on (B) (B) cons consis iste tent nt with with uniq unique ue non non triv trivia iall solu soluti tion on (C) (C) cons consis iste tent nt with with infi infini nite te solu soluti tion on (D) (D) inco incons nsis iste tent nt with with no solu soluti tion on n
Q.58 Q.58
If A1, A3, ..... A2n – 1 are n skew symmetric matrices of same order then B =
(2r 1)(A
2 r 1
)2 r 1 will
r 1
be (A) symmetric (C) neither neither symme symmetric tric nor skew skew symme symmetric tric
Q.59
x 3x 2 2x 1 4x 3x 1 = 0 is The number of real values of x satisfying satisfying 2x 1 7 x 2 17 x 6 12x 1 (A) 3
Q.60
(B) skew symmetric (D) data is adequate
(B) 0
(C) more than 3
(D) 1
1 1 1 3 has no inverse Number Number of real values values of for which the matrix A = 2 3 2 7 (A) 0
Bansal C lasses
(B) 1
(C ) 2
Q. B. on Determinant & Matrices
(D) infinite
[9]
1
If D =
z (y z)
z 1
x2 y(y z)
x x 2y z
x2z
xz
Q.61 Q.61
1
(x y) z2 1
x then, the incorrect statement is y(x y ) xz 2
(A) D is independent of x (C) D is independent of z Q.62 Q.6 2
(B) D is independent of y (D) D is dependent on x, y, z
If every element of a square non non singular matrix A is multiplied by k and the new matrix is denoted by B then | A –1| and | B –1| are related as (A) | A –1| = k | B –1|
(B) | A –1| =
1 k
| B –1|
(C) | A –1| = k n | B –1|
(D) | A –1| = k –n | B –1|
where n is order of matrices.
Q.63 Q.63
If f (x) =
mx
mx p
mx p
n
n p
n p
mx 2n
mx 2n p
mx 2n p
(A) (A) a stra straig ight ht line line para parall llel el to x- axis axis (C) parabola
Q.64 Q.64
Let Let A =
(B) – 1 x 1 ( x 1) 2
If D(x) D(x) =
(B) (B) a stra straig ight ht line line para parall llel el to y- axis axis (D) a straight line with negative slope
1 1 1 4 2 2 2 1 3 and 10B = 5 0 . If B is the inverse of matrix A, then is 1 1 1 1 2 3
(A) – 2
Q.65
then y = f(x) represents represe nts
(D) 5
x3
x 1
x2
( x 1) 3
x
( x 1) 2
( x 1) 3
(A) 5
(C ) 2
(B) – 2
then the coefficient of x in D(x) is (C ) 6
(D) 0
Q.66
The The set of equati equation onss x – y + (cos ) z = 0 3x + y + 2z =0 (cos)x + y + 2z = 0 0 < < 2 , has non- trivial solution(s) (A) for no value of and (B) for all values of and (C) for all values of and only two values of (D) for only one value of and all values of
Q.67
Matrix Matrix A satisfies satisfies A2 = 2A – I where I is the identity matrix then for n 2, An is equal to (n N) (A) nA – I (B) 2 n – 1 A – (n – 1)I (C) nA – (n – 1)I (D) 2 n – 1A – I
Bansal C lasses
Q. B. on Determinant & Matrices
[10]
Q.68
If a, b, c are are real then the value value of determinant determinant
a2 1
ab
ac
ab
b 2 1
bc
ac (A) a + b + c = 0 Q.69 Q.6 9
(B) a + b + c = 1
bc
c
2
1
(C) a + b + c = –1
= 1 if (D) a = b = c = 0
Read the following following mathematical mathematical statements statements carefully: carefully: There can can exist two triangles triangles such that the sides of one triangle triangle are all less than 1 cm while while the I. sides of the other triangle are all bigger than 10 metres, but the area of the first triangle is larger than the area of second triangle. If x, x , y, z are all different real numbers, then II. 1 ( x y)
2
1 ( y z)
2
1 (z x )
2
1 1 1 = x y y z z x
2
.
log3x · log4x · log5x = (log3x · log4x) + (log4x · log5x) + (log5x · log3x) is true for exactly for one real value of x. IV. IV. A matrix has 12 elements. Number of possible orders it can have is six. Now indicat indicatee the the corre correct ct alter alternati natively vely.. (A) exactly one statement is INCORRECT. INCORRECT. (B) exactly two statements are INCORRECT. (C) exactly three statements are INCORRECT. (D) All the four statements are INCORRECT. INCORRECT. III.
Q.70
Q.71 Q.71
The system system of equatio equations ns (sin)x + 2z = 0, (cos)x + (sin)y = 0 , (cos )y + 2z = a has (A) no unique solution (B) a unique solution which is a function of a and (C) a unique solution which is independent of a and (D) a unique solution which is independent of only
Let A =
1 2 3 0 2 0 5 and b = 3 . Which of the following is true? 0 2 1 1
(A) (A) Ax = b has has a uniq unique ue solu soluti tion on.. (C) Ax = b has infini infinitel tely y many many soluti solution ons. s. Q.72
The number number of of positive positive integral integral solution solutionss of the equatio equation n
x3 1
x2y
xy 2
y3 1
xz 2
yz 2
(A) 0 Q.73
(B) (B) Ax = b has has exac exactl tly y thre threee solu soluti tion ons. s. (D) Ax = b is inconsis inconsistent tent..
x 2z y 2 z = 11 is z3 1 (B) 3
(C ) 6
(D) 12
If A, B and C are n × n matrices and det(A) = 2, det(B) = 3 and det(C) = 5, then the value of the det(A2BC –1) is equal to (A)
6 5
Bansal C lasses
(B)
12 5
(C)
18 5
Q. B. on Determinant & Matrices
(D)
24 5
[11]
(1 x ) 2
Q.74
(1 x ) 2 2x 1 x 1 (2 x 2 ) 2 3x 2 x = 0 + (1 x ) 1 5x 1 2x 3x 2 2x 3 2 3x
(1 x ) 2
The equati equation on 2 x 1
3x
x 1
2x
(A) has no real solution (C) has two real real and two non-real solutions
Q.75
The value of the determinan determinantt (A) 9a2 (a + b)
Q.76
(B) has 4 real solutions (D) has infinite number number of solutions solutions , real or non-real non-real
a
a b
a 2 b
a 2 b
a
a b
a b
a 2 b
a
(B) 9b2 (a + b)
is
(C) 3b2 (a + b)
2 1 3 Let three matrices matrices A = 4 1 ; B = 2
4 3 and C =
(D) 7a2 (a + b)
3 4 then 2 3
A(BC) 2 A(BC) 3 ABC + t + ....... + = + tr tr (A) + tr r 2 4 8 (A) 6 Q.77
(B) 9
3 2
2
2 2 1
=0
(B) 2
P is an orthogonal orthogonal matrix matrix and A is a periodic periodic matrix with period 4 and Q = PAP PAP T then X = PTQ2005P will be equal equal to (A) A (B) A2 (C) A3 (D) A4
If x = a + 2b satisfies satisfies the cubic cubic (a, (a, bR) f (x)=
ax
b
b
b
ax
b
b
b
ax
=0, then its other two roots are
(B) real and coincident (D) such that one is real and other imaginary
A is a 2 × 2 matrix such that A (A) –1
Q.81
(D) 1
is
(A) real and different (C) imaginary Q.80
(C ) 3
1
(A) 0
Q.79
(D) none
The number number of positive positive integral integral solution solutionss
1
Q.78
(C) 12
1 = 1 and A2 1 = 1 . The sum of the elements of A, is 1 2 1 0
(B) 0
(C ) 2
(D) 5
Three digit numbers numbers x17, 3y6 and 12z where where x, y, y, z are integers integers from 0 to 9, are divisible by a fixed
x
3
1
constant k. Then the determinant 7
6
z must be divisible by
1
y
2
(A) k
Bansal C lasses
(B) k2
(C) k 3
Q. B. on Determinant & Matrices
(D) None
[12]
Q.82
Q.83 Q.83
In a square square matrix A of order order 3, a i i's are the the sum of of the roots of the equation x 2 – (a + b)x + ab= 0; ai , i + 1 's are the product of the roots, a i , i – 1 's are all unity and the rest of the elements e lements are all zero. The value of the det. (A) is equal to (A) 0 (B) (a + b) 3 (C) a3 – b3 (D) (a2 + b2)(a + b)
28 25 38 Let N = 42 38 65 , then the number of ways is which N can be resolved as a product of two 56 47 83 divisors which are relatively prime is (A) 4 (B) 8
Q.84
(C ) 9
1 1 sin A If A, B, C are are the the angles angles of a triangle triangle and sin A sin 2 A the triangle is (A) a equilateral (C) a right angled triangle
Q.85 Q.85
(D) 16
Let a = Lim x 1
x l n x
1 x l n x
1 1 1 sin B 1 sin C = 0, then 2 2 sin B sin B sin C sin C
(B) an isosceles (D) any triangle ; b = Lim x 0
x 3 16x 4x x2
; c = Lim
l n (1
x 0
sin x ) x
and
( x 1) 3
a b , then the matrix c d is x 1 3sin( x 1) ( x 1)
d = Lim
(A) Idempotent Q.86
Q.87
Q.88 Q.88
(B) Involutary
If the system system of linear equations equations x + 2ay + az = 0 x + 3by + bz = 0 x + 4cy + cz = 0 has a non-zero solution, then a, b, c (A) are in G..P. (C) satisfy a + 2b + 3c = 0
(C) Non singular
(D) Nilpotent
(B) are in H.P. (D) are in A.P.
Give the correct correct order order of initials initials T or F for following statements. Use T if statement is true and F if it is false. Statement-1 Statement-1 : If the graphs of two linear equations equations in two variables are neither parallel nor the same, then there is a unique solution to the system. Statement-2 : If the system of equations ax + by = 0, cx + dy = 0 has a non-zero solution, then it has infinitely many solutions. Statement-3 : The system x + y + z = 1, x = y, y = 1 + z is inconsistent. Statement-4 : If two of the equations in a system of three linear equations are inconsistent, then the whole system is inconsistent. (A) FFTT (B) TTFT (C) TTFF (D) TTTF
Let Let A =
1 x 2 y 2 z 2 2( xy z) 2(zx y)
(A) (1 + xy + yz + zx) 3 (C) (xy + yz + zx) 3
Bansal C lasses
2( xy z) 1 y2 z 2 x 2 2( yz x )
2( zx y) 2( yz x ) then det. A is equal to 1 z2 x2 y2
(B) (1 + x2 + y2 + z2)3 (D) (1 + x 3 + y3 + z3)2
Q. B. on Determinant & Matrices
[13]
Select the correct correct alternatives : (More than one are correct) correct)
Q.89
The set of equation equationss x – y + 3z = 2 , 2x – y + z = 4 , x – 2y + z = 3 has (A) unique soluton only for = 0 (B) unique solution for 8 (C) infinite number of solutions for = 8 (D) no solution for = 8
Q.90
Suppose a1, a2, ....... real numbers, with a 1 0. If a1, a2, a3, ..........are in A.P. then
La1 M (A) A = Ma 4 MNa 5
a2
a3 O
a5
a 6 is singular P si ngular
P
a 7 PQ
a6
(B) the system of equations a1x + a 2y + a3z = 0, a4x + a 5y + a6z = 0, a7x + a 8y + a9z = 0 has infinite number of solutions (C) B =
L a1 Mia N 2
ia 2 O a1
P Q
is non singular ; where i =
1
(D) none of these Q.91
a2
a 2 ( b c) 2
bc
The The determ determin inant ant b 2
b 2 (c a ) 2
ca is divisible by :
c
2
c
2
(a b) 2 ab
(A) a + b + c (C) a2 + b2 + c2
(B) (a + b) (b + c) (c + a) (D) (a b) (b c) (c a)
Q.92
If A and B are 3 × 3 matrices and | A | 0, then which of the following are true? (A) | AB | = 0 | B | = 0 (B) | AB | = 0 B = 0 –1 –1 (C) | A | = | A | (D) | A + A | = 2 | A |
Q.93
The The valu valuee of lying between
1 sin 2 A
cos 2 A
2 sin 4
sin 2 A
1 cos 2 A
2 sin 4
sin 2 A
cos 2 A
1 2 sin 4
(A) A = (C) A = Q.94
Q.95
4
5
4
&
2
,
=
=
2
(B) A =
8
(D) A =
8
If AB = A and BA = B, then (A) A2B = A2 (B) B2A = B2 a
b
The solution solution(s) (s) of the equation equation a
x
a = 0 is/are :
b
b
x
Bansal C lasses
(B) x = a
3 8
6
= ,
=
3 8
(C) ABA = A
x
(A) x = (a + b)
and satisfying the equation
= 0 are :
,
and 0 A
(C ) x = b
Q. B. on Determinant & Matrices
(D) BAB = B
(D) b
[14]
Q.96 Q.96
Q.97 Q.97
If D1 and D2 are two 3 x 3 diagonal matrices, then (A) D1D2 is a diagonal matrix (B) D1D2 = D2D1 (C) D12 + D22 is a diagonal matr atrix (D) none of these 1
a
a2
1
x
b 2
ab
x 2 = 0 , then a2
If
(A) x = a Q.98 Q.9 8
(B) x = b
(C ) x =
(D) x =
a
a b
Which of the following following determinant(s) determinant(s) vanish(es)? vanish(es)? 1 a (B) 1 b c 1 b 1 ca c1
c) (A) 1 c a c a ( c a ) 1 a b a b ( a b) 1 b c
1 ab
b c (b
b ac 0 bc (C) b a ca cb 0 0
Q.99 Q.99
1
If A =
La Mc N
log x x y z
a
bO
d PQ
(D) logy x y z log z x y z
(B) k = –|A|
Q.100 The value value of lying between = 0 & 2
sin 2 2 sin
(A)
log x y
logx z
1
logy z
log z y
1
(where bc 0) satisfies the equations x2 + k = 0, then
(A) a + d = 0
1sin
1 b 1 c 1 a
2
1cos 2 2 cos cos
(C) k = |A|
(D) none of these
= /2 & satisfying the equation :
4sin4 4sin4
= 0 are :
14sin 4
7
(B)
24
5
(C)
24
11
(D)
24
24
sin x q sin x p r sin x 2 1 sin x such that f (x)dx = – 4 then Q.101 If p, q, r, s are in A.P. A.P. and f (x) = q sin x r sin x 0 r sin x s sin x s q sin x p
the common difference differe nce of the A.P. A.P. can be : (A)
1
Q.102 Let A =
(B)
L1 M2 M MN2
2
2O
2
P 1 PQ
2
(C ) 1
(D) none
P 1 2 , then
(A) A2 – 4A – 5I3 = 0 (C) A3 is not invertible
Bansal C lasses
1
(B) A –1 =
1
(A – 4I3) 5 (D) A2 is invertible
Q. B. on Determinant & Matrices
[15]
ANSWER KEY D , B , A 2 0 1 . Q
C , A 1 0 1 . Q
C , A 0 0 1 . Q
C , A 9 9 . Q
, C , B , A 8 9 . Q D
D , A 7 9 . Q
C , B , A 6 9 . Q
C , B , A 5 9 . Q
, C , B , A 4 9 . Q D
, C , B , A 3 9 . Q D
, A 2 . Q C 9
, C , A 1 . Q D 9
, B , A 0 9 . Q C
, B 9 8 . Q D
B 8 8 . Q
B 7 8 . Q
B 6 8 . Q
D 5 8 . Q
B 4 8 . Q
B 3 8 . Q
D 2 8 . Q
A 1 8 . Q
D 0 8 . Q
B 9 7 . Q
A 8 7 . Q
C 7 7 . Q
A 6 7 . Q
7 . Q B 5
7 . Q D 4
7 . Q B 3
7 . Q B 2
7 . Q A 1
B 0 7 . Q
A 9 6 . Q
D 8 6 . Q
C 7 6 . Q
. Q A 6 6
A 5 6 . Q
D 4 6 . Q
A 3 6 . Q
C 2 6 . Q
D 1 6 . Q
D 0 6 . Q
C 9 5 . Q
B 8 5 . Q
B 7 5 . Q
B 6 5 . Q
5 . Q B 5
5 . Q A 4
5 . Q C 3
5 . Q C 2
5 . Q A 1
5 . Q B 0
4 . Q D 9
4 . Q C 8
4 . Q C 7
4 . Q A 6
B 5 4 . Q
C 4 4 . Q
D 3 4 . Q
C 2 4 . Q
C 1 4 . Q
C 0 4 . Q
A 9 3 . Q
C 8 3 . Q
D 7 3 . Q
B 6 3 . Q
B 5 3 . Q
B 4 3 . Q
C 3 3 . Q
C 2 3 . Q
A 1 3 . Q
3 . Q C 0
2 . Q C 9
2 . Q C 8
2 . Q A 7
2 . Q A 6
2 . Q B 5
. Q B 4 2
A 3 2 . Q
C 2 2 . Q
2 . Q A 1
B 0 2 . Q
B 9 1 . Q
D 8 1 . Q
D 7 1 . Q
D 6 1 . Q
A 5 1 . Q
C 4 1 . Q
C 3 1 . Q
D 2 1 . Q
C 1 1 . Q
1 . Q A 0
C 9 . Q
C 8 . Q
. Q C 7
. Q B 6
. Q D 5
. Q A 4
. Q D 3
. Q A 2
. Q A 1
Bansal C lasses
Q. B. on Determinant & Matrices
[16]
