MATHEMATICS
DETERMINANT AN A N D MATRICES
DPP-1 DPP- 1 .................................................................... ............................................................................................ ........................ Page 2
DPP-2 DPP- 2 .................................................................... ............................................................................................ ........................ Page 5
DPP-3 DPP- 3 .................................................................... ............................................................................................ ........................ Page 7
DPP-4 DPP- 4 .................................................................... ............................................................................................ ........................ Page 9
DPP-5 DPP- 5 .................................................................... ............................................................................................ ........................ Page 11
ANSWER KEY ................................................................ ............................................................................. ............. Page 16
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DPP-1
Q.1
Q.2
[STRAIGHT OBJECTIVE TYPE] The value of a for which the system of equations ; a3x + (a +1)3 y + (a + 2)3 z = 0 , ax + (a + 1) y + (a + 2) z = 0 & x + y + z = 0 has a non-zero solution is : (A) 1 (B) 0 (C) ! 1 (D) none of these a11 Let "0 = a 21 a 32
a12 a 22 a 32
a13 a 23 and let "1 denote the determinant formed by the cofactors of elements of a 33
"0 and "2 denote the determinant formed by the cofactor at "1 similarly "n denotes the determinant formed by the cofactors at "n – 1 then the determinant value of "n is (A) Q.3
"20n
(B)
(C)
cos (y ! z) sin (y # z)
(A) 2 sin (x ! y) sin (y ! z) sin (z ! x) (C) 2 cos (x ! y) cos (y ! z) cos (z ! x) a Let a determinant is given byA = p x
then (A) det. B = 6 Q.6
b q y
(B) det. B = – 6
"20
cos ( z ! x) sin ( z # x)
(B) (D)
! 2 sin (x ! y) sin (y ! z) sin (z ! x) ! 2 cos (x ! y) cos (y ! z) cos (z ! x)
p#x c r and suppose that det. A = 6. If B = a # x z a#p
q#y b#y b#q
r#z c#z c#r
(C) det. B = 12(D) det. B = – 12
The values of $% ' for which the following equations sin$x – cos$y + ('+1)z = 0; cos $x + sin$y – '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.7
(D)
The determinant cos (x # y) cos (y # z) cos ( z # x) = sin (x # y )
Q.5
2
"n0
Three distinct points P(3u2, 2u3) ; Q(3v2, 2v3) and R(3w2, 2w3) are collinear then (A) uv + vw + wu = 0 (B) uv + vw + wu = 3 (C) uv + vw + wu = 2 (D) uv + ww + wu = 1 cos (x ! y)
Q.4
n
"20
) R+, n ) I
(D) $ = (2n + 1)
( 2
,'
) R, n ) I
The system of equations x – y cos $ + z cos 2 $ = 0 – x cos $ + y – z cos $ = 0 x cos 2$ – y cos $ + z = 0 has non trivial solution for $ equals (A) n( only, n ) I (C) (2n – 1)
( 2
only, n ) I
(B) n( +
( 4
only, n ) I
(D) all value of $
Determinant & Matrices
[2]
Q.8
The following system of equations 3x – 7y + 5z = 3; 3x + y + 5z = 7 and 2x + 3y + 5z = 5 are (A) consistent with trivial solution (B) consistent with unique non trivial solution (C) consistent with infinite solution (D) inconsistent with no solution
Q.9
The set of homogeneous equations tx + (t+1) y + (t – 1) z = 0 (t +1)x + ty + (t + 2)z = 0 (t – 1)x + (t + 2)y + tz = 0 has non - trivial solutions for (A) three values of t (B) two values of t
Q.10
Q.11
If the system of equations x – 2y + z = a 2x + y – 2z = b and x + 3y – 3z = c have atleast one solution, then the relationship between a, b and c is (A) a + b + c = 0 (B) a – b + c = 0 (C) – a + b + c = 0
Q.13
Q.14
(D) no value of t
(D) a + b – c = 0
Three digit numbers x17, 3y6 and 12z where x, y, z are 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 Q.12
(C) one value of t
(B) k2
(C) k 3
(D) None
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
(C) 9
If the system of linear 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
(B) are in H.P. (D) are in A.P.
(D) 16
Give the correct order of initials T or F for following statements. Use T if statement is true and F if it is false. Statement-1 : If the graphs of two linear equations in two variables are neither parallel nor identical, 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
2
Q.15
ap 2 Let "1 = aq ar 2
(A)
Q.16
2ap 1 apq 2aq 1 and "2 = aqr arp 2ar 1
"1 = "2
(B)
"2 = 2"1
a (p # q) 1 a (q # r ) 1 then a ( r # p) 1
(C)
"1 = 2"2
1 1 # sin x 1 # sin x # cos x Let F (x) = 2 3 # 2 sin x 4 # 3 sin x # 2 cos x then F ' 3 6 # 3 sin x 10 # 6 sin x # 3 cos x (A) – 1
(B) 0
(D) "1 + 2 "2 = 0
/ ( , - * is equal to . 2 +
(C) 1
(D) 2
[MULTIPLE OBJECTIVE TYPE] Q.17
If A11 , A12 , A13 are the cofactors of the elements of the first row of the determinant A = (aij) and " is the value of the determinant then (A) a21 A11 + a22 A12 + a23 A13 = 0 (B) a11 A11 + a 12 A12 + a 13 A13 = " (C) a11 A11 – a12 A12 + a 13 A13 = " (D) a31A11 + a32A12 + a33A13 = 0
# sin x q # sin x r # sin x p
Q.18
If p, q, r, s are in A.P.
and
f (x) =
# sin x r # sin x s # sin x
q
! r # sin x ! 1 # sin x s ! q # sin x p
such that
2
0
f (x)d x = – 4 then the common difference of the A.P. can be :
0
(A)
Q.19
! 1
Let D1 =
(B) b#c c#a a#b
a c b
1
(C) 1
2
b a and D2 = c
(D) 2
( b # c) (c # a ) 2
2
a b2
2
bc ca
(a # b ) 2
c2
ab
The divisor which is common to both D1 and D2 is (A) (a – b) (B) (ab + bc + ca) (C) a + b + c
(D) (c – a)
[INTEGER TYPE] sin 3$ Q.20
If cos 2$ 2
!1
1
4
3 = 0, then find the number of values of $ in [0, 2(].
7
7
Determinant & Matrices
[4]
DPP-2
[STRAIGHT OBJECTIVE TYPE] Q.1
Q.2
If the product of n matrices 61 13 450 112 then the value of n is equal to (A) 26 (B) 27
L1 M0 If A = M MN0
xO
2 1
0P
0
1PQ
P and B =
L3 4 O If A = M P and B = N1 !6Q L2 (A) M N !1
Q.4
Q.5
23 61 33 .......... 61 1 12 450 112 450
L1 ! 2 M0 1 M MN0 0
3O
P
(D) 378
yO 0P
P and AB = I3, then x + y equals
1PQ
(C) 2
(D) none of these
5O
P then X such that A + 2X = B equals
1Q
L3 (B) M N !1
0Q
6a If A = 4c 5
L !2 M6 N
n 3 is equal to the matrix 61 3783 450 1 12 1 12
(C) 377
(B) – 1
(A) 0
Q.3
61 450
5O
P
0Q
(C)
L5 M !1 N
2O
P
(D) none of these
0Q
b3 2 d 12 satisfies the equation x – (a + d)x + k = 0, then bc (D) k = a2 + b2 + c2 + d2 (C) k = ad –
(A) k = bc
(B) k = ad
6 cos 2 7 If A = 4 5 sin 7 cos 7
sin 7 cos 7 3 sin 2 7
6 cos 2 8 1 ; B = 4 2 5 sin 8 cos 8
sin 8 cos 8 3 sin 2 8
1 2
are such that, AB is a null matrix, then which of the following should necessarily be an odd integral multiple of (A) 7
Q.6
( 2
.
Let A + 2B =
(B) 8
61 2 4 6 !3 45! 5 3
(C) 7 – 8
03 62 31 and 2A – B = 42 450 112
then Tr (A) – Tr (B) has the value equal to (A) 0 (B) 1 Q.7
Q.8
!1 !1 1
(D) 7 + 8 53 61 212
(C) 2
The number of solutions of the matrix equation X2 =I other than I, is (A) 0 (B) 1 (C) 2 (where I is the 2 × 2 unit matrix )
61 Given A = 42 5 (A) '
)9
33 61 212 ; I = 450
(D) none
(D) more than 2
03 112 . If A – 'I is a singular matrix then
(B) '2 – 3' – 4 = 0
(C) '2 + 3' + 4 = 0
Determinant & Matrices
(D) '2 – 3' – 6 = 0
[5]
Q.9
Let A =
sin $ 6 1 1 4! sin $ 45 ! 1 ! sin $
(A) Det (A) = 0
Q.10
61 If A = 42 5
(C) Det (A) ) [2, 4]
(B) 2
A=
(D) Det A ) [2, ;)
6 1 45! tan x
(D) – 4
(C) 1
63 ! x 4 A= 4 2 45 ! 2
Number of real values of x for which the matrix
(A) 1
Q.12
(B) Det A ) (0, ;)
23 2 312 , and A – kA – I2 = 0, then value of k is
(A) 4
Q.11
1 3 sin $1 , where 0 : $ < 2(, then 1 12
(B) 2
2 4!x
!4
(C) 3
3 1 1 1 is singular, is ! 1 ! x 12 2
(D) infinite
tan x 3 T – 1 1 12 then let us define a function f (x) = det. (A A ) then which of the following can
not be the value of f
2) $!! ! #!! ! "
n times
(A) f n(x)
Q.13
A is a 2 × 2 matrix such that A (A) – 1
(C) f n – 1(x)
(B) 1
613 45! 112
=
(D) n f (x)
6! 13 61 3 26 1 3 and A = 45 2 12 45! 112 45012 . The sum of the elements of A, is
(B) 0
(C) 2
(D) 5
Q.14
In a square matrix A of order 3 the elements, ai i's are the sum of the roots of the equation x2 – (a + b)x + ab = 0; ai , i + 1 's are the product of the roots, ai , i – 1's are all unity and the rest of the elements 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)
Q.15
Let Dk is the k × k matrix with 0's in the main diagonal, unity as the element of 1 st row and < f (k ) =th column and k for all other entries. If f (x) = x – {x} where {x} denotes the fractional part function then the value of det. (D2) + det. (D3) equals (A) 32 (B) 34 (C) 36 (D) none
Q.16
61 For a matrix A = 40 5
2r ! 13 1 12 , the value of
50
? r @1
61 1003 (A) 40 1 1 5 2
61 (B) 40 5
4950 3 1 12
61 450
2r ! 13 is equal to 1 12
61 50503 (C) 40 1 12 5
Determinant & Matrices
61 (D) 40 5
25003 1 12
[6]
DPP-3
Q.1
60 1 ! 13 A is an involutary matrix given by A = 44 ! 3 4 1 then the inverse of 453 ! 3 4 12 (A) 2A
Q.2
(B)
A
!1
A
(C)
2
62 13 63 Let three matrices A = 4 4 11 ; B = 42 5 2 5
A 2
will be
(D) A2
2
43 312 and C =
6 3 ! 43 45! 2 3 12 then
/ A(BC) 2 , / A(BC)3 , / ABC , * + t * + ....... + ; = * + tr -tr(A) + tr * * r4 8 2 . + . + . + (A) 6
Q.3
(B) 9
Let a = Lim x A1
x ln x
!
1 x ln x
(C) 12
; b = Lim x A0
x 3 ! 16x 4x # x
2
; c = Lim
6a , then the matrix 4c x A !1 3
(A) Idempotent
(B) Involutary
ln (1 # sin x )
x A0
( x # 1)3
d = Lim
(D) none
x
and
b3 d 12 is
(C) Non singular
(D) Nilpotent
Q.4
IfA is an idempotent matrix satisfying, (I – 0.4A) – 1 = I – 7A where I is the unit matrix of the same order as that of A then the value of 7 is equal to (A) 2/5 (B) 2/3 (C) – 2/3 (D) 1/2
Q.5
Consider a matrix A =
63 13 99 45! 6 ! 212 , then (I + A) equals (where I is a unit matrix of order 2)
(A) I + 298A
(B) I + 299A
Q.6
Q.7
(C) I + (299 + 1)A
(D) I + (299 – 1)A
If A is a diagonal matrix of order 3 such that A2 = A, then number of possible matrices A, is (A) 2 (B) 3 (C) 8 (D) 7 Let A =
62 0 40 1 41 ! 2 5
73 01 and B = 112
6! 7 147 77 3 40 1 0 1 . If AB = I, where 4 7 ! 47 ! 27 1 5 2
I is an identity matrix
of order 3 then trace B has value equal to (A) 0 Q.8
(B)
2
(C)
5
1 5
63 2 3 63 and B = 452 7 12 457 then the number of distinct possible real values of 7 equals
Let the matrix A and B be defined as A =
(A) 0
(B) 1
(C) 2
Determinant & Matrices
(D) 5 13 . If det(2A9 B – 1) = – 2, 312
(D) 3
[7]
[COMPREHENSION TYPE] Paragraph for question nos. 9 to 11 A Pythagorean triple is triplet of positive integers (a, b, c) such that a2 + b2 = c2. Define the matrices A, B and C by
A=
61 42 452
2 1 2
33 21 , B = 312
61 2 23 4! 2 ! 1 ! 21 and C = 45 2 2 3 12
6! 1 ! 2 ! 23 42 1 21 45 2 2 3 12
Q.9
If we write Pythagorean triples (a, b, c) in matrix form as [a, b, c] then which of the following matrix product is not a Pythagorean triplet? (A) [3,4,5]A (B) [3,4,5]B (C) [3,4,5]C (D) None of these
Q.10
Which one of the following does not hold good? (A) A – 1 = adj. A (B) (AB) – 1 = adj. (AB) (C) (BC) – 1 = adj. (BC) (D) (ABC) – 1 B adj. (ABC)
Q.11
Tr(A + BT + 3C) equals (A) 17 (B) 15
(C) 19
(D) 18
Paragraph for question nos. 12 to 14 If A is a symmetric and B skew symmetric matrix and A + B is non singular and C = (A + B) – 1(A – B) then
Q.12
Q.13
Q.14
Q.15
CT(A + B)C = (A) A + B
(B) A – B
(C) A
(D) B
CT(A – B)C = (A) A + B
(B) A – B
(C) A
(D) B
CTAC (A) A + B
(B) A – B
(C) A
(D) B
[MULTIPLE OBJECTIVE TYPE] Let A and B are two square idempotent matrices such that AB ± BA in a null matrix, then the value of the det. (A – B) can be equal (A) – 1 (B) 1 (C) 0 (D) 2
Determinant & Matrices
[8]
DPP-4
Q.1
Q.2
[STRAIGHT OBJECTIVE TYPE] If A and B are non singular Matrices of same order then Adj. (AB) is (A) Adj. (A) (Adj. B) (B) (Adj. B) (Adj. A) (C) Adj. A + Adj. B (D) none of these
Lx # ' M x Let A = M MN x
O P x#' x P , then A – 1 exists if x x # ' PQ x
x
(A) x B 0 (C) 3x + ' B 0, ' B 0
(B) ' B 0 (D) x B 0, ' B 0
B 0)
Q.3
Which of the following statements is incorrect for a 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, A – 1 will also be a symmetric matrix (C) If A – 1 = A C A is an idempotent matrix (D) If A – 1 = A C A is an involutary matrix
Q.4
Identify the correct statement : (A) If system of n simultaneous linear equations has a unique solution, then coefficient matrix is singular (B) If system 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) =
Q.5
If A =
1
0 , then F(x) . F(y) = F(x – y) 1 012
43 41 , then A – 1 = 112 (C) A3
(D) A4
60 2 b c 3 If A = 4a b ! c1 is orthogonal, then | abc | is equal to 4a ! b c 1 5 2 (A)
Q.7
63 ! 3 42 ! 3 450 ! 1
03
(B) A2
(A) A Q.6
6cos x ! sin x 4 sin x cos x 4 45 0 0
1 2
(B)
1
(C)
3
Let A = [aij]3 × 3 be such that aij = then
1 6
(D) 1
F3, when i @ j E 0, otherwise D
F det
[Note : {k} denotes fractional part of k.] (A)
2 3
(B)
1 5
(C)
2 5
Determinant & Matrices
(D)
1 3
[9]
Q.8
Let A, B, C, D be (not necessarily square) real matrices such that AT = BCD; BT = CDA; CT = DAB and DT = ABC for the matrix S = ABCD, consider the two statements. I II S3 = S S2 = S 4 (A) II is true but not I (B) I is true but not II (C) both I and II are true (D) both I and II are false.
Q.9
If A is a non singular maxrix satisfying A = AB – BA , then which one of the following holds true (A) det. B = 0 (B) B = 0 (C) det. A = 1 (D) det. (B + I) = det. (B – I)
Q.10
Q.11
0 sin 7 F G ! sin 7 0 Let A = G H ! sin 7 sin 8 ! cos 7 cos8
sin 7 sin 8 I
cos 7 cos8J J , then 0 K
(A) |A| is independent of 7 and 8 (C) A – 1 depends only on 8
(B) A – 1 depends only on 7 (D) none of these
64 6 ! 13 2 1 ,B= Consider the matrices A = 43 0 451 ! 2 5 12
62 40 45! 1
(i) (AB)TC (ii) CTC(AB)T (A) exactly one is defined (C) exactly three are defined
(iii) CTAB and (iv) ATABBTC (B) exactly two are defined (D) all four are defined
6 33 43 1 1 , C = 411 . Out of the given matrix products 45212 212
[PARAGRAPH TYPE] Paragraph for question nos. 12 to 14 Let S be the following set of 2 × 2 matrices : S =
F 6a E A @ 4a 5 D
I b3 J K : a , b 1 , 0 , 1 ) ! H b12 G
Q.12
The number of A in S such that the trace of A is divisible by 2 but det ( A ) is not divisible by 2, is (A) 0 (B) 5 (C) 3 (D) 2 [ Note : The trace of a matrix is the sum of its diagonal entries.]
Q.13
The number of non-zero A in S for which the system of linear equations A
6 x 3 @ 60 3 45 y 12 45012
is inconsistent, is (A) atleast 2 but less than 5 (C) exactly 8 Q.14
Q.15
The number of A in S such that A is either symmetric or skew-symmetric but not both, is (A) 0 (B) 1 (C) 2 (D) 3
[INTEGER TYPE] Find the number of 2 × 2 matrices A whose entries are either 0 or 1 and for which the system A
Q.16
(B) less than 2 (D) greater than 4 but at most 7
6 x 3 @ 603 has at least two distinct solutions. 45 y 12 45012
Let A =
6 cos 7 45! sin 7
sin 7 3 and matrix B is defined such that B = A + 4A2 + 6A3 + 4A4 + A5. cos 7 12
If det (B) = 1, then find the number of values of
7
in [ – 2(, 2(].
Determinant & Matrices
[10]
DPP-5 [
Q.1
Q.2
Let 7, 8, L are the real roots of the equation x3 + ax2 + bx + c = 0 (a, b, c ) R and a B 0). If the system of equations (in u, v and w) given by 7u + 8v + Lw = 0 8u + Lv + 7w = 0 Lu + 7v + 8w = 0 has non-trivial solutions, then a2 equals (A) b (B) 2b (C) 3b (D) 4b
Let A =
6l 4p 41 5
m q 1
n3 r 1 and B = A 2. 1 12
If (l – m)2 + (p – q)2 = 9, (m – n)2 + (q – r)2 = 16, (n – l)2 + (r – p)2 = 25, then the value of det. B equals (A) 36 (B) 100 (C) 144 (D) 169 Q.3
M i = 1, 2, 3, 4 is a diagonal matrix of order 4 such that / 0, ( , * d1 + 2d2 + 4d3 + 8d4 = 16 then the maximum value of f(x) = log(tan x + cot x)(det(A)) where x ) . 2 +
If A = dia. (d1, d2, d3, d4) where di > 0
is equal to (A) 1 Q.4
(A)
N@ "
i
i
@0
1
Let A =
(A)
Q.6
(C) 3
(D) 4
Let {"1, "2, ........., "n} be the the set of all determinants of order 3 that can be made with the distinct real numbers from the set S = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Then which one of the following is correct? n
Q.5
(B) 2
(B)
N@ "
i
i
@9
n
(C)
1
N@ "
i
i
@ 9!
n
(D)
1
N@ "
i
i
@ 36
1
68 ! 13 and det (A4) = 16, then the product of all possible real values of 8 equals 451 28 12
1
(B)
!1
(C) 0 2 [ASSERTION-REASON]
2
Let A =
n
(D) 2
6 cos $ ! sin $ 3 45! sin $ ! cos $12
Statement-1:
A – 1 exists for every $ ) R.
because
Statement-2: A is orthogonal. (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1. (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true.
Determinant & Matrices
[11]
Q.7
Statement-1: If
Q.8
Let A be a 2 × 2 matrix with non-zero entries such that A2 = I, where I is a 2 × 2 identity matrix. Define Tr(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 and statement-2 is correct explanation for statement-1.
A and B are 2 × 2 matrices such that det. (A – B) = 0, then A = B. Statement-2: If A and B are square matrices of same order such that AB = O and B is not a null matrix, then A must be singular. (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1. (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true.
(B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true.
[PARAGRAPH TYPE] Paragraph for question nos. 9 to 11
6 2 ! 2 ! 43 6! 4 ! 3 ! 33 4 1 and B0 = 4 1 0 11 If A0 = 4! 1 3 45 1 ! 2 ! 312 45 4 4 3 12 Bn = adj(Bn – 1), n ) N and I is an identity matrix of order 3 then answer the following questions. Q.9
2
2
3
Q.11
4
(B) – 800
(A) 1000 Q.10
4
det. (A0 + A 0 B0 + A 0 + A 0 B0 + ....... 10 terms) is equal to
B1 + B2 + ........ + B 49 is equal to (A) B0 (B) 7B0
(C) 0
(D) – 8000
(C) 49B0
(D) 49I
For a variable matrix X the equation A0X = B0 will have (A) unique solution (B) infinite solution (C) finitely many solution (D) no solution Paragraph for question nos. 12 to 14
Consider N =
77 7
44 30
4
=
?@ (a ) i
i
1
bi
where b i ' s
) N
and a i 's are prime numbers and
ai < ai + 1 M i. Let two circles be defined as
O
C1 : x 2 and
y
O
2
C2 :
Q.12
The value
N@
i
(A) 23
2
i
6 a1 3 4b # b 1 = 5 1 22
P
[8b1]
6b 3
# bi =
[ 2b 2 ]
is equal to
1
(B) 33
(C) 25
Determinant & Matrices
(D) 37 [12]
Q.13
Radius of the circle which passes through centre of the circle C2 and touches the circle C1, is (A) 1
Q.14
(B)
1
(C)
2
3 2
(D)
1 2
If the line x + 'y = 1 + k ' always passes through a fixed point on the circle C1 for all values of ' then length of the tangent from the point 1, 3 k to the circle C2, is (B) k2
(A) k
(C) 2k
(D) 2k2
[MULTIPLE OBJECTIVE TYPE] Paragraph for question nos. 15 to 18 Consider the matrix function
A(x) =
6 cos!1 x 4 sin !1 x 4 !1 45 cosec x
sin ! x 1 sec! x 1 tan ! x 1
cosec ! x 3 1 tan ! x 1 1 1 cot ! x 1 1
2
and B = A – 1. Also, det.(A(x)) denotes the determinant of square matrix A(x). Q.15
Q.16
Which of the following statement(s) is(are) correct? (A) A( – x) = A(x). (B) A(x) + A( – x) = ( I3 (C) A( – x) = – A(x). (D) A(x) + A( – x) = – (I 3 [Note : I3 denotes an identity matrix of order 3.] Which of the following statement(s) is(are) correct? (A) A(x) is a symmetric matrix. (B) A(x) is a skew symmetric matrix.
(C) Maximum value of det.(A(x)) equals (D) Minimum value of det.(A(x)) equals
(3 8
(3 16
. .
Q.17
Which of the following statement(s) is(are) correct? (A) det.(A(x)) is a continuous function in its domain but not differentiable in its domain. (B) det.(A(x)) is a continuous and differentiable function in its domain. (C) det.(A(x)) is a bounded function. (D) det.(A(x)) is one-one and odd function.
Q.18
Which of the following statement(s) is(are) correct? (A) If a = det. (B) + det. (B2) + det. (B3) + ....... ;, then minimum value of a equals (B) If b = det. adj. (B) + det. adj. (B2) + det. adj. (B3) + ....... 256 . 6 ( ! 256
8
(3 ! 8
;, then maximum value of
(C) If a = det. (B) + det. (B2) + det. (B3) + ....... ;, then maximum value of a equals
16
(3 ! 16
(D) If b = det. adj. (B) + det. adj. (B2) + det. adj. (B3) + ....... ;, then minimum value of b is [Note: adj. (P) denotes adjoint of square Determinant matrix P.] & Matrices
. b is
. 64
(6 ! 64
.
[13]
Q.19
Which of the following is(are) correct? (A) If A and B are two square matrices of order 3 and A is a non-singular matrix such that AB = O, then B must be a null matrix. (B) If A, B, C are three square matrices of order 2 and det. (A) = 2, det.(B) = 3, det. (C) = 4, then the value of det. (3ABC) is 216. 1 (C) If A is a square matrix of order 3 and det. (A) = , then det. (adj. A – 1) is 8. 2 (D) Every skew symmetric matrix is singular.
Q.20
Let A =
L1 M1 N
(A) A – n =
(C) Limit nA;
Q.21
If A – 1 =
0O
P then
1Q
L1 M! n N 1 2
n
!n
0O
1 !n A = (B) Limit nA; n
V n ) N 1PQ
A =
L0 M0 N
0O 0PQ
L0 M !1 N
0O 0PQ
(D) none of these
L1 ! 1 0 O M0 ! 2 1 P M P , then MN0 0 !1PQ
(A) |A| = 2
(C) Adj. A =
(B) A is non-singular
L1 / 2 !1 / 2 0 O M 0 1 1 / 2 P ! M P MN 0 0 !1 / 2PQ
(D) A is skew symmetric matrix
Q.22
If A and B are two 3 × 3 matrices such that their product AB is a null matrix then C (A) det. A B 0 B must be a null matrix. C (B) det. B B 0 A must be a null matrix. (C) If none of A and B are null matrices then atleast one of the two matrices must be singular. (D) If neither det. A nor det. B is zero then the given statement is not possible.
Q.23
Which of the following statement(s) is/are CORRECT? (A) Every skew-symmetric matrix is non-invertible. (B) If A and B are two 3 × 3 matrices such that AB = O then alteast one of A and B must be null matrix. (C) If the minimum number of cyphers in an upper triangular matrix of order n is 5050, then the order of matrix is 101. (D) If A and B are two square matrices of order 3 such that det. A = 5 and det. B = 2, then det. (10AB) equals 104.
Determinant & Matrices
[14]
Q.24
Which of the following statement(s) is(are) correct? (A) If A is square matrix of order 3, then
T 2011
is equal to 0.
(B) If A is a skew - symmetric matrix of order 3, then matrix A4 is symmetric. (C) If
/ 1 3A = - 2 -x .
2 , ! 2 ** and AAT = I, then (x + y) is equal to – 3. y +*
2 1 2
(where I is identity matrix of order 3) (D) If 7, 8, L are the roots of the cubic x + px + q = 0, then the value of the determinant 3
2
is equal to – p3.
7 8 L 8 L 7 L 7 8
[INTEGER TYPE] Q.25
Q.26
Let A be 3 × 3 matrix given by A = [a ij] and B be a column vector such that BTAB is a null matrix for every column vector B. If C = A – AT and a13 = 1, a23 = – 5, a21 = 15, then find the value of det (adj A) + det (adj C). [Note : adj M denotes the adjoint of a square matrix M.]
If 7 and 8 are roots of the equation O1
60 4 25P4 4! 1 5
13 21 11
5
61 42 1 5 22
6 10 0 ! 13 4 0 12 4 4! 1 5
13
5
21 11
6 x 2 ! 5x # 203 = [40] 4 x#2 1 2 1 5 22
then find the value of (1 – 7) (1 – 8).
Q.27
If t is real and
'
=
2
! 3t # 4 , 2 t # 3t # 4 t
then find number of solutions of the system of equations
3x – y + 4z = 3, x + 2y – 3z = – 2, 6x + 5y + 'z = – 3 for a particular value of '.
Determinant & Matrices
[15]
ANSWER KEY DPP-1 Q.1 Q.6 Q.11 Q.16
C D A B
Q.2 Q.7 Q.12 Q.17
B D B ABD
Q.3 Q.8 Q.13 Q.18
A B B AC
Q.4 Q.9 Q.14 Q.19
B C B CD
Q.5 Q.10 Q.15 Q.20
C B D 5
Q.4 Q.9 Q.14
C C D
Q.5 Q.10 Q.15
C A B
Q.4 Q.9 Q.14
C A C
Q.5 Q.10 Q.15
D D ABC
Q.4 Q.9 Q.14
B D C
Q.5 Q.10 Q.15
C A 10
Q.4 Q.9 Q.14 Q.19 Q.24
A C B AB ABC
Q.5 Q.10 Q.15 Q.20 Q.25
B C B ABC 0
DPP-2 Q.1 Q.6 Q.11 Q.16
B C B D
Q.2 Q.7 Q.12
A D D
Q.3 Q.8 Q.13
D B D
DPP-3 Q.1 Q.6 Q.11
A D A
Q.2 Q.7 Q.12
A B A
Q.3 Q.8 Q.13
D B B
DPP-4 Q.1 Q.6 Q.11
B C C
Q.16
4
Q.2 Q.7 Q.12
C B A
Q.3 Q.8 Q.13
C C B
DPP-5 Q.1 Q.6 Q.11 Q.16 Q.21 Q.26
C A D ACD BC 51
Q.2 Q.7 Q.12 Q.17 Q.22 Q.27
C D D AC ABCD 1
Q.3 Q.8 Q.13 Q.18 Q.23
B C A ABCD CD
