Baixe 65878675 - Determinants - Qns e outras Notas de estudo em PDF para Matemática, somente na Docsity!
QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
2 2 2
a a b c b b a c c c a b
(A) 0 (B) a^3 + b^3 + c^3 − 3 abc (C) 3 abc (D) (a + b + c)^3
2. The following system of equations, 3x − 2y + z = 0, λ x − 14y + 15z = 0, x + 2y - 3z = 0 has a solution other than, x = y = z = 0 for λ equal to : (A) 1 (B) 2 (C) 3 (D) 5 3. The roots of the equation,
1 4 20 1 2 5 1 2 5 2
x x
= 0 are :
(A) - 1, - 2 (B) - 1, 2
(C) 1, - 2 (D) 1, 2
4. If
x a x b x a x c x b x c
= 0, then the
value of x is : (A) 0 (B) 1 (C) 2 (D) 3
5. If ω is the cube root of unity, then
1 1 1
2 2 2
ω ω ω ω ω ω
(A) 1 (B) 0
(C) ω (D) ω^2
6. If
x x x
= 0, then x =
(A) 1, 9 (B) - 1, 9
(C) - 1, - 9 (D) 1, - 9
7. The value of the determinant, 7 9 79 4 1 41 5 5 55
is :
(A) - 7 (B) 0
(C) 15 (D) 27
a b c a a b b c a b c c c a b
(A) (a + b + c)^2 (B) (a + b + c)^3 (C) (a + b + c) (ab + bc + ca) (D) None of these
a b a b a b a b a b a b a b a b a b
(A) 3 (a + b) (B) 3 ab (C) 3a + 5b (D) 0
b c a a b c a b c c a b
(A) abc (B) 2 abc (C) 3 abc (D) 4 abc
11. One of the roots of the given equation
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
x a b c b x c a c a a b
= 0, is :
(A) - (a + b) (B) - (b + c) (C) - a (D) - (a + b + c)
12. If 2x + 3y - 5z = 7, x + y + z = 6, 3x - 4y + 2z = 1, then x =
(A)
(B)
(C)
13. x + ky - z = 0, 3x - ky - z = 0 and x - 3y + z = 0 has non-zero solution for k = (A) - 1 (B) 0 (C) 1 (D) 2 14. If ∆ =
a b c a b c a b c
1 1 1 2 2 2 3 3 3
and A 1 , B 1 , C 1
denote the co-factors of a 1 , b 1 , c 1 respectively, then the value of the
determinant,
A B C
A B C
A B C
1 1 1 2 2 2 3 3 3
is :
(A) ∆ (B) ∆^2
(C) ∆^3 (D) 0
15. The number of solutions of equations x + y - z = 0, 3x - y - z = 0 and x - 3y + z = 0 is : (A) 0 (B) 1 (C) 2 (D) Infinite 16. The number of solutions of the equations, x + 4y - z = 0, 3x − 4y − z = 0 and x − 3y + z = 0 is (A) 0 (B) 1 (C) 2 (D) Infinite
b c a b a c a b c b a b c a c
(A) a^3 + b^3 + c^3 - 3 abc (B) 3 abc - a^3 - b^3 - c^3 (C) a^3 + b^3 + c^3 - a^2 b - b^2 c - c^2 a (D) (a + b + c) (a^2 + b^2 + c^2 + ab + bc + ca)
18. If x = cy + bz, y = az + cx, z = bx + ay, where x, y, z are not all zero, then : (A) a^2 + b^2 + c^2 - 2 abc = 0 (B) a^2 + b^2 + c^2 + 2 abc = 0 (C) a^2 + b^2 + c^2 + 2 abc = 1 (D) a^2 + b^2 + c^2 - 2 abc = 1 19. If ω is a cube root of unity, then a root of the following equation, x x x
2 2 2
ω ω ω ω ω ω
= 0, is :
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
(A)
(B)
(C)
(D)
i i i
31. If AB = C, then matrices A, B, C are (A) A 2 × 3 , B 3 × 2 , C 2 × 3 (B) A 3 × 2 , B 2 × 3 , C 3 × 2 (C) A 3 × 3 , B 2 × 3 , C 3 × 3 (D) A 3 × 2 , B 2 × 3 , C 3 × 3 32. If A =
and^ I =^
, then A^2 - 6A = (A) 3 I (B) 5 I (C) - 5 I (D) None of these
33. If A and B are non-singular matrices, then : (A) (AB) -1^ = A -1^ B - (B) AB = BA (C) (AB) ′ = A ′ B ′ (D) (AB) -1^ = B -1^ A - 34. If A =
λ λ
, then for what value of λ , A^2 = O (A) 0 (B) ± 1 (C) - 1 (D) 1
35. If A =
, then :
(A) A ′ = A (B) A ′ = - A
(C) A ′ = 2 A (D) None of these
36. If A, B, C are three square matrices such that AB = AC imples B = C, then the matrix A is always :
(A) A singular matrix (B) A non-singular matrix (C) A orthogonal matrix (D) A diagonal matrix
37. If A =
x −
, B =
and AB = I, then x = (A) - 1 (B) 1 (C) 0 (D) 2
38. The matrix A =
1 2
1 2 1 2
1 − − 2
is :
(A) Unitary (B) Orthogonal (C) Nilpotent (D) Involutary
39. From the following find the correct relation. (A) AB ′ = A′^ B′ (B) AB ′ = B′^ A′
(C) A -1^ = adj A A (D) (AB) -1^ = A -1^ B -
40. If A is a square matrix of order 3, then the true statement is (where I is unit matrix). (A) det (- A) = - det A (B) det A = 0 (C) det (A + I) = 1 + det A (D) det 2A = 2 det A 41. If k is a scalar and I is a unit matrix of order 3, then adj. (k I) = (A) k^3 I (B) k^2 I (C) - k^3 I (D) - k^2 I 42. If A = (1, 2, 3) & B =
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
then AB =
(A)
(B)
(C) [ − 2 − 1 4 ] (D)
43. If A =
i i
( i^ =^ −1 , then A)^ −^1 =
(A)
i i
(B)^
i i
(C)
i i
^
^
(D)
i i 2 i 0
44. If A =
, then AA ′ =
(A) 14 (B)
(C)
(D) None of these
45. If A =
, then A^2 =
(A) Null matrix (B) Unit matrix (C) A (D) 2 A
46. Adjoint of the matrix,
N =
^ −^ −^ −
is :
(A) N (B) 2 N
(C) - N (D) None of these
47. If A is a symmetric matrix, then matrix M ′AM is : (A) Symmetric (B) Slew-symmetric (C) Hermitian (D) Skew-Hermitial 48. If A =
, then A^2 =
(A) 5 A (B) 10 A
(C) 16 A (D) 32 A
49. If A =
and AB = O, then B =
(A)
(B)^
(C)
^ −
(D)^
50. If A and B are square matrices of order 2, then (A + B)^2 = (A) A^2 + 2 AB + B^2 (B) A^2 + AB + BA + B^2 (C) A^2 + 2 BA + B^2 (D) None of these 51. If A = a c d b
, then^ A^ -1^ =
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
A adj. A =
k k
then k is equal to (A) 0 (B) 1 (C) sin α cos α (D) cos 2 α
61. If A =
a ^
^
, then A^4 is equal to :
(A)
a^4
(B)^
a
(C) 4
a^4
(D)^
a
62. The inverse of
is :
(A) −^1
(B)^
(C)
^
^
(D)
^
63. The value of the determinant,
a a nx n x n x nx n x n x
cos( ) cos( ) cos( ) sin ( ) sin ( ) sin ( )
is independent of : (A) n (B) a (C) x (D) None of these
64. The determinant,
a b a b b c b c a b b c
α α α α
if a, b, c are in :
(A) A.P. (B) G.P.
(C) H.P. (D) None of these
65. Let A =
, B =
and C = [ 3 1 2 ]. The expression
which is not definedis : (A) B ′ B (B) CAB (C) A + B ′ (D) A^2 + A
66. Let A =
, then the adjoint
of A is :
(A)
(B)
(C)
(D) None of these
67. If ∆ 1 =
x b b a x b a a x
and ∆ 1 =
x b a x are
given determinants, then :
(A) ∆ 1 = 3 (∆ 2 )^2 (B) d d x
(C)
d d x
(∆ 1 ) = 2 (∆ 2 )^2
(D) ∆ 1 = 3 ∆ 2 3 2/
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
68. If A =
, then A^ (adj.) A =
(A)
(B)^
(C)
(D)^ None of these
69. If a 1 x + b 1 y + c 1 z = 0, a 2 x + b 2 y + c 2 z = 0,
a 3 x + b 3 y + c 3 z = 0 and
a b c a b c a b c
1 1 1 2 2 2 3 3 3
then the given system has : (A) One trivial and one non-trivial solution (B) No solution (C) One solution (D) Infinite solution
70. If A = [ a b] , B = [ − b −a ] and
C =
a −a
^
^
, then the correct statement
is : (A) A = - B (B) A + B = A - B (C) AC = BC (D) CA = CB
71. For positive numbers x, y and z, the numerical value of the determinant, 1 1 1
log log log log log log
x x y y z z
y z x z x y
is :
(A) 0 (B) 1
(C) loge xyz (D) None of these
72. If ∆ =
a b c x y z p q r
, then
k a k b k c k x k y k z k p k q k r is equal to : (A) ∆ (B) k ∆ (C) 3 k ∆ (D) k^3 ∆
73. The inverse of the matrix 3 2 1 4
is
(A)
4 14
2 14 1 14
3 14
−
(B)
3 14
2 14 1 14
4 14
^ −
(C)
4 14
2 14 1 14
3 14
^ −
(D)
3 14
2 14 1 14
3 14
74. The value of n
N
1
Un, if
Un =
n n N N n N N
2 3 2
(A) 0 (B) 1
(C) - 1 (D) None of these
75. Matrix A is such that A^2 = 2A - I, where I is the identity matrix. Then for n ≥ 2, An^ = (A) nA - (n - 1) I (B) nA - I (C) 2 n - 1^ A - (n - 1) I (D) 2 n - 1^ A - I 76. 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
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
2x + y + 3z = 2 , 5x + 5y + 9z = 4 have : (A) Unique solutions (B) Infinitely many solutions (C) Inconsistent (D) None of these
87. The order of [x y z]
a h g h b f g f c
x y z
is : (A) 3 × 1 (B) 1 × 1 (C) 1 × 3 (D) 3 × 3
88. If A and B are two matrices such that AB = B and BA = A, then A^2 + B^2 = (A) 2 AB (B) 2 BA (C) A + B (D) AB
^1
−
(A)
^
^
(B)
(C)
(D)^
90. If A =
^
^
, then A^4 =
(A)
(B)^
(C)
(D)^
^
91. If A =
, then A^2 =
(A)
(B)^
(C)
(D)^
92. If X =
, then the value of Xn is :
(A) 3 n 4 n n n
(B)^
^
n n n n
(C) 3 4
n n n n
(D) None of these
93. If A =
^
^
, then A -1^ =
(A)
(B)^
(C)
(D)
94. The inverse of matrix A =
is : (A) A (B) AT
(C)
(D)
95. If A =
, I is the unit matrix of
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
order 2 & a, b are arbitrary constants, then (aI + bA)^2 is equal to : (A) a^2 I + ab A (B) a^2 I + 2 ab A (C) a^2 I + b^2 A (D) None of these
96. Which of the following is not true? (A) Every skew-symmetric matrix of odd order is non-singular (B) If determinant of a square matrix is non-zero, then it is non-singular (C) Adjoint of a symmetric matrix is symmetric (D) Adjoint of a diagonal matrix is diagonal 97. Which one of the following statements is true? (A) Non-singular square matrix does not have a unique inverse (B) Determinant of a non-singular matrix is zero (C) If A′ = A, then A is a square matrix (D) If A ≠ 0, then A. adj A = A(n - 1)^ where A = [aij]n × n
ANSWERS
1. A 2. D 3. B 4. A 5. B 6. D
- B 8. B 9. D 10. D 11. D 12.C
- C 14. B 15. D 16. B 17. B 18.C
- D 20. B 21. B 22. D 23. A 24.A
- B 26. D 27. C 28. A 29. B 30.A
- D 32. C 33. D 34. B 35. B 36.B
- B 38. C 39. B 40. A 41. B 42.C
- B 44. C 45. B 46. A 47. A 48.C
- D 50. B 51. A 52. C 53. B 54.D
- A 56. B 57. D 58. C 59. B 60.B
- D 62. A 63. A 64. B 65. C 66.D
- B 68. A 69. D 70. C 71. A 72.D
73. A 74. A 75. A 76. D 77. D 78.A
79. B 80. C 81. A 82. B 83. B 84.B
85. C 86. A 87. B 88. C 89. B 90.A
91. D 92. D 93. B 94. A 95. B 96.A
97. C
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