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MCQOPTIONS
This section includes 111 Mcqs, each offering curated multiple-choice questions to sharpen your Discrete Mathematics knowledge and support exam preparation. Choose a topic below to get started.
| 51. |
Consider the following in respect of matrices A and B of same order:1) A2 – B2 = (A + B) (A – B)2) (A – I) (I + A) = O ⇔ A2 = IWhere I is the identity matrix and O is the null matrix.Which of the above is/are correct? |
| A. | 1 only |
| B. | 2 only |
| C. | Both 1 and 2 |
| D. | Neither 1 nor 2 |
| Answer» C. Both 1 and 2 | |
| 52. |
If \(A = \;\left[ {\begin{array}{*{20}{c}} 2&7\\ 1&5 \end{array}} \right]\) then what is A + 3A-1 equal to?Where A is the matrix of order 2. |
| A. | 3I |
| B. | 5I |
| C. | 7I |
| D. | None of the above |
| Answer» D. None of the above | |
| 53. |
If A and B are symmetric matrices, then AB – BA is: |
| A. | Null matrix |
| B. | Symmetric matrix |
| C. | Skew-symmetric matrix |
| D. | None of these |
| Answer» D. None of these | |
| 54. |
If \(A=\begin{bmatrix} 1 & -5 & 7 \\\ 0 & 7 & 9 \\\ 11 & 8 & 9 \end{bmatrix}\), then trace of matrix A is |
| A. | 17 |
| B. | 25 |
| C. | 3 |
| D. | 12 |
| Answer» B. 25 | |
| 55. |
If the system of linear equationsx – 4y + 7z = g3y – 5z = h–2x + 5y – 9z = k is consistent, then: |
| A. | g + 2h + k = 0 |
| B. | g + h + 2k = 0 |
| C. | 2g + h + k = 0 |
| D. | g + h + k = 0 |
| Answer» D. g + h + k = 0 | |
| 56. |
If \(A = \left( {\begin{array}{*{20}{c}} 1&2\\ 2&3 \end{array}} \right)\) and A2 – kA – I2 = 0, where I2 is the 2 × 2 identity matrix, then what is the value of k? |
| A. | 4 |
| B. | -4 |
| C. | 8 |
| D. | -8 |
| Answer» B. -4 | |
| 57. |
Let the numbers 2, b, c be in an A.P. and \({\rm{A}} = \left[ {\begin{array}{*{20}{c}}1&1&1\\2&{\rm{b}}&{\rm{c}}\\4&{{{\rm{b}}^2}}&{{{\rm{c}}^2}}\end{array}} \right].\) If det(A) ∈ [2, 16], then c lies in the interval: |
| A. | [2, 3) |
| B. | (2 + 23⁄4, 4) |
| C. | [4, 6] |
| D. | [3, 2 + 23⁄4] |
| Answer» D. [3, 2 + 23⁄4] | |
| 58. |
A square matrix A is called orthogonal if_______ where A’ is the transpose of A. |
| A. | A = A2 |
| B. | A’ = A - 1 |
| C. | A = A - 1 |
| D. | A = A’ |
| Answer» C. A = A - 1 | |
| 59. |
If A is a 2 × 3 matrix and AB is a 2 × 5 matrix, then B must be a |
| A. | 3 × 5 matrix |
| B. | 5 × 3 matrix |
| C. | 3 × 2 matrix |
| D. | 5 × 4 matrix |
| Answer» B. 5 × 3 matrix | |
| 60. |
If \(B=\left[ \begin{matrix}5 & 2\alpha & 1 \\0 & 2 & 1 \\\alpha & 3 & -1 \\\end{matrix} \right]\) is the inverse of a 3 × 3 matrix A, then the sum of all values of α for which det(A) + 1 = 0, is: |
| A. | 0 |
| B. | -1 |
| C. | 1 |
| D. | 2 |
| Answer» D. 2 | |
| 61. |
Consider the following in respect of a non-singular matrix of order 3:1. A (adj A) = (adj A) A2. |adj A| = |A|Which of the above statements is / are correct? |
| A. | 1 only |
| B. | 2 only |
| C. | Both 1 and 2 |
| D. | Neither 1 nor 2 |
| Answer» B. 2 only | |
| 62. |
Find x and y if \(x + y = \left[ {\begin{array}{*{20}{c}}7&0\\2&5\end{array}} \right],x - y = \left[ {\begin{array}{*{20}{c}}3&0\\0&3\end{array}} \right]\) |
| A. | \(X = \left[ {\begin{array}{*{20}{c}}5&0\\1&4\end{array}} \right]Y = \left[ {\begin{array}{*{20}{c}}2&0\\1&1\end{array}} \right]\) |
| B. | \(X = \left[ {\begin{array}{*{20}{c}}4&5\\1&0\end{array}} \right]Y = \left[ {\begin{array}{*{20}{c}}1&0\\2&1\end{array}} \right]\) |
| C. | \(X = \left[ {\begin{array}{*{20}{c}}0&4\\1&5\end{array}} \right]Y = \left[ {\begin{array}{*{20}{c}}2&1\\0&1\end{array}} \right]\) |
| D. | \(X = \left[ {\begin{array}{*{20}{c}}5&0\\1&4\end{array}} \right]Y = \left[ {\begin{array}{*{20}{c}}2&0\\0&1\end{array}} \right]\) |
| Answer» B. \(X = \left[ {\begin{array}{*{20}{c}}4&5\\1&0\end{array}} \right]Y = \left[ {\begin{array}{*{20}{c}}1&0\\2&1\end{array}} \right]\) | |
| 63. |
If \(\rm A= \begin{bmatrix}\ \ \ \cos\alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix}\), then for any positive integer n, An is: |
| A. | \(\rm \begin{bmatrix} \rm \sin n\alpha & \ \ \ \rm \cos n\alpha \\ \rm \cos n\alpha & \rm -\sin n\alpha \end{bmatrix}\) |
| B. | \(\rm \begin{bmatrix} \rm \cos n\alpha & \rm \sin n\alpha \\ \rm \sin n\alpha & \rm \cos n\alpha \end{bmatrix}\) |
| C. | \(\rm \begin{bmatrix} \rm \cos n\alpha & \ \ \ \rm \sin n\alpha \\ \rm \sin n\alpha & \rm -\cos n\alpha \end{bmatrix}\) |
| D. | \(\rm \begin{bmatrix} \ \ \ \rm \cos n\alpha & \rm \sin n\alpha \\ \rm- \sin n\alpha & \rm \cos n\alpha \end{bmatrix}\) |
| Answer» E. | |
| 64. |
Let \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} {\rm x + y}& \rm y\\ {\rm 2x}&{\rm x - y} \end{array}} \right],\;\rm B = \left[ {\begin{array}{*{20}{c}} 2\\ { - 1} \end{array}} \right]\) and \(\rm C = \left[ {\begin{array}{*{20}{c}} 3\\ 2 \end{array}} \right]\). If AB = C, then what is the value of the determinant of the matrix A? |
| A. | -10 |
| B. | -14 |
| C. | -24 |
| D. | -34 |
| Answer» C. -24 | |
| 65. |
If the matrix AB is a zero matrix, then which one of the following is correct? |
| A. | A must be equal to zero matrix or B must be equal to zero matrix |
| B. | A must be equal to zero matrix and B must be equal to zero matrix |
| C. | It is not necessary that either A is zero matrix or B is zero matrix |
| D. | None of the above |
| Answer» D. None of the above | |
| 66. |
If A is an invertible skew-symmetric matrix, then A-1 is a: |
| A. | Symmetric matrix. |
| B. | Skew-symmetric matrix. |
| C. | Zero matrix. |
| D. | Identity matrix. |
| Answer» C. Zero matrix. | |
| 67. |
Let \({\rm{A}} = \left[ {\begin{array}{*{20}{c}}{{\rm{cos\;\alpha }}}&{ - {\rm{sin\;\alpha }}}\\{{\rm{sin\;\alpha }}}&{{\rm{cos\;\alpha }}}\end{array}} \right],\) (α ∈ R) such that \({A^{32}} = \left[ {\begin{array}{*{20}{c}}0&{ - 1}\\1&0\end{array}} \right]\) then a value of α is: |
| A. | \(\frac{{\rm{\pi }}}{{32}}\) |
| B. | 0 |
| C. | \(\frac{{\rm{\pi }}}{{64}}\) |
| D. | \(\frac{{\rm{\pi }}}{{16}}\) |
| Answer» D. \(\frac{{\rm{\pi }}}{{16}}\) | |
| 68. |
If \(\rm \begin{bmatrix} \rm a+b & \rm b+c & \rm c+a \\ \rm b+c & \rm c+a & \rm a+b\\ \rm c+a & \rm a+b & \rm b+c \end{bmatrix} = \rm k \begin{bmatrix} \rm a & \rm b & \rm c\\ \rm b & \rm c& \rm a\\ \rm c & \rm a & \rm b \end{bmatrix}\) then k is equal to |
| A. | 1 |
| B. | 2 |
| C. | 4 |
| D. | 8 |
| Answer» C. 4 | |
| 69. |
If α and β are the roots of the equation 1 + x + x2 = 0, then the matrix product\(\left[ {\begin{array}{*{20}{c}} 1&\beta \\ \alpha &\alpha \end{array}} \right]\;\left[ {\begin{array}{*{20}{c}} \alpha &\beta \\ 1&\beta \end{array}} \right]\) is equal to |
| A. | \(\left[ {\begin{array}{*{20}{c}} 1&1\\ 1&2 \end{array}} \right]\) |
| B. | \(\left[ {\begin{array}{*{20}{c}} { - 1}&{ - 1}\\ { - 1}&2 \end{array}} \right]\) |
| C. | \(\left[ {\begin{array}{*{20}{c}} 1&{ - 1}\\ { - 1}&2 \end{array}} \right]\) |
| D. | \(\left[ {\begin{array}{*{20}{c}} { - 1}&{ - 1}\\ { - 1}&{ - 2} \end{array}} \right]\) |
| Answer» C. \(\left[ {\begin{array}{*{20}{c}} 1&{ - 1}\\ { - 1}&2 \end{array}} \right]\) | |
| 70. |
If A and B are symmetric matrices of the same order, then (AB' - BA') is: |
| A. | Skew symmetric matrix |
| B. | Symmetric matrix |
| C. | Null matrix |
| D. | Identity matrix |
| Answer» B. Symmetric matrix | |
| 71. |
Let A be an n × n matrix from the set of numbers and A3 - 3A2 + 4A - 6I = 0 where I is an n × n unit matrix. If A-1 exists, then |
| A. | A-1 = A - I |
| B. | A-1 = 3A - 6I |
| C. | A-1 = A + 6I |
| D. | \(A^{-1} = \dfrac{1}{6}(A^2 - 3A + 4I)\) |
| Answer» E. | |
| 72. |
If \(A=\begin{bmatrix}1 & 1 \\\ 1 & 1 \end{bmatrix}\), then \(A^{100}\) is equal to - |
| A. | \(2^{100} A\) |
| B. | \(2^{99}A\) |
| C. | \(100\ A\) |
| D. | \(299 \ A\) |
| Answer» C. \(100\ A\) | |
| 73. |
A square matrix is called a skew-symmetric matrix when: |
| A. | its transpose is an identity matrix |
| B. | its transpose is square matrix |
| C. | its transpose is negative of itself |
| D. | its transpose is equal to itself |
| Answer» D. its transpose is equal to itself | |
| 74. |
If the sum of the matrices \(\rm \begin{bmatrix}\rm x \\\ \rm x \\\ \rm y \end{bmatrix}, \begin{bmatrix} \rm y \\\ \rm y \\\ \rm z \end{bmatrix}\) and \(\begin{bmatrix} \rm z \\\ \rm 0 \\\ \rm 0 \end{bmatrix}\) is matrix \(\begin{bmatrix} \rm 10 \\\ \rm 5 \\\ \rm 5 \end{bmatrix}\), then what is the value of y? |
| A. | -5 |
| B. | 0 |
| C. | 5 |
| D. | 10 |
| Answer» C. 5 | |
| 75. |
From the matrix equation AB = AC we can conclude. B = C provided |
| A. | |A| = 0 |
| B. | |A| ≠ 0 |
| C. | A is symmetric |
| D. | A is square |
| Answer» B. |A| ≠ 0 | |
| 76. |
If \({\rm{f}}\left( {\rm{x}} \right) = \left[ {\begin{array}{*{20}{c}} {\cos {\rm{x}}}&{ - \sin {\rm{x}}}&0\\ {\sin {\rm{x}}}&{\cos {\rm{x}}}&0\\ 0&0&1 \end{array}} \right]\), then which of the following are correct?1. f(θ) × f(ϕ) = f(θ + ϕ)2. The value of the determinant of the matrix f(θ) × f(ϕ) is 13. The determinant of f(x) is an even function.Select the correct answer using the code given below: |
| A. | 1 and 2 only |
| B. | 2 and 3 only |
| C. | 1 and 3 only |
| D. | 1, 2 and 3 |
| Answer» E. | |
| 77. |
If \(A = \left( {\begin{array}{*{20}{c}} 1&2\\ 2&3\\ 3&4 \end{array}} \right)\) and \(B = \left( {\begin{array}{*{20}{c}} 1&2\\ 2&1 \end{array}} \right),\) then which one of the following is correct? |
| A. | Both AB and BA exist |
| B. | Neither AB nor BA exists |
| C. | AB exists but BA does not exist |
| D. | AB does not exist but BA exists |
| Answer» D. AB does not exist but BA exists | |
| 78. |
If \({{\rm{\Delta }}_1} = \left| {\begin{array}{*{20}{c}}{\rm{x}}&{{\rm{sin\theta }}}&{{\rm{cos\theta }}}\\{ - {\rm{sin\theta }}}&{ - {\rm{x}}}&1\\{{\rm{cos\theta }}}&1&{\rm{x}}\end{array}} \right|\) and \({{\rm{\Delta }}_2} = \left| {\begin{array}{*{20}{c}}{\rm{x}}&{{\rm{sin}}2{\rm{\theta }}}&{{\rm{cos}}2{\rm{\theta }}}\\{ - {\rm{sin}}2{\rm{\theta }}}&{ - {\rm{x}}}&1\\{{\rm{cos}}2{\rm{\theta }}}&1&{\rm{x}}\end{array}} \right|,{\rm{x}} \ne 0;\) then for all \({\rm{\theta }} \in \left( {0,\frac{{\rm{\pi }}}{2}} \right):\) |
| A. | Δ1 – Δ2 = -2x3 |
| B. | Δ1 – Δ2 = x(cos 2θ – cos 4θ) |
| C. | Δ1 + Δ2 = -2(x3 + x – 1) |
| D. | Δ1 + Δ2 = -2x3 |
| Answer» E. | |
| 79. |
If \(A=\begin{bmatrix} 0 & 5 \\\ 0 & 0 \end{bmatrix}\) and \(f(x)=1+x+x^2 + ...+x^{16},\) then \(f(A)=\) |
| A. | 0 |
| B. | \(\begin{bmatrix} 1 & 5 \\\ 0 & 1 \end{bmatrix}\) |
| C. | \(\begin{bmatrix} 1 & 5 \\\ 0 & 0 \end{bmatrix}\) |
| D. | \(\begin{bmatrix} 0 & 5 \\\ 1 & 1 \end{bmatrix}\) |
| Answer» C. \(\begin{bmatrix} 1 & 5 \\\ 0 & 0 \end{bmatrix}\) | |
| 80. |
If \(A = \left[ {\begin{array}{*{20}{c}} 0&1\\ 1&0 \end{array}} \right],\) then the matrix A is a/an |
| A. | Singular matrix |
| B. | involuntary matrix |
| C. | Nilpotent matrix |
| D. | Idempotent matrix |
| Answer» C. Nilpotent matrix | |
| 81. |
If A is a symmetric matrix and B is a skew-symmetric matrix such that \({\rm{A}} + {\rm{B}} = \left[ {\begin{array}{*{20}{c}} 2&3\\ 5&{ - 1} \end{array}} \right]\), then A and B are: |
| A. | \(\left[ {\begin{array}{*{20}{c}} 2&{ 5}\\ { 5}&{ - 1} \end{array}} \right],\left[ {\begin{array}{*{20}{c}} 0&{ -1}\\ { 1}&{ 0} \end{array}} \right]\) |
| B. | \(\left[ {\begin{array}{*{20}{c}} 2&{ 4}\\ { 4}&{ - 1} \end{array}} \right],\left[ {\begin{array}{*{20}{c}} 0&{ -1}\\ { 1}&{ 0} \end{array}} \right]\) |
| C. | \(\left[ {\begin{array}{*{20}{c}} 1&{ 4}\\ { 4}&{ - 1} \end{array}} \right],\left[ {\begin{array}{*{20}{c}} 1&{ -1}\\ { 1}&{ 0} \end{array}} \right]\) |
| D. | \(\left[ {\begin{array}{*{20}{c}} 2&{ 4}\\ { 4}&{ - 1} \end{array}} \right],\left[ {\begin{array}{*{20}{c}} 0&{ -1}\\ { 1}&{ 2} \end{array}} \right]\) |
| Answer» C. \(\left[ {\begin{array}{*{20}{c}} 1&{ 4}\\ { 4}&{ - 1} \end{array}} \right],\left[ {\begin{array}{*{20}{c}} 1&{ -1}\\ { 1}&{ 0} \end{array}} \right]\) | |
| 82. |
An ordered pair (α. β) for which the system of linear equations(1 + α)x + βy + z = 2αx + (1 + β)y + z = 3αx + β + 2z = 2 has a unique solution, is |
| A. | (2, 4) |
| B. | (-4, 2) |
| C. | (1, -3) |
| D. | (-3, 1) |
| Answer» B. (-4, 2) | |
| 83. |
Find a matrix X such that 2A + B + X = 0 , where\(A=\begin{bmatrix} -1 & 2 \\\ 3 & 4 \end{bmatrix} \ \text{and} \;\rm B =\ \begin{bmatrix} 3 & -2 \\\ 1 & 5 \end{bmatrix} \ ?\) |
| A. | \(\begin{bmatrix} 1 & 2 \\\ 7 & 13 \end{bmatrix}\) |
| B. | \(\begin{bmatrix} -1 & -2 \\\ -7 & -13 \end{bmatrix}\) |
| C. | \(\begin{bmatrix} 13 & 2 \\\ 7 & 1 \end{bmatrix}\) |
| D. | \(\begin{bmatrix} -13 & -2 \\\ -7 & -1 \end{bmatrix}\) |
| Answer» C. \(\begin{bmatrix} 13 & 2 \\\ 7 & 1 \end{bmatrix}\) | |
| 84. |
Let \(A = \left[ {\begin{array}{*{20}{c}}2&b&1\\b&{{b^2} + 1}&b\\1&b&2\end{array}} \right]\) where b > 0. Then the minimum value of \(\frac{{det\left( {\rm{A}} \right)}}{{\rm{b}}}\)is: |
| A. | \(2\sqrt 3 \) |
| B. | \(- 2\sqrt 3\) |
| C. | \(- \sqrt 3 \) |
| D. | \(\sqrt 3 \) |
| Answer» B. \(- 2\sqrt 3\) | |
| 85. |
If \(A = \left( {\begin{array}{*{20}{c}} 1&2\\ 2&3 \end{array}} \right)\) and \(B = \left( {\begin{array}{*{20}{c}} 1&0\\ 1&0 \end{array}} \right)\) then what is determinant of AB? |
| A. | 0 |
| B. | 1 |
| C. | 10 |
| D. | 20 |
| Answer» B. 1 | |
| 86. |
If the system of linear equationsX – 2y + kz = 12x + y + z = 23x – y – kz = 3has a solution (x, y, z), z ≠ 0, then (x, y) lies on the straight line whose equation is: |
| A. | 3x – 4y – 1 = 0 |
| B. | 4x – 3y – 4 = 0 |
| C. | 4x – 3y – 1 = 0 |
| D. | 3x – 4y – 4 = 0 |
| Answer» C. 4x – 3y – 1 = 0 | |
| 87. |
If \(\begin{bmatrix} 2x + y & 4x \\\ 5x - 7 & 4x \end{bmatrix} = \begin{bmatrix} 7 & 7y -13 \\\ y & x+6 \end{bmatrix}\), then x + y = |
| A. | 3 |
| B. | 4 |
| C. | 5 |
| D. | 6 |
| Answer» D. 6 | |
| 88. |
If a, b, c are non-zero real numbers, then the inverse of the matrix \(A = \left[ {\begin{array}{*{20}{c}} a&0&0\\ 0&b&0\\ 0&0&c \end{array}} \right]\) is equal to |
| A. | \(\left[ {\begin{array}{*{20}{c}} {{a^{ - 1}}}&0&0\\ 0&{{b^{ - 1}}}&0\\ 0&0&{{c^{ - 1}}} \end{array}} \right]\) |
| B. | \(\frac{1}{{abc}}\left[ {\begin{array}{*{20}{c}} {{a^{ - 1}}}&0&0\\ 0&{{b^{ - 1}}}&0\\ 0&0&{{c^{ - 1}}} \end{array}} \right]\) |
| C. | \(\frac{1}{{abc}}\left[ {\begin{array}{*{20}{c}} 1&0&0\\ 0&1&0\\ 0&0&1 \end{array}} \right]\) |
| D. | \(\frac{1}{{abc}}\left[ {\begin{array}{*{20}{c}} a&0&0\\ 0&b&0\\ 0&0&c \end{array}} \right]\) |
| Answer» B. \(\frac{1}{{abc}}\left[ {\begin{array}{*{20}{c}} {{a^{ - 1}}}&0&0\\ 0&{{b^{ - 1}}}&0\\ 0&0&{{c^{ - 1}}} \end{array}} \right]\) | |
| 89. |
If the inverse of the matrix \(\begin{bmatrix} α & 14 & -1 \\\ 2 & 3 & 1 \\\ 6 & 2 & 3 \end{bmatrix}\) does not exist then the value of α is |
| A. | 1 |
| B. | -1 |
| C. | 0 |
| D. | -2 |
| Answer» E. | |
| 90. |
Let \(A = \left[ {\begin{array}{*{20}{c}} 1&1&3\\ 5&2&6\\ { - 2}&{ - 1}&{ - 3} \end{array}} \right]\), then A is |
| A. | Nilpotent |
| B. | Idempotent |
| C. | Scalar |
| D. | None of these |
| Answer» E. | |
| 91. |
If \(B = \left[ {\begin{array}{*{20}{c}} 3&2&0\\ 2&4&0\\ 1&1&0 \end{array}} \right]\), then what is adjoint of B equal to? |
| A. | \(\left[ {\begin{array}{*{20}{c}} 0&0&0\\ 0&0&0\\ { - 2}&{ - 1}&8 \end{array}} \right]\) |
| B. | \(\left[ {\begin{array}{*{20}{c}} 0&0&{ - 2}\\ 0&0&{ - 1}\\ 0&0&8 \end{array}} \right]\) |
| C. | \(\left[ {\begin{array}{*{20}{c}} 0&0&2\\ 0&0&1\\ 0&0&0 \end{array}} \right]\) |
| D. | It does not exist |
| Answer» B. \(\left[ {\begin{array}{*{20}{c}} 0&0&{ - 2}\\ 0&0&{ - 1}\\ 0&0&8 \end{array}} \right]\) | |
| 92. |
If \(A = \left[ {\begin{array}{*{20}{c}} 1&{ - 1}\\ { - 1}&1 \end{array}} \right],\) then the expression A3 - 2A2 is |
| A. | a null matrix |
| B. | an identify matrix |
| C. | equal to A |
| D. | equal to -A |
| Answer» B. an identify matrix | |
| 93. |
Match List - I with List - II and select the correct answer - List - I List - IIA. A square matrix such that A2 = A 1. Niloptent matrixB. A square matrix such that Am = 02. Involutory matrixC. A square matrix such that A2 = I3. Symmetric matrixD. A square matrix such that AT = A4. Idempotent matrix |
| A. | A - 1, B - 3, C - 2, D - 4 |
| B. | A - 3, B - 4, C - 2, D - 1 |
| C. | A - 4, B - 3, C - 2, D - 1 |
| D. | A - 4, B - 1, C - 2, D - 3 |
| Answer» E. | |
| 94. |
If A = \(\left[ {\begin{array}{*{20}{c}}{{\rm{cos\theta }}}&{ - {\rm{sin\theta }}}\\{{\rm{sin\theta }}}&{{\rm{cos\theta }}}\end{array}} \right],\) then the matrix A-50 when \({\rm{\theta }} = \frac{{\rm{\pi }}}{{12}}\) is equal to: |
| A. | \(\left[ {\begin{array}{*{20}{c}}{\frac{1}{2}}&{ - \frac{{\sqrt 3 }}{2}}\\{\frac{{\sqrt 3 }}{2}}&{\frac{1}{2}}\end{array}} \right]\) |
| B. | \(\left[ {\begin{array}{*{20}{c}}{\frac{{\sqrt 3 }}{2}}&{ - \frac{1}{2}}\\{\frac{1}{2}}&{\frac{{\sqrt 3 }}{2}}\end{array}} \right]\) |
| C. | \(\left[ {\begin{array}{*{20}{c}}{\frac{{\sqrt 3 }}{2}}&{\frac{1}{2}}\\{ - \frac{1}{2}}&{\frac{{\sqrt 3 }}{2}}\end{array}} \right]\) |
| D. | \(\left[ {\begin{array}{*{20}{c}}{\frac{1}{2}}&{\frac{{\sqrt 3 }}{2}}\\{ - \frac{{\sqrt 3 }}{2}}&{\frac{1}{2}}\end{array}} \right]\) |
| Answer» D. \(\left[ {\begin{array}{*{20}{c}}{\frac{1}{2}}&{\frac{{\sqrt 3 }}{2}}\\{ - \frac{{\sqrt 3 }}{2}}&{\frac{1}{2}}\end{array}} \right]\) | |
| 95. |
If A = \(\left[ {\begin{array}{*{20}{c}} 1&{ - 1}\\ { - 1}&1 \end{array}} \right]\)and B = \(\left[ {\begin{array}{*{20}{c}} 1&{ 1}\\ { 1}&1 \end{array}} \right]\), then the value of AB is |
| A. | \(\left[ {\begin{array}{*{20}{c}} 1&{ - 1}\\ { - 1}&1 \end{array}} \right]\) |
| B. | \(\left[ {\begin{array}{*{20}{c}} 0&{ 1}\\ { 1}&0 \end{array}} \right]\) |
| C. | \(\left[ {\begin{array}{*{20}{c}} -1&{ 0}\\ { 0}&-1 \end{array}} \right]\) |
| D. | \(\left[ {\begin{array}{*{20}{c}} 0&{ 0}\\ { 0}&0 \end{array}} \right]\) |
| Answer» E. | |
| 96. |
Let \(P=\left[ \begin{matrix}1 & 0 & 0 \\3 & 1 & 0 \\9 & 3 & 1 \\\end{matrix} \right]\) and Q = [qij] be two 3 × 3 matrices such that Q – P5 = I3. Then, \(\frac{{{q}_{21}}+{{q}_{31}}}{{{q}_{32}}}\) is equal to |
| A. | 10 |
| B. | 135 |
| C. | 9 |
| D. | 15 |
| Answer» B. 135 | |
| 97. |
A value of \({\rm{\theta }} \in \left( {0,{\rm{\;}}\frac{{\rm{\pi }}}{3}} \right),{\rm{\;for\;which}}\left| {\begin{array}{*{20}{c}}{1 + {\rm{co}}{{\rm{s}}^2}{\rm{\theta }}}&{{\rm{si}}{{\rm{n}}^2}{\rm{\theta }}}&{4{\rm{\;cos\;}}6{\rm{\theta }}}\\{{\rm{co}}{{\rm{s}}^2}{\rm{\theta }}}&{1 + {\rm{si}}{{\rm{n}}^2}{\rm{\theta }}}&{4{\rm{\;cos\;}}6{\rm{\theta }}}\\{{\rm{co}}{{\rm{s}}^2}{\rm{\theta }}}&{{\rm{si}}{{\rm{n}}^2}{\rm{\theta }}}&{1 + 4{\rm{\;cos\;}}6{\rm{\theta }}}\end{array}} \right| = 0\), is: |
| A. | \(\frac{\pi }{9}\) |
| B. | \(\frac{\pi }{{18}}\) |
| C. | \(\frac{{7\pi }}{{24}}\) |
| D. | \(\frac{{7\pi }}{{36}}\) |
| Answer» B. \(\frac{\pi }{{18}}\) | |
| 98. |
According to Cayley Hamilton Theorem, every _______ matrix satisfies its own characteristic equation. |
| A. | asymmetrical |
| B. | clear |
| C. | square |
| D. | functional |
| Answer» D. functional | |
| 99. |
For how many values of k, is the matrix \(\left[ {\begin{array}{*{20}{c}} 0&{\rm{k}}&4\\ { - {\rm{k}}}&0&{ - 5}\\ { - {\rm{k}}}&{\rm{k}}&{ - 1} \end{array}} \right]\) singular? |
| A. | Only one |
| B. | Only two |
| C. | Only four |
| D. | Infinite |
| Answer» E. | |
| 100. |
If A = \(\left[ {\begin{array}{*{20}{c}} 1&{3 + x}&2\\ {1 - x}&2&{y + 1}\\ 2&{5 - y}&3 \end{array}} \right]\) is a symmetric matrix, then 3x + y is equal to |
| A. | -1 |
| B. | 0 |
| C. | 1 |
| D. | none of these |
| Answer» B. 0 | |