Matrices & Determinants PYQs Set 3 (PDF)

Set 3 of 19 · Questions 61-90 of 565

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 3 (Q61-90)
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61. $If A = 320 - 3 - 3 - 1 441, then (A^{2})^{- 1} =$ (1) $A^{2}$ (2) $2 A$ (3) $A^{3}$ (4) $12 - 2 212 2 - 2 - 1$ 62. $Let A = (a_{ij}) = (log_{5} 128 log_{4} 5 log_{5} 8 log_{4} 25) . If A_{ij} is the cofactor of a_{ij}, C_{ij} = \sum_{k = 1}^{2} a_{ik} A_{j k}, 1 i, j \leq 2, and C = (C_{ij}), then 8 \| C \| is equal to:$ (1) $288$ (2) $222$ (3) $242$ (4) $262$ 63. $If A and B are square matrices of same order and B is a skew symmetric matrix, then A^{’} BA is$ (1) $Null matrix$ (2) $Diagonal matrix$ (3) $Skew symmetric matrix$ (4) $Symmetric matrix$ 64. $If rank of x x x x x^{2} x x x x + 1 is 1, then$ (1) $x = 0 or x = 1$ (2) $x = 1$ (3) $x = 0$ (4) $x \neq = 0$ 65. $If A = a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3} and B = c_{1} a_{1} b_{1} c_{2} a_{2} b_{2} c_{3} a_{3} b_{3} then$ (1) $B = 0$ (2) $B = A^{2}$ (3) $A = - B$ (4) $A = B$ 66. $If A and B are square matrices of order ' n' such that A^{2} - B^{2} = (A - B) (A + B), then which of the following will be true?$ (1) $Either of A or B is zero matrix.$ (2) $A = B$ (3) $AB = BA$ (4) $Either of A or B is an identity matrix.$ 67. $The number of values of θ \in (0, π) for which the system of linear equations x + 3 y + 7 z = 0 - x + 4 y + 7 z = 0 (sin 3 θ) x + (cos 2 θ) y + 2 z = 0 has a non-trivial solution, is:$ (1) $three$ (2) $two$ (3) $four$ (4) $one$ 68. $If A + B = (212120022), AB = (122110121) then A^{2} + B (A + B) =$ (1) $(466342163)$ (2) $(496332474)$ (3) $(6108452496)$ (4) $(344232042)$ 69. $If B = (52 α 1021 α 3 - 1) is the inverse of a 3\times3 matrix A, then the sum of all values of α for which det (A) + 1 = 0, is:$ (1) $0$ (2) $- 1$ (3) $1$ (4) $2$ 70. $If A is a square matrix of order 3 and \| A \| = 5, then ∣ A adj A ∣ is$ (1) $125$ (2) $25$ (3) $625$ (4) $5$ 71. $Consider the simultaneous linear equations AX = B and AY = Q . If A is an invertible matrix and B is the unique solution of AY = Q, then the solution of AX = B is$ (1) $A^{- 1} (B + Q)$ (2) $(A^{- 1})^{2} B$ (3) $A^{- 1} BQ$ (4) $(A^{- 1})^{2} Q$ 72. $Let α be a root of equation x^{2} + x + 1 = 0 and the matrix A = \frac{1}{3} 111 1 α α^{2} 1 α^{2} α^{4}, then the matrix A^{31} is equal to:$ (1) $A$ (2) $A^{2}$ (3) $A^{3}$ (4) $A^{4}$ 73. $The sum of real roots of the equation x 2 - 3 - 6 - 3 x 2 x - 1 x - 3 x + 2 = 0, is equal to :$ (1) $6$ (2) $0$ (3) $1$ (4) $- 4$ 74. $If the system of linear equations x + y + 2 z = 6, 2 x + 3 y + a z = a + 1, - x - 3 y + b z = 2 b, where a, b \in R, has infinitely many solutions, then 7 a + 3 b is equal to :$ (1) $16$ (2) $12$ (3) $22$ (4) $9$ 75. $If X_{4 \times 3}, Y_{4 \times 3} and P_{2 \times 3} are the matrices then the order of the matrix (P (X^{T} Y)^{- 1} P^{T})^{T} is$ (1) $4 \times 3$ (2) $3 \times 4$ (3) $3 \times 3$ (4) $2 \times 2$ 76. $If A is any square matrix of order 3 \times 3 then \| 3 A \| is equal to$ (1) $3 \| A \|$ (2) $1 3 \| A \|$ (3) $27 \| A \|$ (4) $9 \| A \|$ 77. $If 2 (10 3 x) + (y 1 02) = (5168), then the value of x and y are$ (1) $x = 3, y = 3$ (2) $x = - 3, y = 3$ (3) $x = 3, y = - 3$ (4) $x = - 3, y = - 3$ 78. $If x^{3} - 2 x^{2} - 9 x + 18 = 0 and A = 147 2 x 8 369 then the maximum value of A is.$ (1) $96$ (2) $36$ (3) $24$ (4) $120$ 79. $The solutions of the equation 1 + sin^{2} x cos^{2} x 4 sin 2 x sin^{2} x 1 + cos^{2} x 4 sin 2 x sin^{2} x cos^{2} x 1 + 4 sin 2 x = 0, (0 < x < π), are$ (1) $\frac{π}{12}, \frac{π}{6}$ (2) $\frac{π}{6}, \frac{5 π}{6}$ (3) $\frac{5 π}{12}, \frac{7 π}{12}$ (4) $\frac{7 π}{12}, \frac{11 π}{12}$ 80. $If A = (cos 2 θ sin 2 θ - sin 2 θ cos 2 θ) and A + A^{T} = I, where I is the unit matrix of 2 \times 2 and A^{T} is the transpose of A, then the value of θ is equal to$ (1) $\frac{π}{6}$ (2) $\frac{π}{3}$ (3) $π$ (4) $\frac{3 π}{2}$ 81. $The system of linear equations x + y + z = 6, x + 2 y + 3 z = 10 and x + 2 y + az = b has no solution when$ (1) $a = 3, b \neq = 10$ (2) $b = 3, a \neq = 10$ (3) $a = 2, b \neq = 3$ (4) $b = 2, a = 3$ 82. $If A = (2 - 1 - 1 2) such that A^{2} - 4 A + 3 I = 0 where I is a unit matrix of order 2, then A^{- 1} is$ (1) $\frac{1}{3} (2 - 1 - 1 2)$ (2) $\frac{1}{3} (- 1 2 2 - 1)$ (3) $\frac{1}{3} (- 2 1 1 - 2)$ (4) $\frac{1}{3} (2112)$ 83. $Let A = 210 a 35 01 b . If A^{3} = 4 A^{2} - A - 21 I, where I is the identity matrix of order 3 \times 3, then 2 a + 3 b is equal to$ (1) $-9$ (2) $-13$ (3) $-10$ (4) $-12$ 84. $The equations x + y + z = 3, x + 2 y + 2 z = 6 and x + ay + 3 z = b have$ (1) $No solution when a \neq = 3, b is any value$ (2) $Infinite number of solutions when b \neq = 9$ (3) $Unique solution when a \neq = 3, b is any value$ (4) $Unique solution when a = 3 and b\neq9$ 85. $If the system of equations x + y + z = 6 2x + 5y + α z = β x + 2y + 3z = 14 has infinitely many solutions, then α + β is equal to:$ (1) $8$ (2) $36$ (3) $44$ (4) $48$ 86. $If Δ = 111 a b c a^{2} b^{2} c^{2} = K (a - b) (b - c) (c - a), then K is equal to$ (1) $-1$ (2) $1$ (3) $2$ (4) $3$ 87. $If A = b c a a 0 a 0 b b and B = 0 b b a 0 a b c a are two matrices such that A B = 2132867510, then a^{2} + b^{2} + c^{2} =$ (1) $14$ (2) $17$ (3) $22$ (4) $294$ 88. $For which of the following ordered pairs (μ, δ), the system of linear equations x + 2 y + 3 z = 1 3 x + 4 y + 5 z = μ 4 x + 4 y + 4 z = δ is inconsistent ?$ (1) $(4, 6)$ (2) $(3, 4)$ (3) $(4, 3)$ (4) $(1, 0)$ 89. $Let A be a n \times n matrix such that ∣ A ∣ = 2 . If the determinant of the matrix Adj (2 \cdot Adj (2 A^{- 1})) is 2^{84} then n is equal to _________$ 90. $Let for any three distinct consecutive terms a, b, c of an A.P, the lines ax + by + c = 0 be concurrent at the point P and Q (α, β) be a point such that the system of equations x + y + z = 6, 2 x + 5 y + αz = β and x + 2 y + 3 z = 4, has infinitely many solutions. Then (P Q)^{2} is equal to_______$