Matrices & Determinants PYQs Set 4 (PDF)

Set 4 of 19 · Questions 91-120 of 565

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 4 (Q91-120)
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91. $Suppose A and B are two square matrices of same order. If A and B are symmetric matrix, then AB - BA is$ (1) $a symmetric matrix$ (2) $a skew-symmetric$ (3) $a scalar matrix$ (4) $a triangular matrix$ 92. $Let A and B be any two 3x3 symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?$ (1) $A^{4} - B^{4} is a symmetric matrix$ (2) $AB - BA is a symmetric matrix$ (3) $B^{5} - A^{5} is a skew-symmetric matrix$ (4) $AB + BA is a skew-symmetric matrix$ 93. $Let a, b \in R - 0 and I_{2} be the identity matrix of order 2. Then the rank of the (block) matrix (a I_{2} b I_{2} a I_{2} b I_{2}) is$ (1) $2$ (2) $1$ (3) $4$ (4) $3$ 94. $Let M and N be two invertible square matrices over R of order 2 such that N is diagonal. Then MN M^{- 1} is diagonal \dots \dots \dots .$ (1) $for all M$ (2) $only when M is a scalar matrix$ (3) $for all diagonal matrices M$ (4) $M must be a null matrix$ 95. $If A = (a + ib c + id - c + id a - ib) and A^{- 1} = (a + ib - c - id - c + id a - ib) . Find (a^{2} + b^{2} + c^{2} + d^{2}) .$ (1) $1$ (2) $-1$ (3) $i$ (4) $- i$ 96. $If Δ_{1} = x - sin θ cos θ sin θ - x 1 cos θ 1 x and Δ_{2} = x - sin 2 θ cos 2 θ sin 2 θ - x 1 cos 2 θ 1 x, x \neq = 0; then for all θ \in (0, \frac{π}{2}) :$ (1) $∆_{1} - ∆_{2} = - 2 x^{3}$ (2) $∆_{1} - ∆_{2} = x (cos 2 θ - cos 4 θ)$ (3) $∆_{1} \times ∆_{2} = - 2 (x^{3} + x - 1)$ (4) $Δ_{1} + Δ_{2} = - 2 x^{3}$ 97. $Let α and β be real numbers. Consider a 3 \times 3 matrix A such that A^{2} = 3 A + αI . If A^{4} = 21 A + βI, then$ (1) $α = 1$ (2) $α = 4$ (3) $β = 8$ (4) $β = - 8$ 98. $If A is a 3 \times 3 matrix such that \| 5 \cdot adj A \| = 5 then \| A \| is equal to$ (1) $\pm 1$ (2) $\pm 1/5$ (3) $\pm 1/25$ (4) $\pm 5$ 99. $If A = 121 - 1 - 1 0 100, then A^{5} is equal to$ (1) $A$ (2) $Identity matrix$ (3) $Null matrix$ (4) $A^{- 1}$ 100. $The greatest value of c \in R for which the system of linear equations x - cy - cz = 0 cx - y + cz = 0 cx + cy - z = 0 has a non-trivial solution, is$ (1) $- 1$ (2) $\frac{1}{2}$ (3) $2$ (4) $0$ 101. $If A = (2 - 13 - 2), then the inverse of the matrix A^{3} is$ (1) $A$ (2) $1$ (3) $-1$ (4) $- A$ 102. $If A = (cos θ i sin θ i sin θ cos θ), (θ = \frac{π}{24}) and A^{5} = (a c b d) where i = - 1, then which of the following is not true?$ (1) $a^{2} - b^{2} = \frac{1}{2}$ (2) $a^{2} - c^{2} = 1$ (3) $a^{2} - d^{2} = 0$ (4) $0 \leq a^{2} + b^{2} \leq 1$ 103. $If the system of linear equations 2 x + 2 ay + az = 0 2 x + 3 by + bz = 0 2 x + 4 cy + cz = 0 where a, b, c \in R are non-zero and distinct; has a non-zero solution, then$ (1) $a + b + c = 0$ (2) $a, b, c are in A.P.$ (3) $\frac{1}{a}, \frac{1}{b}, \frac{1}{c} are in A.P.$ (4) $a, b, c in G.P.$ 104. $The value of x p p p x q q q x is$ (1) $x (x - p) (x - q)$ (2) $(x - p) (x - q) (x + p + q)$ (3) $(p - q) (x - q) (x - p)$ (4) $pq (x - p) (x - q)$ 105. $Let A = (1021) and B = I + adj (A) + (adj A)^{2} + \dots + (adj A)^{10} . Then, the sum of all the elements of the matrix B is:$ (1) $-124$ (2) $22$ (3) $-88$ (4) $-110$ 106. $If A is a square matrix of order 3 \times 3, then \| KA \| is equal to$ (1) $K \| A \|$ (2) $K^{2} \| A \|$ (3) $3 K \| A \|$ (4) $K^{3} \| A \|$ 107. $If P and Q are symmetric matrices of the same order then PQ - QP is$ (1) $identity matrix$ (2) $zero matrix$ (3) $symmetric matrix$ (4) $skew symmetric matrix$ 108. $If the system of linear equations x + y + 3 z = 0 x + 3 y + k^{2} z = 0 3 x + y + 3 z = 0 has a non-zero solution (x, y, z) for some k \in R then x + (\frac{y}{z}) is equal to:$ (1) $9$ (2) $3$ (3) $- 9$ (4) $- 3$ 109. $If the rank of the matrix A = 1 - 1 0 221 13 k - 1 5 k is 2 and k is a real number, then k is a root of the following quadratic equation$ (1) $x^{2} + 3 x + 2 = 0$ (2) $x^{2} + x - 2 = 0$ (3) $x^{2} + x - 6 = 0$ (4) $x^{2} - x - 6 = 0$ 110. $Let a and b be non-zero real numbers such that ab = 5/2 and given A = (a - bba) and A A^{T} 20 I (l is unit matrix), then the equation whose roots are a and b is$ (1) $x^{2} \mp 10 x + 5 = 0$ (2) $2 x^{2} \pm 10 x + 5 = 0$ (3) $x^{2} - 5 x + \frac{5}{2} = 0$ (4) $x^{2} - 25 x + \frac{5}{2} = 0$ 111. $Let A = {X = (x, y, z)^{T} : P X = 0 and x^{2} + y^{2} + z^{2} = 1}, where P = 1 - 2 1 239 1 - 4 - 1, then the set A :$ (1) $is an empty set.$ (2) $contains more than two elements.$ (3) $contains exactly two elements$ (4) $is a singleton$ 112. $If the ranks of the matrices A = 121010 12 - 1 and B = (122436 4 - 8) are r_{1} and r_{2} respectively then r_{1} - r_{2} =$ (1) $0$ (2) $1$ (3) $2$ (4) $3$ 113. $Let M and N be two 2n \times 2n non-singular skew-symmetric matrices such that MN = NM. If P^{T} denotes the transpose of P, then M^{2} N^{2} (M^{T} N)^{- 1} (M N^{- 1})^{T} is equal to$ (1) $M^{2}$ (2) $- N^{2}$ (3) $- M^{2}$ (4) $MN$ 114. $The system of equations kx + y + z = 1, x + ky + z = k and x + y + zk = k^{2} has no solution if k is equal to:$ (1) $0$ (2) $1$ (3) $-1$ (4) $-2$ 115. $If the system of equations x + y + z = 5 x + 2 y + 3 z = 9 x + 3 y + αz = β has infinitely many solutions, then β - α equals :$ (1) $21$ (2) $8$ (3) $18$ (4) $5$ 116. $If A = (3 - 1 12) then A^{2} - 5 A is equal to$ (1) $I$ (2) $-I$ (3) $7 I$ (4) $- 7 I$ 117. $Let A be a matrix of order 3x3 and det (a) = 2. Then det (det (a) adj (5 adj (A^{3}))) is equal to____.$ (1) $512 x 1 0^{6}$ (2) $256 x 1 0^{6}$ (3) $1024 x 1 0^{6}$ (4) $256 x 1 0^{11}$ 118. $Let A = 0 - 2 3 20 - 1 - 3 10 and B be a matrix such that B (I - A) = I + A . Then the sum of the diagonal elements of B^{T} B is equal to ____ .$ 119. $Let A = (2 - 111) . If the sum of the diagonal elements of A^{13} is 3^{n}, then n is equal to______$ 120. $Consider the matrices : A = (23 - 5 m), B = (20 m) and X = (x y) . Let the set of all m, for which the system of equations A X = B has a negative solution (i.e., x < 0 and y < 0), be the interval (a, b) . Then 8 \int_{a}^{b} ∣ A ∣ d m is equal to_______$