Matrices & Determinants PYQs Set 18 (PDF)

Set 18 of 19 · Questions 511-540 of 565

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 18 (Q511-540)
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511. $Suppose the vectors x_{1}, x_{2} and x_{3} are the solutions of the system of linear equations, A x = b when the vector b on the right side is equal to b_{1}, b_{2} and b_{3} respectively. If x_{1} = 111, x_{2} = 021, x_{3} = 001, b_{1} = 100, b_{2} = 020 and b_{3} = 002, then the determinant of A is equal to :$ (1) $4$ (2) $\frac{1}{2}$ (3) $2$ (4) $\frac{3}{2}$ 512. $Let R = (x 000 y 000 z) be a non-zero 3 \times 3 matrix, where x sin θ = y sin (θ + \frac{2 π}{3}) = z sin (θ + \frac{4 π}{3}) \neq = 0, θ \in (0, 2 π) . For a square matrix M, let trace (M) denote the sum of all the diagonal entries of M . Then, among the statements: (I) Trace (R) = 0 (II) If trace (adj (adj (R)) = 0, then R has exactly one non-zero entry.$ (1) $Both (I) and (II) are true$ (2) $Neither (I) nor (II) is true$ (3) $Only (II) is true$ (4) $Only (I) is true$ 513. $Let A be a real matrix of order 3 \times 3, such that a_{i 1} + a_{i 2} + a_{i 3} = 1, for i = 1, 2, 3 . Then, the sum of all the entries of the matrix A^{3} is equal to:$ (1) $2$ (2) $1$ (3) $3$ (4) $9$ 514. $If the system of linear equations 7 x + 11 y + α z = 13 5 x + 4 y + 7 z = β 175 x + 194 y + 57 z = 361 has infinitely many solutions, then α + β + 2 is equal to$ (1) $4$ (2) $3$ (3) $5$ (4) $6$ 515. $For the system of linear equations 2 x - y + 3 z = 5 3 x + 2 y - z = 7 4 x + 5 y + α z = β which of the following is NOT correct?$ (1) $The system has infinitely many solutions for α = - 5 and β = 9$ (2) $The system has infinitely many solutions for α = - 6 and β = 9$ (3) $The system is inconsistent for α = - 5 and β = 8$ (4) $The system has a unique solution for α \neq = - 5 and β = 8$ 516. $If x + 1 x x x x + λ x x x x + λ^{2} = \frac{9}{8} (103 x + 81), then λ, \frac{λ}{3} are the roots of the equation$ (1) $4 x^{2} + 24 x - 27 = 0$ (2) $4 x^{2} - 24 x - 27 = 0$ (3) $4 x^{2} + 24 x + 27 = 0$ (4) $4 x^{2} - 24 z + 27 = 0$ 517. $If A = \frac{1}{5 ! 6 ! 7 !} 5! 6! 7! 6! 7! 8! 7! 8! 9!, then \|adj(adj(2A))\| is equal to$ (1) $2^{20}$ (2) $2$ (3) $2^{12}$ (4) $2^{16}$ 518. $If A and B are invertible matrices, then which of the following is not correct?$ (1) $adj A = \| A \| A^{- 1}$ (2) $det (A^{- 1}) = [det (A)]^{- 1}$ (3) $(AB)^{- 1} = B^{- 1} A^{- 1}$ (4) $(A + B)^{- 1} = B^{- 1} + A^{- 1}$ 519. $If A is a matrix of order 3 \times 3, then (A^{2})^{- 1} is equal to$ (1) $- A^{2} 2$ (2) $A^{2}$ (3) $(A^{- 1})^{2}$ (4) $(- A)^{- 2}$ 520. $If ω \neq 1 is the complex cube root of unity and matrix H = [ω 0 0 ω], then H70 is equal to -$ (1) $0$ (2) $- H$ (3) $H2$ (4) $H$ 521. $If α, β \neq 0 and f(n) = αn + βn and 3 1 + f (1) 1 + f (2) 1 + f (1) 1 + f (2) 1 + f (3) 1 + f (2) 1 + f (3) 1 + f (4) = K (1 - α)2 (1 - β)2 (α - β)2 , then K is equal to$ (1) $1$ (2) $-1$ (3) $αβ$ (4) $\frac{1}{α β}$ 522. $The value of k \in R, for which the following system of linear equations 3x - y + 4z = 3 x + 2y - 3z = -2 6x + 5y + kz = -3 has infinitely many solutions, is :$ (1) $3$ (2) $-5$ (3) $5$ (4) $-3$ 523. $If a x^{4} + b x^{3} + c x^{2} + d x + e = x^{3} + 3 x x + 1 x - 3 x - 1 - 2 x x + 4 x + 3 x - 4 3 x, then e =$ (1) $- 1$ (2) $1$ (3) $0$ (4) $2$ 524. $The rank of the matrix A = 214101214 is$ (1) $0$ (2) $1$ (3) $2$ (4) $3$ 525. $For the matrix A = 320 - 3 - 3 - 1 441, A^{- 1} =$ (1) $A$ (2) $A^{2}$ (3) $A^{3}$ (4) $A^{4}$ 526. $If A = 147258369, then (A A^{'})^{'} =$ (1) $14321432122325019450$ (2) $1432505012219432194122$ (3) $1432503219414950122256$ (4) $143250327712250122194$ 527. $If a, b, c are non-zero real numbers and if the equations (a - 1) x = y + z, (b - 1) y = z + x, (c - 1) z = x + y have a non-trivial solution, then ab + bc + ca =$ (1) $a^{2} b^{2} c^{2}$ (2) $0$ (3) $abc$ (4) $a + b + c$ 528. $Consider the following statements : (i) If any two rows or columns of a determinant are identical, then the value of the determinant is zero. (ii) If the corresponding rows and columns of a determinant are interchanged, then the value of the determinant does not change. (iii) If any two rows (or columns) of a determinant are interchanged, then the value of the determinant changes in sign. Which of these are correct?$ (1) $(i) and (iii)$ (2) $(i) and (ii)$ (3) $(i), (ii) and (iii)$ (4) $(ii) and (iii)$ 529. $Let A = (23 - 5 1), what is f (A) = ?, where f (x) = x^{3} - 2 x^{2} - 5 .$ (1) $(- 50 42 7036)$ (2) $(50427036)$ (3) $(- 50 - 42 70 - 36)$ (4) $(- 50 - 42 7036)$ 530. $If A = 125 - 2 - 6 0 254 then Adj A =$ (1) $- 24 1730 8 - 6 - 10 2 - 1 2$ (2) $- 24 17 - 30 8 - 6 10 21 - 2$ (3) $- 24 1730 8 - 6 - 10 2 - 1 - 2$ (4) $24 - 17 30 - 8 - 6 - 10 21 - 2$ 531. $Find the value of k, if k - 2 k - 4 k - 8 2 k - 3 2 k - 9 2 k - 27 3 k - 4 3 k - 16 3 k - 64 = 0$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 532. $If x = a, y = b, z = c is the solution of the system of simultaneous linear equations x + y + z = 4, x - y + z = 2, x + 2 y + 2 z = 1, then ab + bc + ca =$ (1) $0$ (2) $-25$ (3) $1$ (4) $-4$ 533. $If A is a matrix of order m \times n and B is a matrix such that A B^{’} and B^{’} A are both defined, the order of the matrix B is$ (1) $m \times m$ (2) $n \times n$ (3) $n \times m$ (4) $m \times n$ 534. $If A = 101 02 - 1 104, A = B + C, B = B^{T} and C = - C^{T}, then C =$ (1) $0 - 0.5 0 0.5 00 000$ (2) $000 00 - 0.5 0 0.5 0$ (3) $0 0.5 - 0.5 - 0.5 00 0.5 00$ (4) $0 - 0.5 0 0.5 0 - 0.5 0 0.5 0$ 535. $If A = (12 - 2 1 13), B = 231121, then (A B)^{'} is equal to$ (1) $(- 3 10 - 2 7)$ (2) $(- 3 - 2 107)$ (3) $(- 3 10 72)$ (4) $(- 3 10 7 - 2)$ 536. $If A = \frac{1}{7} 3 - 6 - 2 - 2 - 3 6 623, then$ (1) $A^{- 1} = A$ (2) $A^{- 1} = A^{T}$ (3) $A^{- 1} does not exist$ (4) $A^{- 1} = - A$ 537. $Solve the system of equations x + 2 y + z = 4, - x + y + z = 0 and x - 3 y + z = 4 .$ (1) $x = 2, y = 0, z = 2$ (2) $x = 2, y = 0, z = - 2$ (3) $x = - 2, y = 2, z = 0$ (4) $x = - 2, y = 0, z = 2$ 538. $If A is 3 \times 4 matrix and B is a matrix such that A^{’} B and B A^{’} are both defined, then B is of the type$ (1) $4 \times 4$ (2) $3 \times 4$ (3) $4 \times 3$ (4) $3 \times 3$ 539. $The total number of 3 \times 3 matrices A having entries from the set {0, 1, 2, 3} such that the sum of all the diagonal entries of A A^{T} is 9, is equal to____$ 540. $If A = 100110111 and M = A + A^{2} + A^{3} + ... + A^{20} then the sum of all the elements of the matrix M is equal to$