Matrices & Determinants PYQs Set 17 (PDF)

Set 17 of 19 · Questions 481-510 of 565

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 17 (Q481-510)
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481. $If A = [12 - 3 k] and A^{2} - 4 A + 10 I = A, then k =$ (1) $0$ (2) $- 4$ (3) $4 and not 1$ (4) $1 or 4$ 482. $If Δ = x - 2 2 x - 3 3 x - 5 2 x - 3 3 x - 4 5 x - 8 3 x - 4 4 x - 5 10 x - 17 = A x^{3} + B x^{2} + C x + D, then B + C is equal to:$ (1) $9$ (2) $- 1$ (3) $1$ (4) $- 3$ 483. $Let α β \neq = 0 and A = β α - β α α α 3 β 2 α . If B = 3 α - α - 2 α - 9 75 3 α - 2 α - 2 β is the matrix of cofactors of the elements of A, then det (A B) is equal to :$ (1) $64$ (2) $216$ (3) $343$ (4) $125$ 484. $If the system of equations (λ - 1) x + (λ - 4) y + λ z = 5, λ x + (λ - 1) y + (λ - 4) z = 7, (λ + 1) x + (λ + 2) y - (λ + 2) z = 9 has infinitely many solutions, then λ^{2} + λ is equal to$ (1) $6$ (2) $10$ (3) $20$ (4) $12$ 485. $Let α, β (α \neq = β) be the values of m, for which the equations x + y + z = 1; x + 2 y + 4 z = m and x + 4 y + 10 z = m^{2} have infinitely many solutions. Then the value of \sum_{n = 1}^{10} (n^{α} + n^{β}) is equal to:$ (1) $3080$ (2) $560$ (3) $3410$ (4) $440$ 486. $If the system of linear equations 3 x + y + β z = 3, 2 x + α y - z = - 3, x + 2 y + z = 4 has infinitely many solutions, then the value of 22 β - 9 α is:$ (1) $31$ (2) $37$ (3) $43$ (4) $49$ 487. $If A = 0 - 1 - 2 i 2 - i 1 + 2 i 07 i - 2 K 0 and A^{- 1} does not exist, then K = (where i = - 1)$ (1) $1 + 2 i$ (2) $-7$ (3) $7$ (4) $1 - 2 i$ 488. $If A = \frac{1}{π} (sin^{- 1} (π x) sin^{- 1} (\frac{x}{π}) tan^{- 1} (\frac{x}{π}) cot^{- 1} (π x)), B = \frac{1}{π} (- cos^{- 1} (π x) sin^{- 1} (\frac{x}{π}) tan^{- 1} (\frac{x}{π}) - tan^{- 1} (π x)) then A - B is equal to$ (1) $I$ (2) $0$ (3) $2 I$ (4) $\frac{1}{2} I$ 489. $If P is a non-singular matrix such that I + P + P^{2} \dots + Pn^{n} = 0 ( 0 denotes the null matrix), then P^{- 1} =$ (1) $Pn^{n}$ (2) $- P^{n}$ (3) $- (1 + P + \dots + P^{n})$ (4) $- (1 + P + \dots + P^{n - 1)})$ 490. $Let S be the set of all λ \in R for which the system of linear equations 2 x - y + 2 z = 2 x - 2 y + λ z = - 4 x + λ y + z = 4 has no solution. Then the set S$ (1) $is a singleton.$ (2) $contains more than two elements$ (3) $is an empty set.$ (4) $contains exactly two elements.$ 491. $If A = (25 - 3 - 7), then A - A^{- 1} =$ (1) $(- 5 0 0 - 5)$ (2) $3 (31023)$ (3) $3 (3 \frac{10}{3} - 2 - 3)$ (4) $5 (1324)$ 492. $Let S denote the set of all real values of λ such that the system of equations λ x + y + z = 1 x + λ y + z = 1 x + y + λ z = 1 is inconsistent, then \sum_{λ \in S} (∣ λ ∣^{2} + ∣ λ ∣) is equal to$ (1) $2$ (2) $12$ (3) $4$ (4) $6$ 493. $For i = 1, 2, 3 and j = 1, 2, 3, if a_{i}^{2} + b_{i}^{2} + c_{i}^{2} = 1, a_{i} a_{j} + b_{i} b_{j} + c_{i} c_{j} = 0 \forall i \neq = j and A = a_{1} b_{1} c_{1} a_{2} b_{2} c_{2} a_{3} b_{3} c_{3} then det (A A^{T}) =$ (1) $0$ (2) $1$ (3) $-1$ (4) $3$ 494. $Let P be a square matrix such that P^{2} = I - P . For α, β, γ, δ \in N, if P^{α} + P^{β} = γ I - 29 P and P^{α} - P^{β} = δ I - 13 P, then α + β + γ - δ is equal to$ (1) $18$ (2) $40$ (3) $22$ (4) $24$ 495. $Let A = 1000412 0 - 1 - 3 . Then the sum of the diagonal elements of the matrix (A + I)^{11} is equal to$ (1) $6144$ (2) $4094$ (3) $4097$ (4) $2050$ 496. $Let A = 2631232112 and P = 157201025 . The sum of the prime factors of P^{- 1} A P - 2 I is equal to$ (1) $26$ (2) $27$ (3) $66$ (4) $23$ 497. $Let α, β, γ be the real roots of the equation x^{3} + a x^{2} + b x + c = 0, (a, b, c \in R and a, b \neq = 0) . If the system of equations (in u, v, w) given by α u + β v + γ w = 0, β u + γ v + α w = 0, γ u + α v + β w = 0 has non-trivial solution, then the value of \frac{a ^{2}}{b} is$ (1) $5$ (2) $3$ (3) $1$ (4) $0$ 498. $If the system of linear equations x + y + z = 5, x + 2 y + 2 z = 6, x + 3 y + λ z = μ, (λ, μ \in R) has infinitely many solutions, then the value of λ + μ is:$ (1) $12$ (2) $9$ (3) $7$ (4) $10$ 499. $Let A = 0 - 6 8, B = 306 5 - 1 - 1 - 7 80 and X = x y z . If D = [α β γ]^{T} is the solution of X^{T} B^{T} = A^{T}, then D^{T} A =$ (1) $0$ (2) $4$ (3) $-2$ (4) $6$ 500. $If (α β γ) 121232 3 - 5 5 = (352) then α^{3} + β^{3} + γ^{3} =$ (1) $8$ (2) $-6$ (3) $6$ (4) $-10$ 501. $If f (x) = cos x 00 1 2 cos x 1 03 2 cos x, then lim_{x \to π} f (x) =$ (1) $-1$ (2) $1$ (3) $0$ (4) $3$ 502. $If the system of equations x + 2 y - 3 z = 2, 2 x + λ y + 5 z = 5, 14 x + 3 y + μ z = 33 has infinitely many solutions, then λ + μ is equal to:$ (1) $13$ (2) $10$ (3) $12$ (4) $11$ 503. $If the system of equations 2 x - y + z = 4, 5 x + λ y + 3 z = 12, 100 x - 47 y + μ z = 212 has infinitely many solutions, then μ - 2 λ is equal to$ (1) $57$ (2) $59$ (3) $55$ (4) $56$ 504. $The maximum value of f (x) = sin^{2} x 1 + sin^{2} x sin^{2} x 1 + cos^{2} x cos^{2} x cos^{2} x cos 2 x cos 2 x sin 2 x, x \in R is:$ (1) $7$ (2) $\frac{3}{4}$ (3) $5$ (4) $5$ 505. $If A is a symmetric matrix and B is a skew-symmetric matrix such that A + B = (25 3 - 1), then A B is equal to :$ (1) $(- 4 - 1 - 1 4)$ (2) $(4 - 1 - 2 - 4)$ (3) $(41 - 2 - 4)$ (4) $(- 4 1 24)$ 506. $Let x, y, z > 1 and A = 1 lo g_{y} x lo g_{z} x lo g_{x} y 2 lo g_{z} y lo g_{x} z lo g_{y} z 3$ (1) $6^{4}$ (2) $2^{8}$ (3) $4^{8}$ (4) $2^{4}$ 507. $Which of the following is false? [AP/Sept. 18, 2020 (I)] 1. If A is a skew symmetric matrix of order 5 \times 5, then the rank of A is less than 5 . 2. If P is a non-zero column matrix and Q is a non-zero row matrix, then the rank of PQ is 1 3. Rank of (123234567) is 2 4. If the lines a_{r} x + b_{r} y + c_{r} = 0 (r = 1, 2, 3) are distinct and intersect at a point, then rank of (a_{1} b_{1} c_{1} a_{2} b_{2} c_{2} a_{3} b_{3} c_{3}) is 3$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 508. $Let P = - 30 90120 20140605611214 and A = 2 - 1 0 7 - ω - ω ω^{2} 1 - ω + 1 where ω = \frac{- 1 + i 3}{2}, and I_{3} be the identity matrix of order 3 . If the determinant of the matrix (P^{- 1} A P - I_{3})^{2} is α ω^{2}, then the value of α is equal to_____$ 509. $Let A = x y z y z x z x y where x, y and z are real numbers such that x + y + z > 0 and x y z = 2 . If A^{2} = I_{3} then the value of x^{3} + y^{3} + z^{3} is$ 510. $Let M denote the set of all real matrices of order 3 \times 3 and let S = -3, -2, -1, 1, 2 . Let S_{1} = (A = (a_{ij}) \in M : A = A^{T} and a_{ij} \in S, \forall i, j), S_{2} = (A = (a_{ij}) \in M : A = - A^{T} and a_{ij} \in S, \forall i, j), S_{3} = (A = (a_{ij}) \in M : a_{11} + a_{22} + a_{33} = 0 and a_{ij} \in S, \forall i, j) . If n (S_{1} \cup_{2} US_{3}) = 125 α, then α equals .$