Matrices & Determinants PYQs Set 2 (PDF)

Set 2 of 19 · Questions 31-60 of 565

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 2 (Q31-60)
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31. $The number of values of k, for which the system of equations : (k + 1)x + 8y = 4k kx + (k + 3)y = 3k - 1 has no solution, is$ (1) $infinite$ (2) $1$ (3) $2$ (4) $3$ 32. $Let the matrix A = 110001010 satisfy A^{n} = A^{n - 2} + A^{2} - I for n \geq 3 . Then the sum of all the elements of A^{50} is:$ (1) $44$ (2) $39$ (3) $52$ (4) $53$ 33. $If A = (cos α - sin α sin α cos α), then A A^{'} =$ (1) $A$ (2) $zero matrix$ (3) $A^{’}$ (4) $I$ 34. $If A = x 11 1 x 1 11 x and B = x 1 1 x, then \frac{d A}{d x} =$ (1) $3 B + 1$ (2) $3 B$ (3) $- 3 B$ (4) $1 - 3 B$ 35. $The values of λ and μ such that the system of equations x + y + z = 6, 3x + 5y + 5z = 26, x + 2y + λz = μ has no solution, are :$ (1) $λ = 3, μ = 5$ (2) $λ = 3, μ \neq 10$ (3) $λ \neq 2, μ = 10$ (4) $λ = 2, μ \neq 10$ 36. $If a matrix A is both symmetric and skew symmetric, then$ (1) $A is diagonal matrix$ (2) $A is a zero matrix$ (3) $A is scalar matrix$ (4) $A is square matrix$ 37. $Let n be the number obtained on rolling a fair die. If the probability that the system x - ny + z = 6 x + (n - 2) y + (n + 1) z = 8 (n - 1) y + z = 1 has a unique solution is \frac{k}{6}, then the sum of k and all possible values of n is :$ (1) $20$ (2) $24$ (3) $21$ (4) $22$ 38. $If 201 13 - 1 1 - 1 1 x y z = 110, then x y z =$ (1) $110 + K 31 - 2, K \in R$ (2) $110 + K 1 - 2 3, K \in R$ (3) $110 + K - 2 13, K \in R$ (4) $10 - 1 + K - 2 13, K \in R$ 39. $If A = (2 - 1 12), B = (1101), C = A B A^{T} and X = A^{T} C^{2} A, then det X is equal to :$ (1) $243$ (2) $729$ (3) $27$ (4) $891$ 40. $Let A be a 3 \times 3 real matrix such that A^{2} (A - 2 I) - 4 (A - I) = O, where I and O are the identity and null matrices, respectively. If A^{5} = α A^{2} + βA + γI, where α, β, and γ are real constants, then α + β + γ is equal to:$ (1) $76$ (2) $12$ (3) $4$ (4) $20$ 41. $If A = (1 - 1102 - 3210) and B = adj A, C = 5 A, then \frac{\| adj B \|}{\| C \|} =$ (1) $1$ (2) $5$ (3) $25$ (4) $-1$ 42. $If Δ = (1 cos θ 1 - cos θ 1 cos θ - 1 - cos θ 1), then Δ lies in the interval$ (1) $[2, 4]$ (2) $(2, 4)$ (3) $[1, 4]$ (4) $[- 1, 1]$ 43. $If A and B are matrices of order 3 and \| A \| = 5, \| B \| = 3, then \| 3 AB \| is$ (1) $425$ (2) $405$ (3) $565$ (4) $585$ 44. $If M and N are square matrices of order 3 , then which one of the following statements is not true?$ (1) $For all symmetric matrices M and N, MN - NM is skew symmetric$ (2) $N^{T} MN is symmetric or skew symmetric according as M is symmetric or skew symmetric$ (3) $For all symmetric matrices M and N, matrix MN is symmetric$ (4) $For any two matrices M and N, adj ( MN ) and adj (NM) need not be equal$ 45. $If (a + b + 2 c ab c 2 a + b + c b c aa + 2 b + c) = 2 then a^{3} + b^{3} + c^{3} - 3 abc =$ (1) $1 - 3 ab - 3 bc - 3 ca$ (2) $0$ (3) $1 - 2 ab - 2 bc - 2 ca$ (4) $1$ 46. $For α, β \in R and a natural number n, let A_{r} = r 2 r 3 r - 2 123 \frac{n ^{2}}{2} + α n^{2} - β \frac{n ( 3 n - 1 )}{2} . Then 2 A_{10} - A_{8} is equal to$ (1) $0$ (2) $4 α + 2 β$ (3) $2 α + 4 β$ (4) $2 n$ 47. $If A = (0110), then A^{2} is equal to$ (1) $(1100)$ (2) $(0011)$ (3) $(0110)$ (4) $(1001)$ 48. $If 1 + sin^{2} θ sin^{2} θ sin^{2} θ cos^{2} θ 1 + cos^{2} θ cos^{2} θ 4 sin 2 θ 4 sin 2 θ 4 sin 2 θ - 1 = 0 and 0 < θ < \frac{π}{2}, then cos 4 θ =$ (1) $\frac{- 1}{2}$ (2) $\frac{1}{2}$ (3) $\frac{3}{2}$ (4) $0$ 49. $Let A be a square matrix of order 3 \times 3, then \| 5 A \| =$ (1) $5 \| A \|$ (2) $125 \| A \|$ (3) $25 \| A \|$ (4) $15 \| A \|$ 50. $If A = (10 0 - 1), P = (1011) and X = A P A^{T}, then A^{T} X^{50} A =$ (1) $(0110)$ (2) $(20 1 - 1)$ (3) $(251 1 - 25)$ (4) $(10501)$ 51. $Let x = ((ab c d) : a, b, c, d \in R) . If f : X \to R is defined by f (A) = det (A) \forall A \in X, then f is$ (1) $one-one but not onto$ (2) $onto but not one-one$ (3) $one-one and onto$ (4) $neither one-one nor onto$ 52. $Let A be the set of all 3 \times 3 determinants with entries 0 or 1 only and B be the subset of A consisting of all determinants with value 1 . If C is the subset of A consisting of all determinants with value -1 , then$ (1) $n (C) = 0$ (2) $n (B) = n (C)$ (3) $A = B \cup C$ (4) $n (B) = 2 n (A)$ 53. $Let A = 123012001 . If u_{1} and u_{2} are column matrices such that A u_{1} = 100 and A u_{2} = 010, then u_{1} + u_{2} is equal to$ (1) $- 1 - 1 0$ (2) $1 - 1 - 1$ (3) $- 1 10$ (4) $- 1 1 - 1$ 54. $Let B = 21 - 1 601 41 - 1 and C = - 1 12 010132 . If a matrix A is such that B A C = I, then A^{- 1} =$ (1) $- 3 02 - 5 92 5146$ (2) $- 3 02 - 5 014 5916$ (3) $- 3 02 - 5 914 - 6 26$ (4) $- 3 02 - 5 914 - 5 26$ 55. $Let the system of linear equations x + y + kz = 2 2 x + 3 y - z = 1 3 x + 4 y + 2 z = k have infinitely many solutions. Then the system (k + 1) x + (2 k - 1) y = 7 (2 k + 1) x + (k + 5) y = 10 has$ (1) $infinitely many solutions$ (2) $unique solution satisfying x - y = 1$ (3) $No solution$ (4) $unique solution satisfying x + y = 1$ 56. $Consider the simultaneous linear equations βx + αy - z = - 1, 3 x - βy + αz = 0 αx + βy + z = 1 . In the usual notation used in Crammer's rule, given that \frac{Δ _{1}}{Δ} = - 1, \frac{Δ _{2}}{Δ} = 1. \frac{Δ _{3}}{Δ} = 2, then (α, β) =$ (1) $(1, 2)$ (2) $(2, 1)$ (3) $(- 1, 2)$ (4) $(1, - 2)$ 57. $Let A be a 3 \times 3 matrix such that A + A^{T} = O . If A 1 - 1 0 = 332, A^{2} 1 - 1 0 = - 3 19 - 24 and det (adj (2 adj (A + I))) = (2)^{α} \cdot (3)^{β} \cdot (11)^{γ}, α, β, γ are non-negative integers, then α + β + γ is equal to ____$ 58. $Let ∣ A ∣ = 6, where A is a 3 \times 3 matrix. If adj (3 adj (A^{2} \cdot adj (2 A))) = 2^{m} \cdot 3^{n}, m, n \in N, then m + n is equal to ____.$ 59. $Let A be a 3x3 real matrix. If det(2Adj(2Adj(Adj(2 A)))) = 2^{41}, then the value of d e t_{(} A_{)}^{2} equal .$ 60. $The number of all 3\times3 matrices A, with entries from the set (- 1, 0, 1) such that the sum of the diagonal elements of A A^{T} is 3, is\dots.$