Matrices & Determinants PYQs Set 15 (PDF)

Set 15 of 19 · Questions 421-450 of 565

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 15 (Q421-450)
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421. $If the system of equations α x + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = β, has infinitely many solutions, then the ordered pair (α, β) is equal to :$ (1) $(1,-3)$ (2) $(-1,3)$ (3) $(1,3)$ (4) $(-l,-3)$ 422. $The number of symmetric matrices of order 3, with all the entries from the set {0,1,2,3,4,5,6,7,8,9} is$ (1) $6^{10}$ (2) $1 0^{6}$ (3) $9$ (4) $1 0^{9}$ 423. $Let a, λ, μ \in R. Consider the system of linear equations ax + 2y = λ 3x - 2y = μ Which of the following statement(s) is(are) correct ?$ (1) $if a = - 3, then the system has infinitely many solutions for all values of λ and μ$ (2) $If a \neq -3, then the system has a a unique solution for all values of λ and μ$ (3) $If λ + μ = 0, the the system has infinitely many solutions for a = -3$ (4) $If λ + μ \neq 0, then the system has no solution for a = -3$ 424. $If the following system of linear equations 2x + y + z = 5 x - y + z = 3 x + y + az= b has no solution, then :$ (1) $a = - \frac{1}{3}, b \neq = \frac{7}{3}$ (2) $a \neq = \frac{1}{3}, b = \frac{7}{3}$ (3) $a \neq = - \frac{1}{3}, b = \frac{7}{3}$ (4) $a = \frac{1}{3}, b \neq = \frac{7}{3}$ 425. $If the trivial solution is the only solution of the system of equations x - ky + z = 0 kx + 3y - kz = 0 3x + y - z = 0 then the set of all values of k is :$ (1) $R - {2, -3}$ (2) $R - { 2 }$ (3) $R - { -3 }$ (4) ${2, -3}$ 426. $The number of square matrices of order 5 with entries from the set (0, 1), such that the sum of all the elements in each row is 1 and the sum of the all the elements in each column is also 1, is$ (1) $225$ (2) $120$ (3) $150$ (4) $125$ 427. $Let A = [x + 1] [x] [x] [x + 2] [x + 3] [x + 2] [x + 3] [x + 3] [x + 4], where [t] denotes the greatest integer less than or equal to det(A) = 192. If , then the set of values of is the interval:$ (1) $[68, 69)$ (2) $[62, 63)$ (3) $[65, 66)$ (4) $[60, 61)$ 428. $If A = [cos θ sin θ - sin θ cos θ], then the matrix A^{-50} when θ = \frac{π}{12}, is equal to :$ (1) $[\frac{1}{2} \frac{3}{2} - \frac{3}{2} \frac{1}{2}]$ (2) $[\frac{3}{2} - \frac{1}{2} \frac{1}{2} \frac{3}{2}]$ (3) $[\frac{3}{2} \frac{1}{2} - \frac{1}{2} \frac{3}{2}]$ (4) $[\frac{1}{2} - \frac{3}{2} \frac{3}{2} \frac{1}{2}]$ 429. $If A = [5 a 3 - b 2] and A adj A = A A^{T}, then 5a + b is equal to$ (1) $5$ (2) $4$ (3) $13$ (4) $- 1$ 430. $Let A = [2 a 30], a \in R be written as P+Q where P is a symmetric matrix and Q is skew symmetric matrix. If det(Q) = 9, then the modulus of the sum of all possible values of P determinant of is equal to :$ (1) $36$ (2) $24$ (3) $45$ (4) $18$ 431. $Let ω \neq 1 be a cube root of unity and S be the set of all non-singular matrices of the form, 1 ω ω^{2} a 1 ω b c 1 where each of a, b and c is either ω or ω2. Then the number of distinct matrices in the set S is$ (1) $2$ (2) $6$ (3) $4$ (4) $8$ 432. $If P = 112 α 34 334 is the adjoint of a 3 \times 3 matrix A and \|A\| = 4, then α is equal to$ (1) $4$ (2) $11 ‘$ (3) $5$ (4) $0$ 433. $Let A = [a_{ij}] be a square matrix of order 3 such that a_{ij} = 2^{j - i}, for all i, j = 1, 2, 3. Then, the matrix A^{2} + A^{3} + ... + A^{10} is equal to :$ (1) $(\frac{3 ^{10} - 3}{2}) A$ (2) $(\frac{3 ^{10} - 1}{2}) A$ (3) $(\frac{3 ^{10} + 1}{2}) A$ (4) $(\frac{3 ^{10} + 3}{2}) A$ 434. $Let A and B be two 3x3 matrices such that AB= I and ∣ A ∣ = \frac{1}{8} then \|adj(Badj(2A))\| is equal to$ (1) $16$ (2) $32$ (3) $64$ (4) $128$ 435. $Let A be a 3 x 3 invertible matrix. If \|adj (24A)\| = adj(3adj(2A))\|, then \|A\|^{2} is equal to :$ (1) $6^{6}$ (2) $2^{12}$ (3) $2^{6}$ (4) $1$ 436. $If A = (\frac{1}{5} - \frac{2}{5} \frac{2}{5} \frac{1}{5}), B = (1 i 01), i = - 1, and Q = A^{T} BA, then the inverse of the matrix A Q^{2021} A^{T} is equal to :$ (1) $(\frac{1}{5} 2021 - 2021 \frac{1}{5})$ (2) $(1 - 2021 i 01)$ (3) $(1 2021 i 01)$ (4) $(10 - 2021 i 1)$ 437. $The equation whose roots are the values of the equation 111 - 3 6 3 x 14 x^{2} = 0 is$ (1) $x^{2} + x + 2 = 0$ (2) $x^{2} + x - 2 = 0$ (3) $x^{2} + 2 x + 2 = 0$ (4) $x^{2} - x - 2 = 0$ 438. $If x, y, z are in arithmetic progression with common difference d, x \neq = 3 d, and the determinant of the matrix 345 4252 k x y z is zero, then the value of k^{2} is -$ (1) $72$ (2) $12$ (3) $36$ (4) $6$ 439. $If Δ = n^{2} (n + 1)^{2} (n + 2)^{2} (n + 1)^{2} (n + 2)^{2} (n + 3)^{2} (n + 2)^{2} (n + 3)^{2} (n + 4)^{2} and 1 - 2 3 - 4 3 x 7 - 5 - 3 = 2Δ + 1, then x =$ (1) $3$ (2) $4$ (3) $5$ (4) $6$ 440. $If f (x) = 100 6 + x x - 3 2 x - 4 36 + x^{2} 3 x^{2} - 27 8 x^{2} - 32, then lim_{x \to 1} \frac{f ( x )}{f ( - x )} =$ (1) $2$ (2) $-1$ (3) $0$ (4) $1$ 441. $The trace of the matrix A = 1011 - 5 78 799 is$ (1) $17$ (2) $25$ (3) $3$ (4) $12$ 442. $The value of (a + 1) (a + 2) (a + 2) (a + 3) (a + 3) (a + 4) a + 2 a + 3 a + 4 111 is$ (1) $- 2$ (2) $(a + 1) (a + 2) (a + 3)$ (3) $0$ (4) $(a + 2) (a + 3) (a + 4)$ 443. $Let x = α, y = β, z = γ be the unique solution of the system of simultaneous linear equations 2 x + 3 y - 2 z + 4 = 0, 3 x - 4 y + 3 z + 5 = 0, k x - 2 y + z + 3 = 0 . If α = - 2 then k =$ (1) $1325$ (2) $5132$ (3) $3152$ (4) $3251$ 444. $The set of all values of t \in R, for which the matrix e^{t} e^{t} e^{t} e^{- t} (sin t - 2 cos t) e^{- t} (2 sin t + cos t) e^{- t} cos t e^{- t} (- 2 sin t - cos t) e^{- t} (sin t - 2 cos t) e^{- t} sin t is invertible, is$ (1) $((2 k + 1) \frac{π}{2}, k ϵ Z)$ (2) $(kπ + \frac{π}{4}, k ϵ Z)$ (3) $(kπ, k ϵ Z)$ (4) $R$ 445. $The number of all possible values of θ, where 0 < θ < π, for which the system of equations (y + z) cos 3θ = (xyz) sin 3θ x sin 3θ = \frac{2 cos 3 θ}{y} + \frac{2 s in 3 θ}{z} (xyz) sin 3θ = (y + 2z) cos 3 θ + y sin 3θ have a solution (x0, y0, z0) with y0 z0 \neq 0, is$ 446. $The number of matrices of order 3x3, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is______.$ 447. $For real numbers α and β, consider the following system of linear equations : x + y - z = 2, x + 2y + αz = 1, 2x - y + z = β If the system has infinite solutions, then α + β is equal to$ 448. $Let A be a 3 \times 3 matrix such that X^{T} A X = O for all nonzero 3 \times 1 matrices X = x y z . If A 111 = 14 - 5, A 121 = 04 - 8, and det (adj (2 (A + I))) = 2^{α} 3^{β} 5^{γ}, α, β, γ \in N, then α^{2} + β^{2} + γ^{2} is ____.$ 449. $Let I be the identity matrix of order 3 \times 3 and for the matrix A = λ 47 25 - 1 362, ∣ A ∣ = - 1 Let B be the inverse of the matrix adj (A adj (A^{2})) . Then ∣ (λ B + I) ∣ is equal to ___.$ 450. $Let A be a 2 \times 2 symmetric matrix such that A (11) = (37) and the determinant of A be 1. If A^{- 1} = α A + β I, where I is an identity matrix of order 2 \times 2, then α + β equals______$