Matrices & Determinants PYQs Set 14 (PDF)

Set 14 of 19 · Questions 391-420 of 565

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 14 (Q391-420)
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391. $Let [λ] be the greatest integer less than or equal to λ. The set of all values of λ for which the system of linear equations x + y + z = 4, 3x + 2y + 5z = 3 9x + 4y + (28 + [λ])z= [λ] has a solution is:$ (1) $R$ (2) $(-\infty,-9) \cup (-9,\infty)$ (3) $[-9,-8)$ (4) $(-\infty,-9) \cup [-8,\infty)$ 392. $If A = 12 a 212 2 - 2 b is a matrix satisfying the equation AAT = 9Ι, where Ι is 3 \times 3 identity matrix, then the ordered pair (a, b) is equal to :$ (1) $(2, - 1)$ (2) $(-2, 1)$ (3) $(2, 1)$ (4) $(-2, - 1)$ 393. $Let λ, μ \in R . If the system of equations 3 x + 5 y + λz = 37 x + 11 y - 9 z = 297 x + 155 y - 189 z = μ has infinitely many solutions, then μ + 2 λ is equal to:$ (1) $24$ (2) $25$ (3) $22$ (4) $27$ 394. $The number of distinct real roots of sin x cos x cos x cos x sin x cos x cos x cos x sin x = 0 in the interval - \frac{π}{4} \leq x \leq \frac{π}{4} is:$ (1) $4$ (2) $1$ (3) $2$ (4) $3$ 395. $Let A = (1 - 2 2 - 5) . Let α, β \in R such that α A^{2} + β A = 2 I . Then α + β is equal to -$ (1) $-10$ (2) $-6$ (3) $6$ (4) $10$ 396. $For the system of linear equations 2x + 4y + 2az = b x + 2y + 3z = 4 2x + 5y + 2z = 8 which of the following is NOT correct?$ (1) $It has unique solution if a = b = 6$ (2) $It has infinitely many solutions if a = 3, b = 6$ (3) $It has infinitely many solutions if a = 3, b = 8$ (4) $It has unique solution if a = b = 8$ 397. $Let P = [a_{ij}] be a 3 \times 3 matrix and let Q = [b_{ij}], where b_{ij} = 2^{i + j} a_{ij} for 1 \leq i, j \leq 3 . If the determinant of P is 2, then the determinant of the matrix Q is$ (1) $2^{10}$ (2) $2^{11}$ (3) $2^{12}$ (4) $2^{13}$ 398. $If A = [2 - 4 - 3 1], then adj (3 A^{2} + 12A) is equal to$ (1) $[72 - 63 - 84 51]$ (2) $[51846372]$ (3) $[51638472]$ (4) $[72 - 84 - 63 51]$ 399. $The number of values of a for which the system of equations : x + y + z = α α x + 2 α y + 3z = -l x + 3 α y + 5z =4 is inconsistent, is$ (1) $0$ (2) $1$ (3) $2$ (4) $3$ 400. $How many 3 \times 3 matrices M with entries from {0, 1, 2} are there, for which the sum of the diagonal entries of M^{T} M is 5 ?$ (1) $198$ (2) $162$ (3) $126$ (4) $135$ 401. $If the system of linear equations 2x + y - z = 7 x - 3y + 2z = 1 x + 4y + δ z = k, where δ, k \in R has infinitely many solutions, then δ + k is equal to:$ (1) $-3$ (2) $3$ (3) $6$ (4) $9$ 402. $If the matrix A = (0 K 2 - 1) satisfies A =_{(} A^{3} + 3 I_{)} = 2 I then the value K of is :$ (1) $\frac{1}{2}$ (2) $- \frac{1}{2}$ (3) $-1$ (4) $1$ 403. $If the system of equations 2x+y-z=5 2x-5y+ λ z = μ x+2y-5z = 7 has infinitely many solutions, then (λ + μ)^{2} + (λ - μ)^{2} is equal to$ (1) $904$ (2) $916$ (3) $912$ (4) $920$ 404. $If x - 4 2 x 2 x 2 x x - 4 2 x 2 x 2 x x - 4 = (A + Bx) (x - A)^{2} then the ordered pair (A, B) is equal to :$ (1) $(-4, 5)$ (2) $(4, 5)$ (3) $(-4, -5)$ (4) $(-4, 3)$ 405. $Let A and B be two square matrices of order 3 such that \| A \| = 3 and \| B \| = 2 . Then (A^{T} A (adj (2 A))^{- 1} (adj (4 B)) (adj (AB))^{- 1} AA^{T}) is equal to :$ (1) $108$ (2) $32$ (3) $81$ (4) $64$ 406. $Let the system of linear equations x + 2y + z = 2, α x + 3y - z = α, -ax + y + 2z = - α be inconsistent. Then α is equal to :$ (1) $\frac{5}{2}$ (2) $- \frac{5}{2}$ (3) $\frac{7}{2}$ (4) $- \frac{7}{2}$ 407. $Let A = (4 α - 2 β) If A^{2} + γ A + 18I = O, then det (a) is equal to _______ .$ (1) $-18$ (2) $18$ (3) $-50$ (4) $50$ 408. $If A = [1 λ 510], A^{- 1} = α A + β I and α + β = - 2, then 4 α^{2} + β^{2} + λ^{2} is equal to:$ (1) $12$ (2) $19$ (3) $14$ (4) $10$ 409. $Let M be a 2 \times 2 symmetric matrix with integer entries. Then M is invertible if$ (1) $the first column of M is the transpose of the second row of M$ (2) $the second row of M is the transpose of first column of M$ (3) $M is a diagonal matrix with nonzero entries in the main diagonal$ (4) $the product of entries in the main diagonal of M is not the square of an integer$ 410. $The system of equations -kx + 3y-14z = 25 -15x + 4y - kz = 3 -4x + y + 3z = 4 is consistent for all k in the set$ (1) $R$ (2) $R-{-11,13}$ (3) $R-{13}$ (4) $R-{-11,11}$ 411. $The number of θ \in_{(} 0, 4 π_{)} for which the system of linear equations 3(sin3 θ)x-y + z = 2 3(cos2 θ)x + 4y + 3z = 3 6x + 7y + 7z = 9 has no solution is :$ (1) $6$ (2) $7$ (3) $8$ (4) $9$ 412. $Which of the following is(are) NOT the square of a 3 \times 3 matrix with real entries?$ (1) $100010 00 - 1$ (2) $100 0 - 1 0 00 - 1$ (3) $- 1 00 0 - 1 0 00 - 1$ (4) $100010001$ 413. $If xyz are not equal and \neq = 0, \neq = 1 the value of lo g x lo g 2 x lo g 3 x lo g y lo g 2 y lo g 3 y lo g z lo g 2 z lo g 3 z is equal to$ (1) $log (xyz)$ (2) $log (6 xyz)$ (3) $0$ (4) $log (x + y + z)$ 414. $Let M = {A = (a c b d) : a, b, c, d \in {\pm 3, \pm 2, \pm 1, 0}} Define f : M \to Z, as f(A) = det(A), for all A \in M where Z is set of all integers. Then the number of A \in M such that f(A) = 15 is equal to$ 415. $Let A be a matrix of order 2x2, whose entries are from the set {0, 1, 2, 3, 4, 5}. If the sum of all the entries of A is a prime number p, 2 < p < 8, then the number of such matrices A is :$ 416. $Let P be a matrix of order 3 × 3 such that all the entries in P are from the set {–1, 0, 1}. Then, the maximum possible value of the determinant of P is _____ .$ 417. $The positive value of the determinant of the matrix A d j (A d j (A)) = 14 - 14 28 2814 - 14 - 14 2814, is__________.$ 418. $Let 5 be the set of values of λ, for which the system of equations 6 λ x - 3y + 3z = 4 λ^{2}, 2x + 6 λ y + 4z = 1 and 3x + 2y + 3 λ z = λ has no solution. Then, 12 Σ_{λ \in S} ∣ λ ∣ is equal to _______.$ 419. $The number of matrices A = [a c b b], where a, b, c, d \in_{- 1, 0, 1, 2, 3, ...., 10}, such that A = A^{- 1}, is _______$ 420. $Let A = 010100001 Then the number of 3x3matrices B with entries from the set {1, 2, 3, 4, 5} and satisfying AB = BA is$