Matrices & Determinants PYQs Set 13 (PDF)

Set 13 of 19 · Questions 361-390 of 565

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 13 (Q361-390)
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361. $Let A be a 2 x 2 matrix with real entries such that A' = α A + 1, where α \in R -_{- 1, 1} . If det (A^{2} -A)= 4, the sum of all possible values of a is equal to$ (1) $0$ (2) $\frac{3}{2}$ (3) $2$ (4) $\frac{5}{2}$ 362. $Let A and B be two 3x3 non-zero real matrices such that AB is a zero matrix. Then$ (1) $The system of linear equations AX = 0 has a unique solution$ (2) $The system of linear equations AX = 0 has infinitely many solutions$ (3) $B is an invertible matrix$ (4) $adj (a) is an invertible matrix$ 363. $Consider the system of linear equations -x + y + 2z = 0 3x - ay + 5z = 1 2x - 2y - az= 7 Let S_{1} be the set of all a \inR for which the system is inconsistent and be the set of all a \inR for which thesystem has infinitely many solutions. If n (S_{1}) and n (S_{2}) denote the number of elements in S_{1} and S_{2} respectively, then$ (1) $n (S_{1}) = 2, n (S_{2}) = 2$ (2) $n (S_{1}) = 1, n (S_{2}) = 0$ (3) $n (S_{1}) = 2, n (S_{2}) = 0$ (4) $n (S_{1}) = 0, n (S_{2}) = 2$ 364. $The ordered pair (a, b), for which the system of linear equations 3x - 2y + z = b 5x - 8y + 9z = 3 2x + y + az = -1 has no solution, is :$ (1) $(3, \frac{1}{3})$ (2) $(- 3, \frac{1}{3})$ (3) $(- 3, - \frac{1}{3})$ (4) $(3, - \frac{1}{3})$ 365. $Let A and B be two 3x3 real matrices such that (A^{2} - B^{2}) is invertible matrix. If A^{5} = B^{5} and A^{3} B^{2} = A^{2} B^{3}, then the value of the determinant of the matrix A^{3} + B^{3} is equal to :$ (1) $2$ (2) $4$ (3) $1$ (4) $0$ 366. $Let P = 323 - 1 0 - 5 - 2 α 0, where α \in R. Suppose Q = [q_{ij}] is a matrix such that PQ = k Ι, where k \in R, k \neq 0 and Ι is the identity matrix of order 3. If q_{23} = - \frac{k}{8} and det (Q) = \frac{k ^{2}}{2}, then$ (1) $α = 0, k = 8$ (2) $4α - k + 8 = 0$ (3) $det (P adj (Q)) = 2^{9}$ (4) $det (Q adj (P)) = 2^{13}$ 367. $The system of linear equations x + λy - z = 0 λx -y -z = 0 x +y - λz = 0 has a non-trivial solution for :$ (1) $Exactly one value of λ.$ (2) $Exactly two values of λ.$ (3) $Exactly three values of λ.$ (4) $Infinitely many values of λ.$ 368. $Let A = [a_{y}] be 3 \times 3 a matrix, where a_{ij} = ⎩ ⎨ ⎧ 1, - x, 2 x + 1, if i = j if ∣ i - j ∣ = 1 otherwise Let a function f : R \to R be defined as f(x) = det(A). Then the sum of maximum and minimum values of f on R is equal to:$ (1) $- \frac{20}{27}$ (2) $\frac{88}{27}$ (3) $\frac{20}{27}$ (4) $- \frac{88}{27}$ 369. $Let the system of linear equations_{X} + y_{+} α z_{=} 2 3x + y + z = 4 x + 2z = 1 have a unique solution (x, y, z). If (a, x), (y, a) and (x, -y*) are collinear points, then the sum of absolute values of all possible values of α is :$ (1) $4$ (2) $3$ (3) $2$ (4) $1$ 370. $Let 210 12 - 1 0 - 1 2 . If \|adj(adj(adj2A))\|= (16)^{n}, then n is equal to$ (1) $8$ (2) $10$ (3) $9$ (4) $12$ 371. $If the system of linear equations 2x + 3y - z = - 2 x + y + z = 4 x-y+\| λ \|z = 4 λ -4 where λ \in R, has no solution, then$ (1) $λ = 7$ (2) $λ = -7$ (3) $λ = 8$ (4) $λ =1$ 372. $Let the matrix A = 001100010 and the matrix B_{0} = A^{49} + 2 A^{98} . If B_{n} = Adj(B_{n - 1}) for all n \geq 1, then det(B_{4}) is equal to :$ (1) $3^{28}$ (2) $3^{30}$ (3) $3^{32}$ (4) $3^{36}$ 373. $Let M and N be two 3 \times 3 matrices such that MN = NM. Further, if M \neq N2 and M2 = N4, then$ (1) $determinant of (M2 + MN2) is 0$ (2) $there is a 3 \times 3 non-zero matrix U such that (M2 + MN2)U is the zero matrix$ (3) $determinant of (M2 + MN2) \geq 1$ (4) $for a 3 \times 3 matrix U, if (M2 + MN2)U equals the zero matrix then U is the zero matrix$ 374. $If α + β + γ = 2 π, then the system of equations x + (cos γ)y+ (cos β)z = 0 (cos γ)x + y + (cos α)z = 0 (cos β)x+ (cos α)y + z = 0 has :$ (1) $no solution$ (2) $infinitely many solution$ (3) $exactly two solutions$ (4) $a unique solution$ 375. $The set of all value of λ for which the system of linear equations : 2x1 - 2x2 + x3 = λx1 2x1 - 3x2 + 2x3 = λx2 - x1 + 2x2 = λx3 has a non-trivial solution,$ (1) $is an empty set$ (2) $is a singleton$ (3) $contains two elements$ (4) $contains more than two elements$ 376. $Two fair dice are thrown. The numbers on them are taken as λ and μ, and a system of linear equations x + y + z = 5 x + 2y + 3z = μ x + 3y + λz= 1 is constructed. If p is the probability that the system has a unique solution and is the probability that the system has no solution, then :$ (1) $p = \frac{1}{6} and q = \frac{1}{36}$ (2) $p = \frac{5}{6} and q = \frac{5}{36}$ (3) $p = \frac{5}{6} and q = \frac{1}{36}$ (4) $p = \frac{1}{6} and q = \frac{5}{36}$ 377. $If the system of linear equations x + ky + 3z = 0 3x + ky - 2z = 0 2x + 4y - 3z = 0 has a non-zero solution (x, y, z), then \frac{x z}{y ^{2}} is equal to :$ (1) $- 30$ (2) $30$ (3) $-10$ (4) $10$ 378. $Let A = 1 α 1 231312 for ,\|A\|= 2. If \|2adj(2adj(2A))\| = 3 2^{n}, then 3n + α is equal to$ (1) $9$ (2) $11$ (3) $12$ (4) $10$ 379. $The number of 3 \times 3 non-singular matrices, with four entries as 1 and all other entries as 0, is$ (1) $5$ (2) $6$ (3) $at least 7$ (4) $less than 4$ 380. $Which of the following values of α satisfy the equation = - 648α ?$ (1) $- 4$ (2) $9$ (3) $- 9$ (4) $4$ 381. $Which of the following matrices can NOT be obtained from the matrix [- 1 1 2 - 1] by a single elementary row operation?$ (1) $[01 1 - 1]$ (2) $[1 - 1 - 1 2]$ (3) $[- 1 - 2 27]$ (4) $[- 1 - 1 23]$ 382. $Let the system of linear equations -x + 2y-9z= 7 -x + 3y + 7z = 9_2x + y + 5z = 8 -3x + y + 13z = A has a unique solution x = α, y = β, z = γ . Then the distance of the point (α, β, γ) from the plane 2x - 2y + z = λ is$ (1) $11$ (2) $7$ (3) $9$ (4) $13$ 383. $The number of values of k for which the linear equations 4x + ky + 2z = 0 kx + 4y + z = 0 2x + 2y + z = 0 posses a non-zero solution is :$ (1) $3$ (2) $2$ (3) $1$ (4) $zero$ 384. $The total number of distinct x \in R for which x 2 x 3 x x^{2} 4 x^{2} 9 x^{2} 1 + x^{3} 1 + 8 x^{3} 1 + 27 x^{3} = 10 is$ 385. $The number of elements in the set {A = (a 0 b d); a, b, d \in {- 1, 0, 1} and (I - A)^{3} = I - A^{3}} where I is 2x2 identity matrix, is :$ 386. $Let f (x) = sin^{2} x 2 + sin^{2} x sin^{2} x - 2 + cos^{2} x cos^{2} x cos^{2} x cos 2 x cos 2 x 1 + cos 2 x, x \in [0, π] Then the maximum value of f(x) is equal to$ 387. $If the system of linear equations 2x - 3y = γ + 5, α x + 5y = β +1, where α, β, γ \in R has infinitely many solutions, then the value of ∣9 α + 3 β + 5 γ ∣ is equal to$ 388. $Let a, b, c, d be in arithmetic progression with common difference λ. If x + a - c x - 1 x - b + d x + b x + c x + c x + a x + b x + c = 2 then value of λ^{2} is equal to ___$ 389. $Let A = (a_{1} a_{2}) and B = (b_{1} b_{2}) be two 2 \times 1 matrices with real entries such that A = X B, where X = \frac{1}{3} (11 - 1 k), and k \in R . If a_{1}^{2} + a_{2}^{2} = \frac{2}{3} (b_{1}^{2} + b_{2}^{2}) and (k^{2} + 1) b_{2}^{2} \neq = - 2 b_{1} b_{2}, then the value of k is$ 390. $Let S be the set containing all 3 x 3 matrices with entries from {-1, 0, 1}. The total number of matrices A S such that the sum of all the diagonal elements of A is 6 is_________.$