Matrices & Determinants PYQs Set 12 (PDF)

Set 12 of 19 · Questions 331-360 of 565

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 12 (Q331-360)
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331. $Let A = (α 6 - 1 β), α > 0, such that det (A) = 0 and α + β = 1 . If I denotes 2 \times 2 identity matrix, then the matrix (I + A)^{8} is:$ (1) $(257514 - 64 - 127)$ (2) $(7661530 - 255 - 509)$ (3) $(10252024 - 511 - 1024)$ (4) $(46 - 1 - 1)$ 332. $For the system S of linear equations x + y + z = 3, 2 x + 2 y - z = 3, x + y + λz = 1 the incorrect option among the following statements is$ (1) $S has infinitely many solutions, if λ = - 1$ (2) $S has no solution, if λ = - 2$ (3) $S has no unique solution for any real λ$ (4) $S is consistent for all λ \in R$ 333. $Let a \in R and A be a matrix of order 3 \times 3 such that det (A) = - 4 and A + I = 12 a a 11 102, where I is the identity matrix of order 3 \times 3 . If det ((a + 1) adj ((a - 1) A)) is 2^{m} 3^{n}, m, n \in {0, 1, 2, \dots, 20}, then m + n is equal to:$ (1) $14$ (2) $17$ (3) $15$ (4) $16$ 334. $The probability that a randomly chosen 2x2 matrix with all the entries from the set of first 10 primes, is singular, is equal to :$ (1) $\frac{133}{1 0 ^{4}}$ (2) $\frac{18}{1 0 ^{3}}$ (3) $\frac{19}{1 0 ^{3}}$ (4) $\frac{271}{1 0 ^{4}}$ 335. $Let A = 246 2 + p 6 + 2 p 12 + 3 p 2 + p + q 8 + 3 p + 2 q 20 + 6 p + 3 q . If det (adj (adj (3 A))) = 2^{m} \cdot 3^{n}, m, n \in N, then m + n is equal to$ (1) $22$ (2) $20$ (3) $24$ (4) $26$ 336. $For a system of simultaneous linear equations, if A X = 112, Adj A = 111 - 1 11 - 1 - 1 1 and det A > 0, then X =$ (1) $- 1 02$ (2) $112$ (3) $0 - 1 - 1$ (4) $211$ 337. $Let a, b, c \in R be all non-zero and satisfy a^{3} + b^{3} + c^{3} = 2 . If the matrix A = (ab c b c a c ab) satisfies A^{T} A = 1, then a value of abc can be:$ (1) $2$ (2) $\frac{1}{3}$ (3) $- \frac{1}{3}$ (4) $\frac{2}{3}$ 338. $Let A = 0 p p 2 q q - q r - r r . If A A^{T} = I_{3} then ∣ p ∣ is:$ (1) $\frac{1}{5}$ (2) $\frac{1}{3}$ (3) $\frac{1}{2}$ (4) $\frac{1}{6}$ 339. $If A = (3121), then A^{2} + x A + y I = 0 for (x, y) =$ (1) $(- 1, 3)$ (2) $(- 4, 1)$ (3) $(1, 3)$ (4) $(4, 1)$ 340. $Let the system of equations 2 x + 3 y + 5 z = 9, 7 x + 3 y - 2 z = 8, 12 x + 3 y - (4 + λ) z = 16 - μ have infinitely many solutions. Then the radius of the circle centered at (λ, μ) and touching the line 4 x = 3 y is:$ (1) $\frac{7}{5}$ (2) $\frac{21}{5}$ (3) $7$ (4) $\frac{17}{5}$ 341. $If (x_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) are the vertices of a triangle whose area is k square units, then x_{1} x_{2} x_{3} y_{1} y_{2} y_{3} 444^{2} is$ (1) $32 k^{2}$ (2) $16 k^{2}$ (3) $64 k^{2}$ (4) $48 k^{2}$ 342. $Let A = (i - i - i i), i = - 1 . Then, the system of linear equations A^{8} (x y) = (864) has:$ (1) $A unique solution$ (2) $Infinitely many solutions$ (3) $No solution$ (4) $Exactly two solutions$ 343. $Let A = 00 - 1 0 - 1 0 - 1 00, B = 010100001 then (A^{- 1} B)^{- 1} + (A B^{- 1})^{- 1} =$ (1) $100 0 - 2 0 002$ (2) $00 - 2 - 2 00 0 - 2 0$ (3) $- 2 00 00 - 2 0 - 2 0$ (4) $0 - 2 0 00 - 2 - 2 00$ 344. $Let the numbers 2, b, c be in an A.P. and A = 124 1 b b^{2} 1 c c^{2} . If det (A) \in (2, 16), then c lies in the interval :$ (1) $[2, 3]$ (2) $(2 + 3^{\frac{3}{4}}, 4)$ (3) $[4, 6]$ (4) $(3, 2 + 2^{\frac{3}{4}})$ 345. $A value of b for which the rank of the matrix A = 14 b 9 1429 - 1 - 3 2 b 0123 is 3, is$ (1) $-2$ (2) $-4$ (3) $-6$ (4) $3$ 346. $If A is a square matrix of order 3, then Adj (Adj A^{2}) =$ (1) $\| A \|^{2}$ (2) $\| A \|^{4}$ (3) $\| A \|^{8}$ (4) $\| A \|^{16}$ 347. $P is a 3 \times 3 square matrix and Tr (P) \neq = 0 . If Tr (P - P^{T}) + Tr (P + P^{T}) + \frac{Tr ( P )}{Tr ( P ^{T} )} + Tr (P) \times Tr (P^{T}) = 0 then Tr (P) = then Tr (P) =$ (1) $0$ (2) $-1$ (3) $4$ (4) $3$ 348. $If x, y, z \in R, then the value of the determinant A = (5^{x} + 5^{- x})^{2} (6^{x} + 6^{- x})^{2} (7^{x} + 7^{- x})^{2} (5^{x} - 5^{- x})^{2} (6^{x} - 6^{- x})^{2} (7^{x} - 7^{- x})^{2} 111 is$ (1) $10$ (2) $12$ (3) $1$ (4) $0$ 349. $The symmetric part of the matrix A = 162 28 - 2 427 is$ (1) $0 - 2 - 1 - 2 0 - 2 - 1 - 2 0$ (2) $123480307$ (3) $02 - 1 - 2 0 - 2 120$ (4) $143480307$ 350. $If A = (2335), then the determinant of the matrix (A^{2025} - 3 A^{2024} + A^{2023}) is$ (1) $12$ (2) $28$ (3) $24$ (4) $16$ 351. $Let AX = D be a system of three linear non-homogeneous equations. If \| A \| = 0 and rank (A) = rank ([AD]) = α, then$ (1) $AX=D will have infinite number of solutions when α = 3$ (2) $AX = D will have unique solution when α < 3$ (3) $AX = D will have infinite number of solutions when α < 3$ (4) $AX = D will have no solution when α < 3$ 352. $If A and B are symmetric matrices of same order such that AB + BA = X and AB - BA = Y, then (XY)^{T} =$ (1) $XY$ (2) $X^{T} Y^{T}$ (3) $- YX$ (4) $- Y^{T} X^{T}$ 353. $If there exists a k^{th} order non-singular sub matrix in a matrix P of order m \times n, then the rank (ρ) of P$ (1) $satisfies k\leqρ\leqm$ (2) $satisfies k < ρ < n$ (3) $satisfies k\leqρ\leqmin{m,n}$ (4) $is equal to k + 1$ 354. $If the system of equations 2 x + 3 y - z = 0, x + ky - 2 z = 0 and 2 x - y + z = 0 has non-trivial solution (x, y, z) then \frac{x}{y} + \frac{y}{z} + \frac{z}{x} + k is equal to :$ (1) $\frac{3}{4}$ (2) $\frac{1}{2}$ (3) $- \frac{1}{4}$ (4) $- 4$ 355. $Investigate the values of λ and μ for the system x + 2 y + 3 z = 6, x + 3 y + 5 z = 9, 2 x + 5 y + λz = μ and match the values in List-I with the items in List-II. List-I (A) λ = 8, μ \neq = 15 (B) λ \neq = 8, μ \in R (C) λ = 8, μ = 15 List-II 1. Infinitely many solutions 2. No solution 3. Unique solution A B C$ (1) $1 2 3$ (2) $2 3 1$ (3) $3 1 2$ (4) $3 2 1$ 356. $For the system of equations x+y+z=6 x + 2y + α z = 10 x + 3y + 5z = β which one of the following is NOT true?$ (1) $System has no solution for α = 3, β = 24$ (2) $System has a unique solution for α = -3, β = 14$ (3) $System has infinitely many solutions for α = 3, β = 14$ (4) $System has a unique solution for α = 3, β \neq = 14$ 357. $If the system of equations x + y + az = b 2x + 5y + 2z = 6 x + 2y + 3z = 3 has infinitely many solutions, then 2a + 3b is equal to$ (1) $25$ (2) $20$ (3) $23$ (4) $28$ 358. $It S is the set of distinct values of ‘b’ for which the following system of linear equations x + y + z = 1 x + ay + z = 1 ax + by + z = 0 has no solution, then S is :$ (1) $an empty set$ (2) $an infinite set$ (3) $a finite set containing two or more elements$ (4) $a singleton$ 359. $If A is a 3 x 3 matrix and \|A\|= 2, then \|3adj(\|3A\| A^{2})\| is equal to$ (1) $3^{12} . 6^{11}$ (2) $3^{12} . 6^{10}$ (3) $3^{10} . 6^{11}$ (4) $3^{11} . 6^{10}$ 360. $If the system of equations x - 2 y + 3 z = 9 2 x + y + z = b x - 7 y + az = 24, has infinitely many solutions, then a - b is equal to:$