Matrices & Determinants PYQs Set 10 (PDF)

Set 10 of 19 · Questions 271-300 of 565

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 10 (Q271-300)
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271. $If 3 x x 1 = 3421 then x is equal to$ (1) $8$ (2) $4$ (3) $\pm 22$ (4) $2$ 272. $The following system of linear equations 3 x + 3 y + 2 z = 9 3 x + 2 y + 2 z = 9 x - y + 4 z = 8$ (1) $does not have any solution$ (2) $has a unique solution$ (3) $has a solution (α, β, γ) satisfying α + β^{2} + γ^{3} = 12$ (4) $has infinitely many solutions$ 273. $For an invertible matrix A if A (adj A) = (100010), then ∣ A ∣ is$ (1) $100$ (2) $-100$ (3) $10$ (4) $-10$ 274. $Let A = 5 - sin^{2} θ cos^{2} θ sin^{2} θ - 5 1 cos^{2} θ 15 . Then maximum value of det (A) is$ (1) $-125$ (2) $200$ (3) $- \frac{255}{2}$ (4) $145$ 275. $If A and B are two matrices such that AB = B and BA = A then A^{2} + B^{2} =$ (1) $AB$ (2) $A + B$ (3) $2 BA$ (4) $2 AB$ 276. $If A = (1 + 2 i - i i 1 - 2 i), where i = - 1, then A (adj A) =$ (1) $- 2 I$ (2) $2 I$ (3) $5 I$ (4) $4 I$ 277. $If the inverse of the matrix α 26 1432 - 1 13 does not exist, then the value of α is$ (1) $1$ (2) $-1$ (3) $0$ (4) $-2$ 278. $If A = (12 - 1 - 1), B = (14 1 - 1), then (A + B)^{- 1} is$ (1) $(- \frac{1}{2} - \frac{3}{2} 0 \frac{1}{2})$ (2) $(\frac{1}{2} \frac{3}{2} 0 - \frac{1}{2})$ (3) $(\frac{1}{2} - \frac{3}{2} 0 \frac{1}{2})$ (4) $(\frac{1}{2} \frac{3}{2} 0 \frac{1}{2})$ 279. $If A = (x 1 01) and B = (8701) and A^{3} = B, then x =$ (1) $-2 or 3$ (2) $-2$ (3) $2 or -3$ (4) $2$ 280. $If A and B are non-singular matrices and det (AB) = (det A) (det B), then ((det A) (det B)) B^{- 1} A^{- 1} =$ (1) $Adj (BA)$ (2) $Adj (A) + Adj (B)$ (3) $Adj (AB)$ (4) $(Adj B) (Adj A)$ 281. $For the system of linear equations x + y + z = 6 αx + βy + 7 z = 3 x + 2 y + 3 z = 14 Which of the following is NOT true?$ (1) $If α = β = 7, then the system has no solution$ (2) $If α = β and α \neq = 7 then the system has a unique solution.$ (3) $There is a unique point (α, β) on the line x + 2 y + 18 = 0 for which the system has infinitely many solutions.$ (4) $For every point α, β\neq7, 7 on the line x - 2 y + 7 = 0 the system has infinitely many solutions.$ 282. $Let A be a 3 \times 3 real matrix such that A 101 = 2101, A - 1 01 = 4 - 1 01, A 010 = 2010 . Then, the system (A - 3 I) x y z = 123 has$ (1) $unique solution$ (2) $exactly two solutions$ (3) $no solution$ (4) $infinitely many solutions$ 283. $If X = x y z is a solution of the system of equations A X = B, where adj A = 4 - 5 1 20 - 2 253 and B = 402, then ∣ x + y + z ∣ is equal to :$ (1) $2$ (2) $\frac{3}{2}$ (3) $1$ (4) $3$ 284. $The system of linear equations x + y + z = 2 2 x + 3 y + 2 z = 5 2 x + 3 y + (a^{2} - 1) z = a + 1$ (1) $is inconsistent when a = 4$ (2) $has a unique solution for (a) = 3$ (3) $has infinitely many solutions for a = 4$ (4) $is inconsistent when (a) = 3$ 285. $If the value of a third order determinant is 16 , then the value of the determinant formed by replacing each of its elements by its cofactor is$ (1) $96$ (2) $256$ (3) $48$ (4) $1$ 286. $If A = (α^{2} 55 - α) and det (A^{10}) = 1024, then α =$ (1) $-2$ (2) $-1 .$ (3) $-3$ (4) $0$ 287. $Let A be a 3 \times 3 matrix such that \| adj (adj (adjA)) \| = 81 . If S = (n \in Z : (\| adj (adjA) \|)^{\frac{( n - 1 ) ^{2}}{2}} = \| A \|^{(3 n^{2} - 5 n - 4)}) then \sum (A^{(n^{2} + n)}) is equal to$ (1) $820$ (2) $866$ (3) $750$ (4) $732$ 288. $If A is a matrix of order 3 , such that A (adj A) = 10 I, then \| adj A \| =$ (1) $1$ (2) $10$ (3) $100$ (4) $10 I$ 289. $Let A and B be two symmetric matrices of order 3. Statement-1 : A(BA) and (AB)A are symmetric matrices. Statement-2 : AB is symmetric matrix if matrix multiplication of A with B is commutative.$ (1) $Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.$ (2) $Statement-1 is true, Statement-2 is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.$ (3) $Statement-1 is true, Statement-2 is false.$ (4) $Statement-1 is false, Statement-2 is true.$ 290. $Consider the system of linear equation x + y + z = 4 μ, x + 2 y + 2 λz = 10 μ, x + 3 y + 4 λ^{2} z = μ^{2} + 15, where λ, μ \in R . Which one of the following statements is NOT correct?$ (1) $The system has unique solution if λ \neq = \frac{1}{2} and μ\neq1,15$ (2) $The system is inconsistent if λ = \frac{1}{2} and μ \neq = 1$ (3) $The system has infinite number of solutions if λ = \frac{1}{2} and μ = 15$ (4) $The system is consistent if λ \neq = \frac{1}{2}$ 291. $If (2312) A = (1001), then the matrix A is$ (1) $(2 - 3 - 1 2)$ (2) $(- 2 3 1 - 2)$ (3) $(23 - 1 2)$ (4) $(2312)$ 292. $If k is one of the roots of the equation x^{2} - 25 x + 24 = 0 such that A = 131221 13 k is non-singular matrix, then A^{- 1} =$ (1) $- \frac{1}{46} 90 - 138 2 - 94 462 80 - 8$ (2) $- \frac{1}{92} 45 - 69 1 - 47 231 40 - 4$ (3) $- \frac{1}{46} 45 - 69 1 - 47 231 40 - 4$ (4) $- \frac{1}{92} 90 - 138 2 - 94 462 80 - 8$ 293. $If α \neq = a, β \neq = b, γ \neq = c and (α b c a β c abγ) = 0, then \frac{a}{α - a} + \frac{b}{β - b} + \frac{γ}{γ - c} is equal to:$ (1) $3$ (2) $0$ (3) $1$ (4) $2$ 294. $If A = 112231356, then (Adj (Adj A))^{- 1} =$ (1) $\frac{1}{6} 85 - 5 - 6 13 3 - 2 1$ (2) $\frac{1}{6} 134 - 5 - 9 03 1 - 2 1$ (3) $\frac{1}{36} 134 - 5 - 9 03 1 - 2 1$ (4) $\frac{1}{12} 43 - 5 - 3 42 221$ 295. $Let αβγ = 45; α, β, γ \in R . If x (α, 1, 2) + y (1, β, 2) + z (2, 3, γ) = (0, 0, 0) for some x, y, z \in R, xyz \neq = 0, then 6 α + 4 β + γ is equal to______$ 296. $Let S = {m \in Z : A^{m^{2}} + A^{m} = 3 I - A^{- 6}}, where A = (21 - 1 0) . Then n (S) is equal to ______.$ 297. $Let A = (31 - 4 - 1) and B be two matrices such that A^{100} = 100 B + I . Then the sum of all the elements of B^{100} is _______.$ 298. $If the system of linear equations 2x + y - z = 3 x - y - z = α 3x + 3y + βz = 3 has infinitely many solution, then α + β - α β is equal to .$ 299. $Let A be a 3 \times 3 matrix of non-negative real elements such that A (111) = 3 (111) . Then the maximum value of det (A) is_____$ 300. $If A = (& 0 & & - tan (\frac{θ}{2}) & tan (\frac{θ}{2}) & & math on & & 0) and (I_{2} + A) (I_{2} - A)^{- 1} = (& a & & - b & b & & a) then 13 (a^{2} + b^{2})$