Matrices & Determinants PYQs Set 9 (PDF)

Set 9 of 19 · Questions 241-270 of 565

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 9 (Q241-270)
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241. $If A and B are symmetric matrices of the same order, then which one of the following is NOT true?$ (1) $A + B is symmetric$ (2) $A - B is symmetric$ (3) $AB + BA is symmetric$ (4) $AB - BA is symmetric$ 242. $The equations x - y + 2 z = 4 3 x + y + 4 z = 6 x + y + z = 1 have$ (1) $unique solution$ (2) $infinitely many solutions$ (3) $no solution$ (4) $two solutions$ 243. $Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A^{2} is 1 then the possible number of such matrices is:$ (1) $6$ (2) $1$ (3) $4$ (4) $12$ 244. $Let λ be a real number for which the system of linear equations: x + y + z = 6 4 x + λy - λz = λ - 2 3 x + 2 y - 4 z = - 5 has infinitely many solutions. Then λ is a root of the quadratic equation :$ (1) $λ^{2} + 3 λ - 4 = 0$ (2) $λ^{2} - 3 λ - 4 = 0$ (3) $λ^{2} + λ - 6 = 0$ (4) $λ^{2} - λ - 6 = 0$ 245. $If for the matrix, A = (1 - α α β), AA^{⊤} = I_{2} then the value of α^{4} + β^{4} is$ (1) $1$ (2) $3$ (3) $2$ (4) $4$ 246. $If x + 1 x + 2 x + 3 x + 2 x + 3 x + 4 x + a x + b x + c = 0, then a, b, c are$ (1) $equal$ (2) $in A.P.$ (3) $in G.P.$ (4) $in H.P.$ 247. $If A, B and C are the angles of a triangle, then the system of equations - x + ycos C + zcos B = 0, xcos C - y + z cos A = 0 and xcos B + ycos A - z = 0$ (1) $Only zero solution$ (2) $A non-zero solution for all △ ABC$ (3) $Only zero solution but for certain values of A, B and C$ (4) $A non-zero solution if △ ABC is an equilateral triangle and not for all triangles.$ 248. $If (1011) \cdot (1021) \cdot (1031) \dots (10 n - 1 1) = (10781), then the inverse of (10 n 1) is:$ (1) $(11201)$ (2) $(10 - 13 1)$ (3) $(10 - 12 1)$ (4) $(11301)$ 249. $If A = (1432), B = (21 - 1 2), then ∣ A B B^{'} ∣ =$ (1) $50$ (2) $100$ (3) $-250$ (4) $250$ 250. $Let α be a solution of x^{2} + x + 1 = 0, and for some a and b in R, (4 a b) 1 - 1 - 2 16 - 1 - 14 132 - 8 = (000) . If \frac{4}{α ^{4}} + \frac{m}{α ^{a}} + \frac{n}{α ^{b}} = 3, then m + n is equal to ________.$ (1) $11$ (2) $3$ (3) $8$ (4) $7$ 251. $All the real values of p, q so that the system of equations: 2 x + py + 6 z = 8, x + 2 y + qz = 5 and x + y + 3 z = 4 may have no solution are$ (1) $p = 2, q \neq = 3$ (2) $p = 2, q = \frac{15}{2}$ (3) $p \neq = 2, q = 3$ (4) $p = 3; q = \frac{15}{4}$ 252. $If A is a 3 \times 3 nonsingular matrix and if \| A \| = 3, then ((2 A)^{- 1}) =$ (1) $3$ (2) $24$ (3) $\frac{1}{24}$ (4) $\frac{1}{3}$ 253. $If A = 500 5 α α 0 α 5 α 5 and det (A^{2}) = 25, then ∣ α ∣ =$ (1) $5$ (2) $5^{2}$ (3) $1$ (4) $\frac{1}{5}$ 254. $The values of α, for which 11 2 α + 3 \frac{3}{2} \frac{1}{3} 3 α + 1 α + \frac{3}{2} α + \frac{1}{3} 0 = 0, lie in the interval$ (1) $(- 2, 1)$ (2) $(- 3, 0)$ (3) $(- \frac{3}{2}, \frac{3}{2})$ (4) $(0, 3)$ 255. $Let A + 2 B = 16 - 5 2 - 3 3 031 and 2 A - B = 220 - 1 - 1 1 562 . If Tr (A) denotes the sum of all diagonal elements of the matrix A, then Tr (A) - Tr (B) has value equal to$ (1) $1$ (2) $2$ (3) $0$ (4) $3$ 256. $Let B, C be n \times n matrices such that A = B + C, BC = CB and C^{2} is a null matrix. Then, B^{2020}, [B + (2021) C] =$ (1) $A^{2020}$ (2) $null zero matrix of order n \times n$ (3) $A^{2021}$ (4) $B^{2021}$ 257. $Consider the system of linear equations : x1 + 2x2 + x3 = 3 2x1 + 3x2 + x3 = 3 3x1 + 5x2 + 2x3 = 1 The system has$ (1) $exactly 3 solutions$ (2) $a unique solution$ (3) $no solution$ (4) $infinite number of solutions$ 258. $If θ \in (0, \frac{π}{2}), then (sin θ + cosec θ)^{2} (cos θ + sec θ)^{2} (tan θ + cot θ)^{2} (sin θ - cosec θ)^{2} (cos θ - sec θ)^{2} (tan θ - cot θ)^{2} 202020202020 =$ (1) $1$ (2) $-1$ (3) $0$ (4) $2020$ 259. $An ordered pair (α, β) for which the system of linear equations (1 + α) x + βy + z = 2 ax + (1 + β) y + z = 3 αx + β y + 2 z = 2 has a unique solution, is:$ (1) $(2, 4)$ (2) $(- 3, 1)$ (3) $(- 4, 2)$ (4) $(1, - 3)$ 260. $For the system of linear equations ax + y + z = 1, x + ay + z = 1, x + y + az = β, which one of the following statements is NOT correct?$ (1) $It has infinitely many solutions if α=2 and β = - 1$ (2) $It has no solution if α = - 2 and β=1$ (3) $x + y + z = \frac{3}{4} if α=2 and β=1$ (4) $It has infinitely many solutions if α = 1 and β=1$ 261. $If A_{n} = (1 - n n n 1 - n) then ∣ A_{1} ∣ + ∣ A_{2} ∣ + \dots + ∣ A_{2021} ∣ =$ (1) $-2021$ (2) $(2021)^{2}$ (3) $- (2021)^{2}$ (4) $4042$ 262. $If a_{1}, a_{2}, a_{3}, \dots, a_{9} are in A.P., then the value of a_{1} a_{4} a_{7} a_{2} a_{5} a_{8} a_{3} a_{6} a_{9} is$ (1) $a_{1} + a_{9}$ (2) $log_{e} (log_{e} e)$ (3) $1$ (4) $\frac{9}{2} (a_{1} + a_{9})$ 263. $If x_{1}, x_{2}, x_{3} as well as y_{1}, y_{2}, y_{3} are in geometric progression with the same common ratio, then the points (x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3}) are$ (1) $vertices of an equilateral triangle$ (2) $vertices of a right angled triangle$ (3) $vertices of a right angled isosceles triangle$ (4) $collinear$ 264. $What is the rank of the matrix 111111111 ?$ (1) $2$ (2) $1$ (3) $3$ (4) $0$ 265. $For 3\times3 matrices M and N, which of the following statement(s) is (are) NOT correct ?$ (1) $NT M N is symmetric or skew symmetric, according as M is symmetric or skew symmetric$ (2) $M N - N M is skew symmetric for all symmetric matrices M and N$ (3) $M N is symetric for all symmetric matrices M and N$ (4) $(adj M) (adj N) = adj(MN) for all invertible matrices M and N$ 266. $If A is a square matrix of order 3, then adj (adj A^{2}) is equal to$ (1) $\| A \|^{2}$ (2) $\| A \|^{4}$ (3) $\| A \|^{8}$ (4) $\| A \|^{16}$ 267. $The number of singular matrices of order 2, whose elements are from the set 2, 3, 6, 9, is ________.$ 268. $Let A be a square matrix of order 3 such that det (A) = - 2 and det (3 adj (- 6 adj (3 A))) = 2^{m + n} \cdot 3^{mn}, m > n . Then 4 m + 2 n is equal to .$ 269. $Let M be any 3 \times 3 matrix with entries from the set 0, 1, 2 . The maximum number of such matrices, forwhich the sum of diagonal elements of M^{⊤} M is seven, is$ 270. $Let A be a non-singular matrix of order 3 . If det (3 adj (2 adj ((det A) A))) = 3^{- 13} \cdot 2^{- 10} and det (3 adj (2 A)) = 2^{m} \cdot 3^{n}, then \| 3 m + 2 n \| is equal to$