Matrices & Determinants PYQs Set 7 (PDF)

Set 7 of 19 · Questions 181-210 of 565

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 7 (Q181-210)
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181. $Consider the following system of questions αx + 2 y + z = 1 2 αx + 3 y + z = 1 3 x + αy + 2 z = β For some α, β ϵ R . then which of the following is NOT correct.$ (1) $It has no solution if α = - 1 and β \neq = 2$ (2) $It has no solution for α=-1 and for all β ϵ R$ (3) $It has no solution for α = 3 and for all β\neq2$ (4) $It has a solution for all α \neq = - 1 and β = 2$ 182. $If a_{1}, a_{2}, \dots, a_{9} are in GP, then lo g a_{1} lo g a_{4} lo g a_{7} lo g a_{2} lo g a_{5} lo g a_{8} lo g a_{3} lo g a_{6} lo g a_{9} is equal to$ (1) $log (a_{1}, a_{2}, \dots a_{n})$ (2) $1$ (3) $(log a_{q})^{9}$ (4) $0$ 183. $If A = 112173397 then Tr (A^{2} - A) =$ (1) $0$ (2) $-12$ (3) $152$ (4) $125$ 184. $Let A and B be 3 \times 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Thenthe system of linear equations (A^{2} B^{2} - B^{2} A^{2}) X = O where X is a 3 \times 1 column matrix of unknownvariables and O is a 3 \times 1 null matrix, has :$ (1) $a unique solution$ (2) $exactly two solutions$ (3) $infinitely many solutions$ (4) $no solution$ 185. $If A is a 3 \times 3 matrix and the matrix obtained by replacing the elements of A with their corresponding cofactors is 14 - 2 - 2 - 5 4 1 - 2 1 then a possible value of the determinant of A is$ (1) $4$ (2) $3$ (3) $2$ (4) $1$ 186. $If A = 1 b 3 a 2 d 3 c 4 is a symmetric matrix and B = 0 - 5 6 50 c b - 7 0 is a skew symmetric matrix, then A B =$ (1) $4852 - 59 271943 4822 - 67$ (2) $4832 - 11 261943 3622 - 67$ (3) $1232 - 11 267943 3650 - 67$ (4) $1232 - 11 321943 4122 - 67$ 187. $Let A = (a_{ij}) be an n \times n matrix defined by a_{ij} = (k^{i}, \forall i = j 0, otherwise) . If m = trace of A and lim_{k \to 1} \frac{n - m}{1 - k} = 171 then the value of n is$ (1) $18$ (2) $23$ (3) $35$ (4) $42$ 188. $If A and B are the two real values of k for which the system of equations x + 2 y + z = 1, x + 3 y + 4 z = k . x + 5 y + 10 z = k^{2} is consistent, then A + B =$ (1) $3$ (2) $4$ (3) $5$ (4) $7$ 189. $If the system of equations 3 x - 2 y + z = 0, λx - 14 y + 15 z = 0 and x + 2 y - 3 z = 0 has a solution other than x = y = z = 0, then λ is equal to$ (1) $1$ (2) $2$ (3) $3$ (4) $5$ 190. $If the determinant of the adjoint of a (real) matrix of order 3 is 25 , then the determinant of the inverse of the matrix is$ (1) $0.2$ (2) $\pm 5$ (3) $\frac{1}{5 625}$ (4) $\pm 0.2$ 191. $If A = 0001202403844128, then the rank of A is$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 192. $In the matrix - 1 - 4 - 7 x - 5 y 3 - 6 9, if the cofactors of - 6 and - 7 are respectively 22 and 27, then 5 x + y =$ (1) $0$ (2) $-1$ (3) $-2$ (4) $-4$ 193. $If 3 A + 4 B^{'} = (70 - 10 6 1731) and 2 B - 3 A^{'} = - 1 4 - 5 180 - 7 then B =$ (1) $1 - 1 2 314$ (2) $- 1 4 - 5 - 18 - 16 - 7$ (3) $1 - 1 2 - 3 14$ (4) $1 - 1 2 31 - 4$ 194. $The characteristic equation of a matrix A is λ^{3} - 5 λ^{2} - 3 λ + 2 = 0, then \| adj (A) \| =$ (1) $4$ (2) $9$ (3) $25$ (4) $\frac{1}{2}$ 195. $The value of x + y x x - y y + z y y - z z + x z z - x =$ (1) $2 (x + y + z)^{2}$ (2) $2 (x + y + z)^{3}$ (3) $(x + y + z)^{3}$ (4) $0$ 196. $If P and Q are square matrices such that P^{2006} = 0 and PQ = P + Q, then det (Q) will be$ (1) $0$ (2) $1 only$ (3) $-1 only$ (4) $\pm 1$ 197. $Let f (x) = \int \frac{7 x ^{10} + 9 x ^{8}}{( 1 + x ^{2} + 2 x ^{9} ) ^{2}} d x, x > 0, lim_{x \to 0} f (x) = 0 and f (1) = \frac{1}{4} . If A = 0 \frac{1}{4} α^{2} 0 f^{'} (1) 4 111 and B = adj (adj A) be such that ∣ B ∣ = 81, then α^{2} is equal to$ (1) $2$ (2) $1$ (3) $3$ (4) $4$ 198. $If a, b and c are in A.P., then the value of x + 2 x + 4 x + 6 x + 3 x + 5 x + 7 x + a x + b x + c is$ (1) $0$ (2) $a + b + c$ (3) $x - (a + b + c)$ (4) $9 x^{2} + a + b + c$ 199. $The sum of the distinct values of x for which the matrix A = 11 x 1 x 1 x 11 has no inverse, is$ (1) $4$ (2) $3$ (3) $2$ (4) $-1$ 200. $The total number of matrices A = 0 2 x 2 x 2 y y - y 1 - 1 1, (x, y \in R, x \neq = y) for which A^{T} A = 3 I_{3} is:$ (1) $2$ (2) $3$ (3) $6$ (4) $4$ 201. $If A = (1324), then A^{- 1} =$ (1) $(- 2 1 43)$ (2) $(2143)$ (3) $\frac{- 1}{2} (4 - 3 - 2 1)$ (4) $\frac{1}{2} (4 - 3 - 2 1)$ 202. $If the vertices of a triangle are (- 2, 6), (3, - 6) and (1, 5), then the area of the triangle is$ (1) $40 sq. units$ (2) $15.5 sq. units$ (3) $30 sq. units$ (4) $35 sq, units$ 203. $For a 3\times3 matrix M, let trace (M) denote the sum of all the diagonal elements of M. Let A be a 3\times3 matrix such that \| A \| = \frac{1}{2} and trace (A) = 3 . If B = adj (adj (2 A)), then the value of \| B \| + trace (B) equals:$ (1) $56$ (2) $132$ (3) $174$ (4) $280$ 204. $Let A = (\frac{1}{2} 0 - 2 1) and P = (cos θ sin θ - sin θ cos θ), θ > 0 . If B = P A P^{T}, C = P^{T} B^{10} P and the sum of the diagonal elements of C is \frac{m}{n}, where g cd (m, n) = 1, then m + n is:$ (1) $127$ (2) $258$ (3) $65$ (4) $2049$ 205. $If A = (001010100), then A^{4} is equal to$ (1) $2 A$ (2) $I$ (3) $4 A$ (4) $A$ 206. $235352523 + 1749111121113169 =$ (1) $32$ (2) $-67$ (3) $93$ (4) $-22$ 207. $If the system of linear system x - 4 y + 7 z = g 3 y - 5 z = h - 2 x + 5 y - 9 z = k is consistent, then:$ (1) $g + 2 h + k = 0$ (2) $g + h + 2 k = 0$ (3) $2 g + h + k = 0$ (4) $g + h + k = 0$ 208. $Let A = cos θ 0 sin θ 010 - sin θ 0 cos θ . If for some θ \in (0, π), A^{2} = A^{T}, then the sum of the diagonal elements of the matrix (A + I)^{3} + (A - I)^{3} - 6 A is equal to_______.$ 209. $Let A be a 3 \times 3 matrix and det (A) = 2 . If n = det (adj (adj (\dots \dots (adj A) 2024 - times))) Then the remainder when n is divided by 9 is equal to______$ 210. $The number of 3 \times 2 matrices A , which can be formed using the elements of the set -2, -1, 0, 1, 2 such that the sum of all the diagonal elements of A^{T} A is 5 , is$