Matrices & Determinants PYQs Set 8 (PDF)

Set 8 of 19 · Questions 211-240 of 565

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 8 (Q211-240)
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211. $If A = (1 tan \frac{α}{2} - tan \frac{α}{2} 1) and AB = I then B =$ (1) $cos^{2} \frac{α}{2} \cdot I$ (2) $cos^{2} \frac{α}{2} \cdot A^{T}$ (3) $sin^{2} \frac{α}{2} \cdot A$ (4) $cos^{2} \frac{α}{2} \cdot A$ 212. $Let P and Q be 3 \times 3 matrices P \neq Q. If P3 = Q3 and P2Q = Q2P, then determinant of (P2 + Q2) is equal to :$ (1) $- 2$ (2) $1$ (3) $0$ (4) $- 1$ 213. $Let A be a matrix of order 3 \times 3 and \| A \| = 5 . If \| 2 adj (3 Aadj (2 A)) \| = 2^{α} \cdot 3^{β} \cdot 5^{γ}, α, β, γ \in N, then α + β + γ is equal to$ (1) $26$ (2) $27$ (3) $25$ (4) $28$ 214. $The set of all points that forms a triangle of area 15 sq. units with the points (1, - 2) and (- 5, 3) lies on$ (1) $5 x + 6 y + 23 = 0$ (2) $(5 x + 6 y - 23) (5 x + 6 y + 37) = 0$ (3) $25 x^{2} + 36 y^{2} + 24 x - 30 y - 227 = 0$ (4) $5 x + 6 y - 37 = 0$ 215. $The matrix A = (\frac{1}{2} \frac{- 1}{2} \frac{1}{2} \frac{- 1}{2}) is$ (1) $Unitary$ (2) $Orthogonal$ (3) $Nilpotent$ (4) $Involuntary$ 216. $If matrix D_{1} = diag (a, b, c), matrix D_{2} = diag (3, 3, 3) and A is a skew symmetric matrix of 3^{rd} order, then Tr (D_{1} D_{2} A + D_{1} D_{2} + D_{1} A + D_{2} A) - Tr (D_{1} + D_{2}) =$ (1) $2 a + 2 b + 2 c - 9$ (2) $3 a + 3 b + 3 c - 9$ (3) $3 a + 3 b + 3 c$ (4) $a^{3} + b^{3} + c^{3}$ 217. $If A = 1 a b 0 - 1 c 001 is such that A^{2} = I, then$ (1) $b = \frac{ac}{2}$ (2) $b = - \frac{ac}{2}$ (3) $b = \frac{a + c}{2}$ (4) $b = ac$ 218. $Let A = (011 k), k \in R and A^{3} = (ab c d) . If d = 228, then b + c =$ (1) $52$ (2) $74$ (3) $2$ (4) $100$ 219. $If the system of linear equations 2 x + 2 y + 3 z = a 3 x - y + 5 z = b x - 3 y + 2 z = c where, a, b, c are non-zero real numbers, has more than one solution, then :$ (1) $b - c + c = 0$ (2) $b - c - a = 0$ (3) $a + b + c = 0$ (4) $b + c - a = 0$ 220. $If A = 1 - 1 1 212324, then (A^{2} - 5 A) A^{- 1} =$ (1) $4 - 1 1 242321$ (2) $- 4 - 1 1 2 - 4 2 32 - 1$ (3) $- 4 23 - 1 - 4 2 12 - 1$ (4) $- 1 41 - 2 - 2 4 1 - 3 - 2$ 221. $Let A = (a_{ij}) be 3 \times 3 matrix such that A 010 = 001, A 413 = 010 and A 212 = 100, then a_{23} equals:$ (1) $-1$ (2) $2$ (3) $1$ (4) $0$ 222. $The system of linear equations x + y + z = 62 x + 5 y + az = 36 x + 2 y + 3 z = b has$ (1) $unique solution for a = 8 and b = 16$ (2) $infinitely many solutions for a = 8 and b = 16$ (3) $unique solution for a = 8 and b = 14$ (4) $infinitely many solutions for a = 8 and b = 14$ 223. $If the cofactors of the elements 3, 7 and 6 of the matrix 142 2 - 1 4 376 are a, b and c respectively, then (a b c) 142 + (a b c) 376 =$ (1) $-1$ (2) $1$ (3) $0$ (4) $3$ 224. $Let S_{1} and S_{2} be respectively the sets of all a ϵ R - (0) for which the system of linear equations ax + 2 ay - 3 az = 1 (2 a + 1) x + (2 a + 3) y + (a + 1) z = 2 (3 a + 5) x + (a + 5) y + (a + 2) z = 3 has unique solution and infinitely many solutions, Then$ (1) $n S_{1} =2 and S_{2} is an infinite set$ (2) $S_{1} is an infinite set an n S_{2} =2$ (3) $S_{1} =Φ and S_{2} = R-{0}$ (4) $S_{1} =R-0 and S_{2} =Φ$ 225. $If A = 201120012 then ∣ adj A ∣ =$ (1) $1/9$ (2) $0$ (3) $81$ (4) $9$ 226. $If the minimum and the maximum values of the function f : (\frac{π}{4}, \frac{π}{2}) \to R, defined by f (θ) = - sin^{2} θ - cos^{2} θ 12 - 1 - sin^{2} θ - 1 - cos^{2} θ 10 11 - 2 are m and M respectively, then the ordered pair (m, M) is equal to:$ (1) $(0, 22)$ (2) $(0, 4)$ (3) $(- 4, 4)$ (4) $(- 4, 0)$ 227. $If O (A) = 2 \times 3, O (B) = 3 \times 2, and O (C) = 3 \times 3, which one of the following is not defined?$ (1) $CB + A^{’}$ (2) $BAC$ (3) $C (A + B^{’})^{’}$ (4) $C (A + B^{’})$ 228. $Let α and β be the roots of the equation x^{2} + x + 1 = 0 . Then for y \neq = 0 in R, y + 1 α β α y + β 1 β 1 y + α is equal to :$ (1) $y (y^{2} - 1)$ (2) $y (y^{2} - 3)$ (3) $y^{3}$ (4) $y^{3} - 1$ 229. $If A = (cos θ sin θ - sin θ cos θ), then A \cdot A^{’} is$ (1) $I$ (2) $A$ (3) $- A$ (4) $A^{2}$ 230. $If x^{2} + 3 x x - 1 x - 3 x + 1 2 - x x - 3 x - 3 x + 4 3 x = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4}, then (a_{1} + a_{3}) + 2 (a_{0} + a_{2} + a_{4}) =$ (1) $-1$ (2) $0$ (3) $1$ (4) $-29$ 231. $For α, β ϵ R, suppose the system of linear equations x - y + z = 5 2 x + 2 y + αz = 8 3 x - y + 4 z = β has infinitely many solutions. The α and β are the roots of$ (1) $x^{2} - 10 x + 16 = 0$ (2) $x^{2} + 18 x + 56 = 0$ (3) $x^{2} - 18 x + 56 = 0$ (4) $x^{2} + 14 x + 24 = 0$ 232. $If det (AB) = (det A) (det B) and A is a non-singular matrix of order 3 \times 3, then det (adj A) =$ (1) $det (A)$ (2) $(det (A))^{- 1}$ (3) $(det (A))^{2}$ (4) $(det (A))^{3}$ 233. $Let A = 100 0 α β 0 β α and ∣2 A ∣^{3} = 2^{21} where α, β \in Z . Then a value of α is$ (1) $3$ (2) $5$ (3) $17$ (4) $9$ 234. $Let θ = \frac{π}{5} and A = (cos θ - sin θ sin θ cos θ) . If B = A + A^{4} then det (B) :$ (1) $lies in (2, 3)$ (2) $is zero$ (3) $is one$ (4) $lies in (1, 2)$ 235. $If A = (2 - 2 - 2 2), then A^{n} = 2^{k} A, where k =$ (1) $2^{n - 1}$ (2) $n + 1$ (3) $n - 1$ (4) $2 (n - 1)$ 236. $If A is a square matrix of order 3 such that det (A) = 3 and det (adj (- 4 adj (- 3 adj (3 adj ((2 A)^{- 1}))))) = 2^{m} 3^{n}, then m + 2 n is equal to :$ (1) $2$ (2) $3$ (3) $6$ (4) $4$ 237. $Let A be a square matrix such that A A^{T} = I . Then \frac{1}{2} A ((A + A^{T})^{2} + (A - A^{T})^{2}) is equal to$ (1) $A^{2} + I$ (2) $A^{3} + I$ (3) $A^{2} + A^{T}$ (4) $A^{3} + A^{T}$ 238. $If f (x) = 2 cos^{4} x 3 + 2 cos^{4} x 2 cos^{4} x 2 sin^{4} x 2 sin^{4} x 3 + 2 sin^{4} x 3 + sin^{2} 2 x sin^{2} 2 x sin^{2} 2 x then \frac{1}{5} f^{'} (0) is equal to ______$ (1) $0$ (2) $1$ (3) $2$ (4) $6$ 239. $Let A be a 3 x 3 matrix having entries from, the set {-1, 0, 1}. The number of all such matrices A having sum of all the entries equal to 5, is ________$ 240. $Let S be the set of all integer solutions, (x, y, z) of the system of equations x - 2 y + 5 z = 0 - 2 x + 4 y + z = 0 - 7 x + 14 y + 9 z = 0 such that 15 \leq x^{2} + y^{2} + z^{2} \leq 150 . Then, the number of elements in the set S is equal to:$