Matrices & Determinants PYQs Set 6 (PDF)

Set 6 of 19 · Questions 151-180 of 565

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 6 (Q151-180)
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151. $The system of linear equations λx + 2 y + 2 z = 5 2 λx + 3 y + 5 z = 8 4 x + λy + 6 z = 10 has$ (1) $Infinite solutions when λ = 8$ (2) $Infinite solutions when λ = 2$ (3) $no solutions when λ = 8$ (4) $no solutions when λ = 2$ 152. $Let A = (20 - 1 2) . If B = I -^{3} C_{1} (adj A) +^{3} C_{2} (adj A)^{2} -^{3} C_{3} (adj A)^{3}, then the sum of all elements of the matrix B is$ (1) $-1$ (2) $-3$ (3) $-4$ (4) $-5$ 153. $If A = (2 - k 1 2 3 - k) is singular matrix, then the value of 5 k - k^{2} is equal to$ (1) $-4$ (2) $4$ (3) $6$ (4) $-6$ 154. $If the system of equations x + (2 sin α) y + (2 cos α) z = 0 x + (cos α) y + (sin α) z = 0 x + (sin α) y - (cos α) z = 0 has a non-trivial solution, then α \in (0, \frac{π}{2}) is equal to :$ (1) $\frac{11 π}{24}$ (2) $\frac{5 π}{24}$ (3) $\frac{7 π}{24}$ (4) $\frac{3 π}{4}$ 155. $If A = 113234343, then A^{- 1} =$ (1) $- \frac{1}{4} - 7 9 - 5 6 - 1 2 111$ (2) $- \frac{1}{4} - 7 9 - 5 - 6 6 - 2 - 1 - 1 1$ (3) $\frac{1}{4} - 7 9 - 5 6 - 6 2 - 1 - 1 1$ (4) $- \frac{1}{4} - 7 9 - 5 6 - 6 2 - 1 - 1 1$ 156. $If x, y, z are all different and not equal to zero and (1 + x 1111 + y 1111 + z) = 0 then the value of x^{- 1} + y^{- 1} + z^{- 1} is equal to$ (1) $xyz$ (2) $x^{- 1} y^{- 1} z^{- 1}$ (3) $- x - y - z$ (4) $-1$ 157. $The inverse of the matrix A = 200030004 is$ (1) $\frac{1}{24} 200030004$ (2) $200030004$ (3) $\frac{1}{24} 100010001$ (4) $\frac{1}{2} 00 0 \frac{1}{3} 0 00 \frac{1}{4}$ 158. $If A = (cos θ sin θ - sin θ cos θ), then the matrix A^{- 50} when θ = \frac{π}{12}, is equal to :$ (1) $(\frac{1}{2} \frac{3}{2} - \frac{3}{2} \frac{1}{2})$ (2) $(\frac{3}{2} \frac{1}{2} - \frac{1}{2} \frac{3}{2})$ (3) $(\frac{3}{2} - \frac{1}{2} \frac{1}{2} \frac{3}{2})$ (4) $(\frac{1}{2} - \frac{3}{2} \frac{3}{2} \frac{1}{2})$ 159. $If 2 A + 3 B = (2 - 14325) and A + 2 B = (503162), then B =$ (1) $(8 - 12 - 110 - 1)$ (2) $(812 - 110 - 1)$ (3) $(81 - 2 - 110 - 1)$ (4) $(8121101)$ 160. $If A = 100011110, then A^{- 1} =$ (1) $A - 2 A^{2}$ (2) $2 A - A^{2}$ (3) $2 A^{2} + A$ (4) $2 A + A^{2}$ 161. $Let P = (100310931) and Q = (q_{ij}) be two 3\times3 matrices such that Q - P^{5} = I_{3} . Then \frac{q _{21} + q _{31}}{q _{32}} is equal to :$ (1) $10$ (2) $135$ (3) $15$ (4) $9$ 162. $If the system of equations α 11 - 1 - α - 1 - 1 - 1 - α x y z = α - 1 α - 1 α - 1 is inconsistent, then α =$ (1) $1$ (2) $-2$ (3) $-1$ (4) $2$ 163. $If the solution of the system of simultaneous equations \frac{1}{x} + \frac{2}{y} - \frac{3}{z} - 1 = 0, \frac{2}{x} - \frac{4}{y} + \frac{3}{z} - 1 = 0 and \frac{3}{x} + \frac{6}{y} - \frac{6}{z} - 4 = 0 is x = α, y = β, z = γ then α^{2} + γ^{2} =$ (1) $5 β$ (2) $β^{2}$ (3) $3 β$ (4) $2 β^{2}$ 164. $If f (x) = x^{3} 3 x^{2} + 2 x^{3} - x 2 x^{2} + 1 2 x 4 1 + 3 x x^{3} + 6 x^{2} - 2 for all x \in R, then 2 f (0) + f^{'} (0) is equal to$ (1) $48$ (2) $24$ (3) $42$ (4) $18$ 165. $If A = (α 2 2 α) and ∣ A^{3} ∣ = 125, then α =$ (1) $\pm 1$ (2) $\pm 2$ (3) $\pm 3$ (4) $\pm 5$ 166. $If a - b - c 2 b 2 c 2 a b - c - a 2 c 2 a 2 b c - a - b = (a + b + c) (x + a + b + c)^{2}, x \neq = 0 and a + b + c \neq = 0, then x is equal to :$ (1) $abc$ (2) $- (a + b + c)$ (3) $2 (a + b + c)$ (4) $- 2 (a + b + c)$ 167. $If Δ = 111 a b c a^{2} b^{2} c^{2} and Δ_{1} = 1 b c a 1 c a b 1 ab c then$ (1) $Δ_{1} \neq = Δ$ (2) $Δ_{1} = Δ$ (3) $Δ_{1} = - Δ$ (4) $Δ_{1} = 3Δ$ 168. $If f (x) = (sin x cos x tan x x^{3} x^{2} x 2 x 1 x), then Lt x \to 0 \frac{f ( x )}{x ^{2}} =$ (1) $1$ (2) $0$ (3) $3$ (4) $2$ 169. $Let A = (m p n q), d = ∣ A ∣ \neq = 0 and ∣ A - d (adj A) ∣ = 0 . Then$ (1) $(1 + d)^{2} = (m + q)^{2}$ (2) $1 + d^{2} = (m + 1)^{2}$ (3) $(1 + d)^{2} = m^{2} + q^{2}$ (4) $1 + d^{2} = m^{2} + q^{2}$ 170. $Let A = (cos α sin α - sin α cos α), (α \in R) such that A^{32} = (01 - 1 0) . Then a value of α is:$ (1) $\frac{π}{32}$ (2) $0$ (3) $\frac{π}{64}$ (4) $\frac{π}{16}$ 171. $Let A be a 3 \times 3 matrix with det (A) = 4 . Let R_{i} denote the i^{th} row of A . If a matrix B is obtained byperforming the operation R_{2} \to 2 R_{2} + 5 R_{3} on 2 A then det (B) is equal to:$ (1) $64$ (2) $16$ (3) $80$ (4) $128$ 172. $Let λ \in R . The system of linear equations 2 x_{1} - 4 x_{2} + λ x_{3} = 1 x_{1} - 6 x_{2} + x_{3} = 2 λ x_{1} - 10 x_{2} + 4 x_{3} = 3 is inconsistent for:$ (1) $exactly two value of λ .$ (2) $exactly one positive value of λ$ (3) $every value of λ$ (4) $exactly one negative value of λ$ 173. $If A = 103 - 2 2 - 2 2 - 3 4, then A \cdot adj (A) is equal to$ (1) $511151115$ (2) $500050005$ (3) $800080008$ (4) $000000000$ 174. $If f (x) = x x + 2 x + 6 x + 1 x + 4 x + 9 x + 3 x + 7 x + 13, then f (5) =$ (1) $-15$ (2) $10$ (3) $-2$ (4) $0$ 175. $If A and B are square matrices of the same order such that (A + B) (A - B) = A^{2} - B^{2}, then (AB A^{- 1})^{2} =$ (1) $A^{2} B^{2}$ (2) $B^{2}$ (3) $A^{2}$ (4) $I$ 176. $Let A = (a_{ij}) and B = (b_{ij}) be two 3 \times 3 real matrices such that b_{ij} = (3)^{i + j - 2} a_{j i}, where i, j = 1, 2, 3 . If the determinant of B is 81, then the determinant of A is:$ (1) $\frac{1}{9}$ (2) $3$ (3) $\frac{1}{81}$ (4) $\frac{1}{27}$ 177. $If A = (a_{11} a_{12} a_{13} a_{21} a_{22} a_{23}) and B = (b_{11} b_{12} b_{13} b_{21} b_{22} b_{23}), then which one of the following is True?$ (1) $A^{T} BB^{T} A = B^{T} AA^{T} B$ (2) $The orders of A^{T} BB^{T} A and B^{T} AA^{T} B are equal$ (3) $The orders of A + B, A^{T} B, BA^{T} are equal$ (4) $Rank of A and B are equal$ 178. $If A = (22 - 32), B = (0 - 110) then (B^{- 1} A^{- 1})^{- 1} =$ (1) $(2 - 223)$ (2) $(22 - 23)$ (3) $(2 - 322)$ (4) $(1 - 1 - 23)$ 179. $Let A be a 2 \times 2 real matrix and I be the identity matrix of order 2 . If the roots of the equation \| A - xI \| = 0 be -1 and 3 , then the sum of the diagonal elements of the matrix A^{2} is.$ 180. $Let A = 211010101, B = [B_{1}, B_{2}, B_{3}], where B_{1}, B_{2}, B_{3} are column matrices, and A B_{1} = 100, A B_{2} = 230, A B_{3} = 321 . If α = ∣ B ∣ and β is the sum of all the diagonal elements of B, then α^{3} + β^{3} is equal to$