Matrices & Determinants PYQs Set 5 (PDF)

Set 5 of 19 · Questions 121-150 of 565

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 5 (Q121-150)
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121. $Consider the system of linear equations x + y + z = 5, x + 2 y + λ^{2} z = 9 x + 3 y + λz = μ, where λ, μ \in R . Then, which of the following statement is NOT correct?$ (1) $System has infinite number of solution if λ = 1 and μ = 13$ (2) $System is inconsistent if λ = 1 and μ \neq = 13$ (3) $System is consistent if λ \neq = 1 and μ = 13$ (4) $System has unique solution if λ \neq = 1 and μ \neq = 13$ 122. $If A is a non-singular matrix such that (A - 2 I) (A - 3 I) = 0, then \frac{1}{5} A + \frac{6}{5} A^{- 1} =$ (1) $O$ (2) $I$ (3) $2 I$ (4) $3 I$ 123. $Let A, B be two 3 \times 3 matrices and C be a 3 \times 3 unit matrix such that AB - C is a non-singular matrix. Let D = (AB - C)^{- 1} . Then, consider the following statements. Statement I : det (BA) = det (BA - C) det (BDA) Statement II : ABD = DAB Which of the above statements is (are) true?$ (1) $Statement I is true, but Statement II is false$ (2) $Statement II is true, but Statement I is false$ (3) $Both Statement I and Statement II are true$ (4) $Both Statement I and Statement II are false$ 124. $The system of linear equations 3 x - 2 y - kz = 10 2 x - 4 y - 2 z = 6 x + 2 y - z = 5 m is inconsistent if :$ (1) $k = 3, m = \frac{4}{5}$ (2) $k \neq = 3, m \in R$ (3) $k \neq = 3, m \neq = \frac{4}{5}$ (4) $k = 3, m \neq = \frac{4}{5}$ 125. $If 0 b - 3 20 c a 40 is a skew-symmetric matrix, then (a b b a) (b c c b) =$ (1) $(0000)$ (2) $(1001)$ (3) $(2 - 8 - 8 2)$ (4) $(2882)$ 126. $Let A = 2 b 1 b b^{2} + 1 b 1 b 2 where b > 0 . Then the minimum value of \frac{d e t ( A )}{b} is:$ (1) $23$ (2) $- 23$ (3) $- 3$ (4) $3$ 127. $If A = 121112051, then a_{11} A_{21} + a_{12} A_{22} + a_{13} A_{23} =$ (1) $1$ (2) $0$ (3) $-1$ (4) $2$ 128. $If - 1 a 3 25 c b 67 is a symmetric matrix, then a b c b c a c a b =$ (1) $0$ (2) $-121$ (3) $143$ (4) $-143$ 129. $If A = 121 - 1 11 1 - 3 1, 10 B = 4 - 5 1 20 - 2 2 α 3 and B is the inverse of A, then the value of α is$ (1) $2$ (2) $0$ (3) $5$ (4) $4$ 130. $Let M and m respectively be the maximum and the minimum values of f (x) = 1 + sin^{2} x sin^{2} x sin^{2} x cos^{2} x 1 + cos^{2} x cos^{2} x 4 sin 4 x 4 sin 4 x 1 + 4 sin 4 x, x \in R . Then M^{4} - m^{4} is equal to:$ (1) $1280$ (2) $1295$ (3) $1215$ (4) $1040$ 131. $If x (32) + y (1 - 1) = (155) then the value of x and y are$ (1) $x = - 4, y = - 3$ (2) $x = 4, y = 3$ (3) $x = - 4, y = 3$ (4) $x = 4, y = - 3$ 132. $If A = (3546) and B = (x 0 0 y), x, y \in N, then$ (1) $There is exactly one such matrix B such that AB = I$ (2) $There is no matrix B such that AB = BA$ (3) $There exist only a finite number of matrices B such that AB = BA$ (4) $There exist infinite number of matrices B such that AB = BA$ 133. $Consider the matrix .\nGiven below are two statements :\nStatement I: is the inverse of the matrix .\nStatement II: .\nIn the light of the above statements, choose the correct answer from the options given below$ (1) $Statement I is false but Statement II is true$ (2) $Both Statement I and Statement II are false$ (3) $Statement I is true but Statement II is false$ (4) $Both Statement I and Statement II are true$ 134. $If the system of linear equations x - 2 y + kz = 1 2 x + y + z = 2 3 x - y - kz = 3 has a solution (x, y, z), z \neq = 0, then (x, y) lies on the straight line whose equation is :$ (1) $3 x - 4 y - 1 = 0$ (2) $4 x - 3 y - 4 = 0$ (3) $4 x - 3 y - 1 = 0$ (4) $3 x - 4 y - 4 = 0$ 135. $If A = 111 13 - 1 243, B = adj (A) and C = 3 A then \frac{∣ adj B ∣}{∣ C ∣} is$ (1) $8$ (2) $4$ (3) $2$ (4) $16$ 136. $If A = 0 x 0 x 59 167 x is singular, then the possible values of x are$ (1) $0, 12, - 12$ (2) $0, 1, - 1$ (3) $0, 4, - 4$ (4) $0, 5, - 5$ 137. $The area of triangle with vertices (K, 0), (4, 0), (0, 2) is 4 square units, then value of K is$ (1) $8$ (2) $0 or -8$ (3) $0$ (4) $0 or 8$ 138. $If A = (2294) and I = (1001), then 10 A^{- 1} is equal to:$ (1) $4 I - A$ (2) $6 I - A$ (3) $A - 4 I$ (4) $A - 6 I$ 139. $If 2 a 2 b 2 c x_{1} x_{2} x_{3} y_{1} y_{2} y_{3} = \frac{ab c}{2} \neq = 0, then the area of the triangle whose vertices are (\frac{x _{1}}{a}, \frac{y _{1}}{a}), (\frac{x _{2}}{b}, \frac{y _{2}}{b}) and (\frac{x _{3}}{c}, \frac{y _{3}}{c}) is$ (1) $\frac{1}{8} abc$ (2) $\frac{1}{8}$ (3) $\frac{1}{4} abc$ (4) $\frac{1}{4}$ 140. $If A = (α 2 2 α) and ∣ A^{3} ∣ = 27, then α =$ (1) $\pm 2$ (2) $\pm 5$ (3) $\pm 1$ (4) $\pm 7$ 141. $If ω is an imaginary cube root of unity, then the value of 1 ω 1 ω^{2} 1 ω 1 - ω^{4} 1 + ω^{5} ω^{2} is$ (1) $-4$ (2) $ω^{2} - 4$ (3) $ω^{2}$ (4) $4$ 142. $For the system of linear equations: x - 2 y = 1, x - y + kz = - 2, ky + 4 z = 6, k \in R consider the following statements: (A)The system has unique solution if k \neq = 2, k \neq = - 2. (B)The system has unique solution if k = - 2 (C) The system has unique solution ifk=2 DThe system has no-solution ifk=2. (E)The system has infinite number of solutions ifk\neq-2. Which of the following statements are correct?$ (1) $(B) and (E) only$ (2) $(C) and D only$ (3) $(A) and (D) only$ (4) $(A) and (E) only$ 143. $If A = (2 - 4 - 3 1) then (A^{T})^{2} + (12 A)^{T} =$ (1) $5 (8 - 9 125)$ (2) $5 (8 - 12 - 9 5)$ (3) $(4060 - 45 25)$ (4) $(40 - 45 - 60 25)$ 144. $The values of λ and μ for which the system of linear equations x + y + z = 2 x + 2 y + 3 z = 5 x + 3 y + λz = μ has infinitely many solutions are, respectively$ (1) $5 and 7$ (2) $6 and 8$ (3) $4 and 9$ (4) $5 and 8$ 145. $The constant term of the polynomial x + 3 x x + 2 x x + 1 2 x x + 2 x - 1 3 x + 1 is$ (1) $2$ (2) $0$ (3) $1$ (4) $-1$ 146. $The system 2 x + 3 y + z = 5, 3 x + y + 5 z = 7 and x + 4 y - 2 z = 3 has$ (1) $unique solution$ (2) $finite number of solution$ (3) $infinite solutions$ (4) $no solution$ 147. $If (1 - 1 11) (x y) = (24), then the values of x and y respectively are$ (1) $- 3, - 1$ (2) $1,3$ (3) $3,1$ (4) $- 1, 3$ 148. $Let A and B be two invertible matrices of order 3 \times 3 . If det (AB A^{T}) = 8 and det (A B^{- 1}) = 8, then det (B A^{- 1} B^{T}) is equal to:$ (1) $\frac{1}{4}$ (2) $1$ (3) $\frac{1}{16}$ (4) $16$ 149. $For some α, β \in R, let A = (α 212) and B = (111 β) be such that A^{2} - 4 A + 2 I = B^{2} - 3 B + I = O . Then (det (adj (A^{3} - B^{3})))^{2} is equal to____.$ 150. $Let I be an identity matrix of order 2 \times 2 and P = (25 - 1 - 3) . Then the value of n \in N for which P^{n} = 5 I - 8 P is equal to$