Matrices & Determinants PYQs Set 16 (PDF)

Set 16 of 19 · Questions 451-480 of 565

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 16 (Q451-480)
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451. $If Δ = 100510671 and Δ^{'} = 13400613100, then$ (1) $Δ^{2} - 3 Δ^{'} = 0$ (2) $(Δ + Δ^{'})^{2} - 3 (Δ + Δ^{'}) + 2 = 0$ (3) $(Δ + Δ^{'})^{2} + 3 (Δ + Δ^{'}) + 5 = 0$ (4) $Δ + 3 Δ^{'} + 1 = 0$ 452. $If A is a 3 \times 3 matrix and ∣ A ∣ = \frac{1}{2} then (A^{- 1} (Adj (Adj A)))^{- 1} =$ (1) $8$ (2) $\frac{1}{8}$ (3) $\frac{1}{2}$ (4) $2$ 453. $If the system of equations 2 x + 3 y - z = 5, x + α y + 3 z = - 4, 3 x - y + β z = 7 has infinitely many solutions, then 13 α β is equal to$ (1) $1110$ (2) $1120$ (3) $1210$ (4) $1220$ 454. $Let Δ = A x B y C z x^{2} y^{2} z^{2} 111 and Δ_{1} = A x y z B y z x C z x y then$ (1) $Δ_{1} = - Δ$ (2) $Δ_{1} = Δ$ (3) $Δ_{1} = 2Δ$ (4) $Δ_{1} \neq = Δ$ 455. $If the system of equations 2 x + λ y + 3 z = 5, 3 x + 2 y - z = 7, 4 x + 5 y + μ z = 9 has infinitely many solutions, then (λ^{2} + μ^{2}) is equal to:$ (1) $30$ (2) $26$ (3) $22$ (4) $18$ 456. $Among the statements : statement I: If 1 cos α cos β cos α 1 cos γ cos β cos γ 1 = 0 cos α cos β cos α 0 cos γ cos β cos γ 0, then cos^{2} α + cos^{2} β + cos^{2} γ = \frac{3}{2}, and statement II: If x^{2} + x 2 x^{2} + 3 x - 1 x^{2} + 2 x + 3 x + 1 3 x 2 x - 1 x - 2 3 x - 3 2 x - 1 = p x + q, then p^{2} = 196 q^{2},$ (1) $only II is true$ (2) $both are false$ (3) $both are true$ (4) $only I is true$ 457. $If A = \frac{1}{2} (1 - 3 31), then$ (1) $A^{30} - A^{25} = 2 I$ (2) $A^{30} + A^{25} + A = I$ (3) $A^{30} + A^{25} - A = I$ (4) $A^{30} = A^{25}$ 458. $Let B = (1135) and A be a 2 \times 2 matrix such that A B^{- 1} = A^{- 1} . If B C B^{- 1} = A and C^{4} + α C^{2} + β I = O, then 2 β - α is equal to$ (1) $16$ (2) $2$ (3) $8$ (4) $10$ 459. $If 11 x 2 x - 2 1 - 1 11 is singular, then the value of x is$ (1) $2$ (2) $3$ (3) $1$ (4) $0$ 460. $If the system of equations 11 x + y + λ z = - 5, 2 x + 3 y + 5 z = 3, 8 x - 19 y - 39 z = μ has infinitely many solutions, then λ^{4} - μ is equal to :$ (1) $51$ (2) $45$ (3) $47$ (4) $49$ 461. $If Δ_{1} = 111 a^{2} b^{2} c^{2} a^{3} b^{3} c^{3} and Δ_{2} = b c c a ab b + c c + a a + b 111 then \frac{Δ _{1}}{Δ _{2}} =$ (1) $ab + bc + ca$ (2) $abc$ (3) $2 (ab + bc + ca)$ (4) $(a + b + c)^{2}$ 462. $Let x^{2} + x + 1 3 x^{2} - 1 x^{2} + 5 x + 1 x + 1 x + 2 2 x + 3 2 x - 3 x - 1 x + 4 = a x^{4} + b x^{3} + c x^{2} + d x + e be an identity in x . If a, b, c, d are known, then the value of e is$ (1) $29$ (2) $24$ (3) $16$ (4) $9$ 463. $Let A = - 2 x 2 x 13 11 - 1 . If the roots of the equation det A = 0 are l, m then l^{3} - m^{3} =$ (1) $35$ (2) $-35$ (3) $19$ (4) $-19$ 464. $If A and B are two square matrices of the same order such that A B = B and B A = A, then A^{2} + B^{2} is always equal to$ (1) $A + B$ (2) $I$ (3) $2 BA$ (4) $2 AB$ 465. $The following system of linear equations 7 x + 6 y - 2 z = 0 3 x + 4 y + 2 z = 0 x - 2 y - 6 z = 0, has$ (1) $infinitely many solution, x, y, z satisfying y = 2 z$ (2) $infinitely many solution, x, y, z satisfying x = 2 z$ (3) $Only the trivial solution$ (4) $no solution.$ 466. $If b and c are non-zero real numbers, A = 1 b c b 23 c 34 and B = 0 - b - c b 0 - 2 c 20 then det (A + B) =$ (1) $3$ (2) $1$ (3) $-1$ (4) $0$ 467. $1 - 1 3 210322^{2022202120242023} is equal to$ (1) $849 4 - 1 6 11313$ (2) $849 4 - 1 6 13312$ (3) $849 4 - 1 6 13313$ (4) $849416111313$ 468. $Let a - 2 b + c = 1 . If f (x) = x + a x + b x + c x + 2 x + 3 x + 4 x + 1 x + 2 x + 3, then$ (1) $f (- 50) = - 1$ (2) $f (50) = 1$ (3) $f (50) = - 501$ (4) $f (- 50) = 501$ 469. $If the system of linear equations x - 2 y + z = - 4, 2 x + α y + 3 z = 5, 3 x - y + β z = 3 has infinitely many solutions, then 12 α + 13 β is equal to$ (1) $60$ (2) $64$ (3) $54$ (4) $58$ 470. $The set of all values of λ for which the system of linear equations x - 2 y - 2 z = λ x x + 2 y + z = λ y - x - y = λ z has a non-trivial solution :$ (1) $is a singleton$ (2) $contains exactly two elements$ (3) $is an empty set$ (4) $contains more than two elements$ 471. $Let α, β, γ be real numbers. If 7 β 3 52 γ α 111 132 = α + β - 2 α + β - 2 γ α + 2 β + 3 γ then 100 + \frac{2 α + 11 β}{γ} =$ (1) $27$ (2) $-25$ (3) $225$ (4) $-227$ 472. $If A = 1 - sin θ - 1 sin θ 1 - sin θ 1 sin θ 1, then for all θ \in (\frac{3 π}{4}, \frac{5 π}{4}), det (A) lies in the interval:$ (1) $(1, \frac{5}{2})$ (2) $(\frac{5}{2}, 4)$ (3) $(0, \frac{3}{2})$ (4) $(\frac{3}{2}, 3)$ 473. $If a d g b e h c f i is a skew symmetric matrix and b, c, f are non-zero real numbers then \frac{b}{c} =$ (1) $\frac{d h}{f g}$ (2) $\frac{df}{g h}$ (3) $\frac{- df}{g h}$ (4) $\frac{- d h}{f g}$ 474. $Let A, B, C be 3 \times 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements. (S1) A^{13} B^{26} - B^{26} A^{13} is symmetric (S2) A^{26} C^{13} - C^{13} A^{26} is symmetric Then,$ (1) $Only S2 is true$ (2) $Only S1 is true$ (3) $Both S1 and S2 are false$ (4) $Both S1 and S2 are true$ 475. $If (1 tan θ - tan θ 1) (1 - tan θ tan θ 1)^{- 1} = (a b - b a), then$ (1) $a = 1, b = 1$ (2) $a = sin 2 θ and b = cos 2 θ$ (3) $a = cos 2 θ and b = sin 2 θ$ (4) $a = 0 and b = 0$ 476. $If A = \frac{1}{π} (sin^{- 1} (π x) sin^{- 1} (\frac{x}{π}) tan^{- 1} (\frac{x}{π}) cot^{- 1} (π x)), B = \frac{1}{π} (- cos^{- 1} (π x) sin^{- 1} (\frac{x}{π}) tan^{- 1} (\frac{x}{π}) - tan^{- 1} (π x)) then A - B is equal to$ (1) $\frac{1}{2} I$ (2) $I$ (3) $O$ (4) $2 I$ 477. $If A = e^{t} e^{t} e^{t} e^{- t} cos t - e^{- t} cos t - e^{- t} sin t 2 e^{- t} sin t e^{- t} sin t - e^{- t} sin t + e^{- t} cos t - 2 e^{- t} cos t then A is:$ (1) $invertible for all t \in R .$ (2) $invertible only if t = π$ (3) $not invertible for any t \in R .$ (4) $invertible only if t = \frac{π}{2}$ 478. $Let P = 323 - 1 0 - 5 - 2 α 0, where α \in R . Suppose Q = [q_{ij}] is a matrix satisfying P Q = k I_{3} for some nonzero k \in R . If q_{23} = - \frac{k}{8} and ∣ Q ∣ = \frac{k ^{2}}{2}, then α^{2} + k^{2} is equal to$ 479. $Let A = [a_{ij}], a_{ij} \in Z \cap [0, 4], 1 \leq i, j \leq 2 . The number of matrices A such that the sum of all entries is a prime number p \in (2, 13) is ______$ 480. $If the system of equations 2 x + 7 y + λ z = 3, 3 x + 2 y + 5 z = 4, x + μ y + 32 z = - 1 has infinitely many solutions, then (λ - μ) is equal to_______$