Matrices & Determinants PYQs Set 19 (PDF)

Set 19 of 19 · Questions 541-565 of 565

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 19 (Q541-565)
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541. $If a system of three linear equations in three unknowns, which is in the matrix equation form of AX = D, is inconsistent, then rank of \frac{rank of A}{rank of AD} is$ (1) $less than one$ (2) $greater than or equal to one$ (3) $one$ (4) $greater than one$ 542. $The set of real values of α for which the system of linear equations x + (sin α) y + (cos α) z = 0, and x + (cos α) y + (sin α) z = 0 - x + (sin α) y - (cos α) z = 0 has a non-trivial solution is$ (1) $\frac{nπ}{2} + (- 1)^{n} \frac{π}{4} + \frac{π}{8} ( n is an integer)$ (2) $\frac{nπ}{2} + (- 1)^{n} \frac{π}{8} ( n is an integer)$ (3) $\frac{nπ}{2} + (- 1)^{n} \frac{π}{8} - \frac{π}{8} ( n is an integer)$ (4) $\frac{nπ}{2} + (- 1)^{n} \frac{π}{4} - \frac{π}{8} ( n is an integer)$ 543. $The value of the determinant b^{2} - ab ab - a^{2} b c - a c b - c a - b c - a b c - a c b^{2} - ab ab - a^{2} is$ (1) $abc$ (2) $a + b + c$ (3) $0$ (4) $ab + bc + ca$ 544. $If [x] is the greatest integer less than or equal to x and \| x \| is the modulus of x, then the system of three equations 2 x + 3 \| y \| + 5 [z] = 0, x + \| y \| - 2 [z] = 4, x + \| y \| + [z] = 1 has$ (1) $a unique solution$ (2) $finitely many solutions$ (3) $infinitely many solution$ (4) $no solution$ 545. $The maximum value of the determinant of the matrix 1 + sin^{2} x sin^{2} x sin^{2} x cos^{2} x 1 + cos^{2} x cos^{2} x 4 sin 2 x 4 sin 2 x 1 + 4 sin 2 x is$ (1) $0$ (2) $2$ (3) $4$ (4) $6$ 546. $sin α sin β sin γ cos α cos β cos γ sin (α + δ) sin (β + δ) sin (γ + δ) =$ (1) $0$ (2) $1$ (3) $1 + sin α sin β sin γ$ (4) $1 - (sin α - sin β) (sin β - sin γ) (sin γ - sin α)$ 547. $If A = 2 - 1 4 121303, B = 313221130, then det (2 B^{- 1} A^{- 1}) is equal to$ (1) $\frac{1}{6}$ (2) $\frac{- 1}{24}$ (3) $\frac{1}{3}$ (4) $\frac{- 1}{6}$ 548. $If the matrix A = 122212221 satisfies the matrix equation A^{2} - 4 A - 5 I = 0, then A^{- 1} =$ (1) $\frac{1}{5} - 3 - 2 2 232 2 - 2 - 3$ (2) $\frac{1}{5} - 3 22 2 - 3 2 22 - 3$ (3) $\frac{1}{5} - 3 2 - 2 2 - 3 - 2 223$ (4) $\frac{1}{5} - 3 22 2 - 3 2 223$ 549. $The rank of the matrix 456 2 k 3 1 - x 1 1 + x is 1, then,$ (1) $k = \frac{5}{2}, x = \frac{1}{5}$ (2) $k = \frac{5}{2}, x \neq = \frac{1}{5}$ (3) $k = \frac{1}{5}, x = \frac{5}{2}$ (4) $k \neq = \frac{5}{2}, x = \frac{1}{5}$ 550. $The system of equations 4 x + y + 2 z = 5, x - 5 y + 3 z = 10, 9 x - 3 y + 7 z = 20 has$ (1) $no solution$ (2) $unique solution$ (3) $two solutions$ (4) $infinite number of solutions$ 551. $What is the value of a a - b b + c b b - c c + a c c - a a + b ?$ (1) $a^{3} + b^{3} + c^{3} + 3 abc$ (2) $a^{3} + b^{3} + c^{3} - 3 abc$ (3) $a^{3} + b^{3} + c^{3} - 6 abc$ (4) $a^{3} + b^{3} + c^{3} + 6 abc$ 552. $If x = α, y = β and z = γ is the unique solution of the system of equations 5 x - 7 y + 3 z = 0, 7 x + 10 y - 8 z = 3 and 2 x + 3 y - 4 z + 4 = 0, then β is equal to$ (1) $\frac{1}{2}$ (2) $2$ (3) $-2$ (4) $\frac{- 1}{2}$ 553. $If a, b, c are distinct positive real numbers, then the value of the determinant a b c b c a c a b is$ (1) $< 0$ (2) $> 0$ (3) $0$ (4) $\geq0$ 554. $Inverse of a diagonal non-singular matrix is$ (1) $diagonal matrix$ (2) $scalar matrix$ (3) $skew symmetric matrix$ (4) $zero matrix$ 555. $If p and q are two distinct real values of λ for which the system of equations (λ - 1) x + (3 λ + 1) y + 2 λz = 0 (λ - 1) x + (4 λ - 2) y + (λ + 3) z = 02 x + (3 λ + 1) y + 3 (λ - 1) z = 0 has non-zero solution, then p^{2} + q^{2} - pq =$ (1) $15$ (2) $9$ (3) $3$ (4) $6$ 556. $Let A = 101010010 . Then A^{2025} - A^{2020} is equal to :$ (1) $A^{6} - A$ (2) $A^{5}$ (3) $A^{5} - A$ (4) $A^{6}$ 557. $Let for some real numbers α and β, a = α - i β . If the system of equations 4ix + (l + i) y = 0 and 8 (cos \frac{2 π}{3} + i sin \frac{2 π}{3}) x + a ˉ y = 0 has more than one solution then \frac{α}{β} is equal to$ (1) $- 2 + 3$ (2) $2 - 3$ (3) $2 + 3$ (4) $- 2 - 3$ 558. $Let A = [1 - 1 24] . If A^{- 1} = α I + β A, α, β \in R, I is a 2x2 identity matrix, then 4_{(} α - β_{)} is equal to :$ (1) $5$ (2) $\frac{8}{3}$ (3) $2$ (4) $4$ 559. $The values of a and b, for which the system of equations 2x + 3y + 6z = 8 x + 2y + az = 5 3x + 5y + 9z = b has no solution, are:$ (1) $a = 3, b \neq 13$ (2) $a \neq 3, b \neq 13$ (3) $a \neq 3, b = 3$ (4) $a = 3, b = 13$ 560. $If f (x) = x - 3 x - 5 1 2 x^{2} - 18 2 x^{2} - 50 2 3 x^{3} - 81 4 x^{3} - 500 3, then f (1) f (3) + f (3) f (5) + f (5) f (1) is equal to$ (1) $f (1)$ (2) $f (3)$ (3) $f (1) + f (3)$ (4) $f (1) + f (5)$ 561. $The value of sin^{2} 1 4^{\circ} sin^{2} 6 6^{\circ} tan 13 5^{\circ} sin^{2} 6 6^{\circ} tan 13 5^{\circ} sin^{2} 1 4^{\circ} tan 13 5^{\circ} sin^{2} 1 4^{\circ} sin^{2} 6 6^{\circ} is$ (1) $1$ (2) $- 1$ (3) $2$ (4) $0$ 562. $If A = 1 - 2 3 0 - 1 4 - 2 21, then A^{- 1} =$ (1) $A^{2} - 2 A - 4 I$ (2) $A^{2} - A - 3 I$ (3) $\frac{1}{2} (A^{2} + A + 2 I)$ (4) $A^{2} + A - 2 I$ 563. $If A, B are two non singular matrices of order 3, \| B \| = k, a positive integer, then match the items of list-I with the items of list-II. List-I (A) (k^{- 1} A^{- 1}) (B) AdjA-1 (C) BA B^{- 1} = I, \Rightarrow B A^{k} B^{- 1} = (D) AdjAdjA-1= List-II I. B A^{k} + A^{k} B II. BAdj(B)\|B\| III. \frac{1}{\| B \| ^{3} \| A \|} IV. \frac{1}{\| A \|} (A^{- 1}) V. \frac{1}{\| A \| ^{2}} The correct answer is A B C D$ (1) $III V II IV$ (2) $III IV I II$ (3) $I V II IV$ (4) $III IV II I$ 564. $Choose the correct option about the matrices given below A = cos \frac{π}{4} - sin \frac{π}{4} 0 sin \frac{π}{4} cos \frac{π}{4} 0 001, B = 100 0 cos \frac{π}{3} - sin \frac{π}{3} 0 sin \frac{π}{3} cos \frac{π}{3}, C = cos \frac{π}{6} 0 - sin \frac{π}{6} 01 cos \frac{π}{6} sin \frac{π}{6} 00, D = cos \frac{π}{2} - sin \frac{π}{2} 0 sin \frac{π}{2} cos \frac{π}{2} 0 001$ (1) $A^{2020} = I$ (2) $B^{2020} = I$ (3) $D^{2019} = I$ (4) $B^{2022} = I$ 565. $If P = [1 1/2 01], then P^{s 0} is:$ (1) $[12501]$ (2) $[10501]$ (3) $[10251]$ (4) $[15001]$