Matrices and Determinants-JEE-Main-PYQ’s

Mathematics Questions

Let $A, B$ and $C$ be three $2 \times 2$ matrices with real entries such that $B = (I + A)^{-1}$ and $A + C = I$ . If $BC = \begin{bmatrix} 1 & -5 \\ -1 & 2 \end{bmatrix}$ and $CB \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 12 \\ -6 \end{bmatrix}$ , then $x_1 + x_2$ is

(A) 4

(B) 2

(D) -2

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Let $P = [p_{ij}]$ and $Q = [q_{ij}]$ be two square matrices of order 3 such that $q_{ij} = 2^{(i+j-1)} p_{ij}$ and $\det(Q) = 2^{10}$ . Then the value of $\det(\text{adj}(\text{adj } P))$ is:

(A) 81

(B) 16

(D) 32

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Let $f(x) = \int \frac{7x^{10} + 9x^8}{(1 + x^2 + 2x^9)^2} dx, x > 0, \lim_{x \to 0} f(x) = 0$ and $f(1) = \frac{1}{4}$ .
If $A = \begin{bmatrix} 0 & 0 & 1 \\ \frac{1}{4} & f'(1) & 1 \\ \alpha^2 & 4 & 1 \end{bmatrix}$ and $B = \text{adj}(\text{adj } A)$ be such that $|B| = 81$ , then $\alpha^2$ is equal to

(A) 2

(B) 4

(D) 1

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The system of linear equations
$x + y + z = 6$
$2x + 5y + az = 36$
$x + 2y + 3z = b$
has :

(A) unique solution for $a = 8$ and $b = 16$

(B) infinitely many solutions for $a = 8$ and $b = 14$

(D) unique solution for $a = 8$ and $b = 14$

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Among the statements :
I: If $\begin{vmatrix} 1 & \cos \alpha & \cos \beta \\ \cos \alpha & 1 & \cos \gamma \\ \cos \beta & \cos \gamma & 1 \end{vmatrix} = \begin{vmatrix} 0 & \cos \alpha & \cos \beta \\ \cos \alpha & 0 & \cos \gamma \\ \cos \beta & \cos \gamma & 0 \end{vmatrix}$ , then $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = \frac{3}{2}$ , and
II: If $\begin{vmatrix} x^2 + x & x + 1 & x - 2 \\ 2x^2 + 3x - 1 & 3x & 3x - 3 \\ x^2 + 2x + 3 & 2x - 1 & 2x - 1 \end{vmatrix} = px + q$ , then $p^2 = 196q^2$ ,

(A) both are true

(B) both are false

(D) only II is true

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Let $n$ be the number obtained on rolling a fair die. If the probability that the system
$x - ny + z = 6$
$x + (n - 2)y + (n + 1)z = 8$
$(n - 1)y + z = 1$
has a unique solution is $\frac{k}{6}$ , then the sum of $k$ and all possible values of $n$ is :

(A) 22

(B) 20

(D) 21

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If $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ is a solution of the system of equations $AX = B$ , where
$\text{adj } A = \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$ , then $|x + y + z|$ is equal to :

(A) 3

(B) 2

(D) 1

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If $A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}$ , then the determinant of the matrix $(A^{2025} - 3A^{2024} + A^{2023})$ is

(A) 12

(B) 24

(D) 16

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If the system of equations
$3x + y + 4z = 3$
$2x + \alpha y - z = -3$
$x + 2y + z = 4$
has no solution, then the value of $\alpha$ is equal to:

(A) 13

(B) 4

(D) 23

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For the matrices $A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} -29 & 49 \\ -13 & 18 \end{bmatrix}$ , if
$(A^{15} + B) \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ , then among the following which one is true?

(A) $x = 16, y = 3$

(B) $x = 5, y = 7$

(D) $x = 18, y = 11$

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Matrices and Determinants-JEE-Main-PYQ’s

Mathematics Questions

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