JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $A$ be a symmetric matrix such that $|A| = 2$ and $\begin{bmatrix} 2 & 1 \\ 3 & \frac{3}{2} \end{bmatrix} A = \begin{bmatrix} 1 & 2 \\ \alpha & \beta \end{bmatrix}$ . If the sum of the diagonal elements of $A$ is $s$ , then $\frac{\beta s}{\alpha^2}$ is equal to

Choose the correct answer:

Explanation

Solution

1. Matrix $A$ assume karna:

Maana $A = \begin{bmatrix} x & y \\ y & z \end{bmatrix}$ (Symmetric matrix).

Given $|A| = xz - y^2 = 2$ .

2. Matrix Multiplication:

\begin{bmatrix} 2 & 1 \\ 3 & \frac{3}{2} \end{bmatrix} \begin{bmatrix} x & y \\ y & z \end{bmatrix} = \begin{bmatrix} 2x+y & 2y+z \\ 3x+\frac{3}{2}y & 3y+\frac{3}{2}z \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ \alpha & \beta \end{bmatrix}

Ab elements ko compare karte hain:

$2x + y = 1$
$2y + z = 2$
$3x + \frac{3}{2}y = \alpha \implies \frac{3}{2}(2x + y) = \alpha \implies \frac{3}{2}(1) = \alpha \implies \alpha = \frac{3}{2}$
$3y + \frac{3}{2}z = \beta \implies \frac{3}{2}(2y + z) = \beta \implies \frac{3}{2}(2) = \beta \implies \beta = 3$

3. Values of $x, y, z$ and $s$ :

Pehli do equations se: $y = 1 - 2x$ aur $z = 2 - 2y = 2 - 2(1 - 2x) = 4x$ .

$xz - y^2 = 2$ mein put karte hain:

x(4x) - (1 - 2x)^2 = 2 \implies 4x^2 - (1 + 4x^2 - 4x) = 2

4x - 1 = 2 \implies 4x = 3 \implies x = \frac{3}{4}

$z = 4(\frac{3}{4}) = 3$ .

Sum of diagonal elements $s = x + z = \frac{3}{4} + 3 = \frac{15}{4}$ .

4. Final Calculation:

\frac{\beta s}{\alpha^2} = \frac{3 \cdot \frac{15}{4}}{(\frac{3}{2})^2} = \frac{\frac{45}{4}}{\frac{9}{4}} = 5