JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

For $\alpha, \beta \in \mathbb{R}$ , suppose the system of linear equations

x - y + z = 5

2x + 2y + \alpha z = 8

3x - y + 4z = \beta

has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of:

Choose the correct answer:

Explanation

Solution

Infinitely many solutions ke liye $D = D_1 = D_2 = D_3 = 0$ .

D = \begin{vmatrix} 1 & -1 & 1 \\ 2 & 2 & \alpha \\ 3 & -1 & 4 \end{vmatrix} = 0 \implies 1(8+\alpha) + 1(8-3\alpha) + 1(-2-6) = 0

8 + \alpha + 8 - 3\alpha - 8 = 0 \implies 2\alpha = 8 \implies \alpha = 4

Ab $D_1 = 0$ ke liye:

D_1 = \begin{vmatrix} 5 & -1 & 1 \\ 8 & 2 & 4 \\ \beta & -1 & 4 \end{vmatrix} = 0 \implies 5(8+4) + 1(32-4\beta) + 1(-8-2\beta) = 0

60 + 32 - 4\beta - 8 - 2\beta = 0 \implies 6\beta = 84 \implies \beta = 14

Roots $4$ aur $14$ hain, toh equation:

(x - 4)(x - 14) = x^2 - 18x + 56 = 0

Sahi Option: (3)