JEE 2022 — Mathematics PYQ

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Question

JEE 2022 — Mathematics PYQ

JEE | Mathematics | 2022

For the system of linear equations $\alpha x + y + z = 1$ , $x + \alpha y + z = 1$ , $x + y + \alpha z = \beta$ , which one of the following statements is NOT correct?

Choose the correct answer:

Explanation

$\therefore \Delta = \begin{vmatrix} \alpha & 1 & 1 \\ 1 & \alpha & 1 \\ 1 & 1 & \alpha \end{vmatrix} = 0$

$\Rightarrow \alpha (\alpha^2 - 1) - 1 (\alpha - 1) + (1 - \alpha) = 0$

$\Rightarrow \alpha^3 - \alpha - \alpha + 1 + 1 - \alpha = 0$

$\Rightarrow \alpha^3 - 3\alpha + 2 = 0 \Rightarrow 2, 1, -2$

(1) If $\alpha = 2 \Rightarrow \Delta \neq 0$

$\therefore$ Unique solution exist

(2) If $\alpha = -2, \beta = 1$

$\Rightarrow \Delta = 0$ and $\Delta_n \neq 0$

$\therefore$ No solution exist

(3) If $\alpha = 2, \beta = 1$

$\therefore 2x + y + z = 1$

$x + 2y + z = 1$

$x + y + 2z = 1$

On adding, we get

$4 (x + y + z) = 3$

$\Rightarrow x + y + z = \frac{3}{4}$

(4) For $\alpha = 1, \beta = 1$ , it has inifinite solutions.