JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Consider the following system of equations:

\alpha x + 2y + z = 1

2\alpha x + 3y + z = 1

3x + \alpha y + 2z = \beta

for some $\alpha, \beta \in \mathbb{R}$ . Then which of the following is NOT correct?

Choose the correct answer:

Explanation

System of equation is,
$α x + 2 y + z = 1$
$2 α x + 3 y + z = 1$
$3 x + α y + 2 z = β$

Now, $D = α 2 α 3 am p; 2 am p; 3 am p; α am p; 1 am p; 1 am p; 2$
$\Rightarrow D = α^{2} - 2 α - 3$

For no solution, $D = 0$
$\Rightarrow α^{2} - 2 α - 3 = 0$
$\Rightarrow (α + 1) (α - 3) = 0 \Rightarrow α = - 1, 3$

Here, $D_{1} = 11 β am p; 2 am p; 3 am p; α am p; 1 am p; 1 am p; 2$
$\Rightarrow D_{1} = 2 - β$

$D_{2} = - 1 - 2 3 am p; 1 am p; 1 am p; β am p; 1 am p; 1 am p; 2$
$\Rightarrow D_{2} = 2 - β$

$D_{3} = - 1 - 2 3 am p; 2 am p; 3 am p; - 1 am p; 1 am p; 1 am p; β$
$\Rightarrow D_{3} = β - 2$

So, equation has no solution for,
$β \neq = 2, α = - 1$ and $β \neq = 2, α = 3$