JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

If the system of equations

$2x + y - z = 5$

$2x - 5y + \lambda z = \mu$

$x + 2y - 5z = 7$

has infinitely many solutions, then $(\lambda + \mu)^2 + (\lambda - \mu)^2$ is equal to:

Choose the correct answer:

Explanation

Infinitely many solutions ke liye determinant $\Delta = 0$ hona chahiye:

$\begin{vmatrix} 2 & 1 & -1 \\ 2 & -5 & \lambda \\ 1 & 2 & -5 \end{vmatrix} = 0 \implies 2(25 - 2\lambda) - 1(-10 - \lambda) - 1(4 + 5) = 0$

$50 - 4\lambda + 10 + \lambda - 9 = 0 \implies -3\lambda + 51 = 0 \implies \lambda = 17$ .
$\mu$ ke liye $\Delta_x, \Delta_y, \text{ or } \Delta_z$ bhi $0$ hona chahiye: $\begin{vmatrix} 5 & 1 & -1 \\ \mu & -5 & 17 \\ 7 & 2 & -5 \end{vmatrix} = 0 \implies 5(25 - 34) - 1(-5\mu - 119) - 1(2\mu + 35) = 0$

$-45 + 5\mu + 119 - 2\mu - 35 = 0 \implies 3\mu + 39 = 0 \implies \mu = -13$ .
Final value: $(\lambda + \mu)^2 + (\lambda - \mu)^2 = 2(\lambda^2 + \mu^2) = 2(17^2 + (-13)^2) = 2(289 + 169) = 2(458) = 916$ .

Sahi Option: (2) 916

Choose the correct answer:

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