JEE 2022 — Mathematics PYQ

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Question

JEE 2022 — Mathematics PYQ

JEE | Mathematics | 2022

If the system of linear equations

$2x + y - z = 7$

$x - 3y + 2z = 1$

$x + 4y + \delta z = k$ ,

where $\delta, k \in R$ has infinitely many solutions, then $\delta + k$ is equal to:

Choose the correct answer:

Explanation

Solution

For infinitely many solutions, the determinant of the coefficient matrix ( $\Delta$ ) must be zero.

1. Find $\delta$ using $\Delta = 0$ :

$\Delta = \begin{vmatrix} 2 & 1 & -1 \\ 1 & -3 & 2 \\ 1 & 4 & \delta \end{vmatrix} = 0$

$2(-3\delta - 8) - 1(\delta - 2) - 1(4 + 3) = 0$

$-6\delta - 16 - \delta + 2 - 7 = 0$

$-7\delta - 21 = 0 \implies \delta = -3$ .

2. Find $k$ using $\Delta_x, \Delta_y,$ or $\Delta_z = 0$ :

Using $\Delta_z = \begin{vmatrix} 2 & 1 & 7 \\ 1 & -3 & 1 \\ 1 & 4 & k \end{vmatrix} = 0$ :

$2(-3k - 4) - 1(k - 1) + 7(4 + 3) = 0$

$-6k - 8 - k + 1 + 49 = 0$

$-7k + 42 = 0 \implies k = 6$ .

3. Final Calculation:

$\delta + k = -3 + 6 = 3$ .

Correct Option: (B)