If the system of linear equations 2x + y - z = 7 x - 3y + 2z = 1 x + 4y + = k , where , k ∈ R has infinitely many solutions, then + k is equal to

Correct answer: B. 3 For infinitely many solutions, ∆ = | cases 2 & 1 & - 1 1 & - 3 & 2 1 & 4 & cases | = 0 ⇒ = - 3 and ∆ _ 1 = | cases 7 & 1 & - 1 1 & - 3 & 2 K & 4 & - 3 cases | = 0 ⇒ k = 6 ⇒ + K = 3

JEE Main 202229 JuneMorning Shift

Exam: JEE Main
Year: 2022
Date: 29 June
Shift: Morning Shift
Subject: Mathematics
Chapter: Determinants

JEE Main 2022 · 29 June · Morning Shift · Mathematics · Determinants

If the system of linear equations 2x + y - z = 7 x - 3y + 2z = 1 x...

If the system of linear equations

2x + y - z = 7

x - 3y + 2z = 1

x + 4y + \deltaz = k

, where

\delta, k \in R

has infinitely many solutions, then

\delta + k

is equal to

Options

A $- 3$
B $3$
C $6$
D $9$

Correct answer

B. $3$

Step-by-step solution

For infinitely many solutions, $∆ = \left|$ \begin{cases}2 & 1 & - 1 \\ 1 & - 3 & 2 \\ 1 & 4 & \delta\end{cases} $\right| = 0$

$⇒ \delta = - 3$

and ${∆}_{1} = \left|$ \begin{cases}7 & 1 & - 1 \\ 1 & - 3 & 2 \\ K & 4 & - 3\end{cases} $\right| = 0 ⇒ k = 6$

$⇒ \delta + K = 3$

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