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System of Linear Equations

2x-3y+4z=0, ...

Question

$2 x - 3 y + 4 z = 0$ , $5 x - 2 y - z = 0$ , $21 x - 8 y + a z = 0$ has infinite solutions then $a =$

- 5

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- 4

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2

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Solution

The correct option is A $- 5$
The determinant of coefficient matrix must be zero.
$| A | = ∣ ∣ ∣ ∣ \begin{matrix} 2 & - 3 & 4 5 & - 2 & - 1 21 & - 8 & a \end{matrix} ∣ ∣ ∣ ∣$ = $0$
$\Rightarrow 2 [- 2 a - 8] + 3 [5 a + 21] + 4 [- 40 + 42] = 0$
$\Rightarrow - 4 a - 16 + 15 a + 63 + 8 = 0$
$\Rightarrow 11 a + 55 = 0$
$\Rightarrow a = - 5$
So, option A is the correct answer.

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