Number of real values of 3-x 2 2 2 4-x 1 -2 -4 -1-x is singular- then

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Greatest Integer Function

Number of rea...

Question

Number of real values of $∣ ∣ ∣ ∣ \begin{matrix} 3 - x & 2 & 2 2 & 4 - x & 1 - 2 & - 4 & - 1 - x \end{matrix} ∣ ∣ ∣ ∣$ is singular, then

1

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3

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2

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infinite

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Solution

The correct option is B $3$
A matrix is singular if and only if it has a determinant of $0$ .
$∣ ∣ ∣ ∣ \begin{matrix} 3 - x & 2 & 2 2 & 4 - x & 1 - 2 & - 4 & - 1 - x \end{matrix} ∣ ∣ ∣ ∣ = 0$
$\Rightarrow (3 - x) [(4 - x) (- 1 - x) + 4] - 2 [- 2 - 2 x + 2] + 2 [- 8 + 8 - 2 x] = 0$
$\Rightarrow (3 - x) [- 4 - 4 x + x + x^{2} + 4] - 2 [- 2 x] + 2 [- 2 x] = 0$
$\Rightarrow (3 - x) [x^{2} - 3 x] + 4 x - 4 x = 0$
$\Rightarrow x (x - 3) (x - 3) = 0$
$∴ x = 0, 3, 3$
Hence real roots are $3$

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Number of real values of ∣∣ ∣∣3−x2224−x1−2−4−1−x∣∣ ∣∣ is singular, then

Number of real values of $∣ ∣ ∣ ∣ \begin{matrix} 3 - x & 2 & 2 2 & 4 - x & 1 - 2 & - 4 & - 1 - x \end{matrix} ∣ ∣ ∣ ∣$ is singular, then