For how many value (s) of x in the closed interval [ - 4, , , - 1… - Vidyadip

MCQ

For how many value $(s)$ of $x$ in the closed interval $[ - 4,\,\, - 1]$ is the matrix $\left[ {\begin{array}{*{20}{c}}3&{ - 1 + x}&2\\3&{ - 1}&{x + 2}\\{x + 3}&{ - 1}&2\end{array}} \right]$ singular

A
$2$
B
$0$
C
$3$
✓
$1$

✓

Answer

Correct option: D.

$1$

d
(d) For the given matrix to be singular, we must have, $\left| {\,\begin{array}{*{20}{c}}3&{ - 1 + x}&2\\3&{ - 1}&{x + 2}\\{x + 3}&{ - 1}&2\end{array}\,} \right|\, = 0$
==> $\left| {\,\begin{array}{*{20}{c}}3&{ - 1 + x}&2\\0&{ - x}&x\\x&{ - x}&0\end{array}\,} \right|\, = 0$, $\,[{R_2} \to {R_2} - {R_1},\,{R_3} \to {R_3} - {R_1}]$
==> $\left| {\,\begin{array}{*{20}{c}}{x + 4}&{ - 1 + x}&2\\0&{ - x}&x\\0&{ - x}&0\end{array}\,} \right|\, = 0$, $[{C_1} \to {C_1} + {C_2} + {C_3}]$
==> $(x + 4)\,(0 + {x^2}) = 0 \Rightarrow x = - 4,\,0$
Note that only $ - 4 \in [ - 4,\, - 1]$.

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