MCQ
The determinant $\left| {\,\begin{array}{*{20}{c}}{4 + {x^2}}&{ - 6}&{ - 2}\\{ - 6}&{9 + {x^2}}&3\\{ - 2}&3&{1 + {x^2}}\end{array}\,} \right|$ is not divisible by
- A$x$
- B${x^3}$
- C$14 + {x^2}$
- ✓${x^5}$
✓
Answer
Correct option: D.
${x^5}$
d
(d) $\left| {\,\begin{array}{*{20}{c}}{4 + {x^2}}&{ - 6}&{ - 2}\\{ - 6}&{9 + {x^2}}&3\\{ - 2}&3&{1 + {x^2}}\end{array}} \right| = {x^4}(14 + {x^2})$ $ = x.{x^3}(14 + {x^2})$
(d) $\left| {\,\begin{array}{*{20}{c}}{4 + {x^2}}&{ - 6}&{ - 2}\\{ - 6}&{9 + {x^2}}&3\\{ - 2}&3&{1 + {x^2}}\end{array}} \right| = {x^4}(14 + {x^2})$ $ = x.{x^3}(14 + {x^2})$
Hence, the determinant is divisible by $x$,${x^3}$ and $(14 + {x^2})$,
but not divisible by ${x^5}$.
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