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3 and 4 . determinant and metrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths3 and 4 . determinant and metrices1 Mark
MCQ
Let $A = \left[ {\begin{array}{*{20}{c}}
1&2&3\\
2&2&{ - 1}\\
3&0&k
\end{array}} \right]$ and $f(x) = {x^3} - 2{x^2} - \alpha x + \beta  = 0$ . If $A$ satisfies $f(x)=0$ ,then
  • A
    $k = 1, \alpha = 14$
  • B
    $\alpha  = 13,\beta  = 22$
  • ✓
    $k =  - 1,\beta  = 22$
  • D
    $\alpha  =  - 14,\beta  =  - 22$
✓

Answer

Correct option: C.
$k =  - 1,\beta  = 22$
c
Sum of roots $=$ sum of diagonal elements

$\Rightarrow 2=1+2+k \Rightarrow k=-1$

and product of roots $=$ value of the determinant.

$\Rightarrow-\beta=\left|\begin{array}{ccc}{1} & {2} & {3} \\ {2} & {2} & {-1} \\ {3} & {0} & {\mathrm{k}}\end{array}\right|=-2 \mathrm{k}-24$

$\Rightarrow \beta=22$

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