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Determinants — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDeterminants1 Mark
MCQ
There are two value of a which makes the determinant $\triangle=\begin{vmatrix}1&-2&5\\2&\text{a}&-1\\0&4&2\text{a}\end{vmatrix}$ equal to $86$. The sum of these two values is :
  • A
    $4$
  • B
    $5$
  • ✓
    $-4$
  • D
    $9$
✓

Answer

Correct option: C.
$-4$
$\triangle=\begin{vmatrix}1&-2&5\\2&\text{a}&-1\\0&4&2\text{a}\end{vmatrix}=86$
$\Rightarrow 1\left(2 a^2+4\right)-2(-4 a-20)=86$
$\Rightarrow 2 a^2+4+8 a+40=86$
$\Rightarrow 2 a^2+8 a-42=0$
$\Rightarrow a^2+4 a-21=0$
$\Rightarrow a^2+7 a-3 a-21=0$
$\Rightarrow a(a+7)-3(a+7)=0$
$\Rightarrow a=-7,3$
Sum of the two values of $a=-7+3=-4$
Hence, the correct option is $(c)$

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