Question

Find the integral value of x, if \[\begin{vmatrix}x^2 & x & 1 \\ 0 & 2 & 1 \\ 3 & 1 & 4\end{vmatrix} = 28 .\]

Answer 1

\[\text{ Given }: \begin{vmatrix}x^2 & x & 1 \\ 0 & 2 & 1 \\ 3 & 1 & 4\end{vmatrix} = 28\]
\[ \Rightarrow x^2 \left( 8 - 1 \right) - x\left( 0 - 3 \right) + 1\left( 0 - 6 \right)\]
\[ \Rightarrow 8 x^2 - x^2 + 3x - 6 = 28\]
\[ \Rightarrow 7 x^2 + 3x - 6 = 28\]
\[ \Rightarrow 7 x^2 + 3x - 34 = 0\]
\[ \Rightarrow \left( 7x + 17 \right) \left( x - 2 \right) = 0\]
\[ \Rightarrow x = 2\]
Integral value of x is 2. Thus,

\[x = \frac{- 17}{7}\] is not an integer.