Question

If $\left|\begin{array}{ccc}x-1& 5x& 7\\ x^2-1& x-1& 8\\ 2x& 3x& 0\end{array}\right|=a{x}^3+b{x}^2+cx+d$ then value of $c$ is given by

• A. $-1$
• B. $12$
• C. $15$
• D. $17$

Answer 1

Answer: (D)

$17$

Differentiating both the sides of (1), we get
$\left|\begin{array}{ccc}1& 5& 0\\ x^2-1& x-1& 8\\ 2x& 3x& 0\end{array}\right|+\left|\begin{array}{ccc}x-1& 5x& 7\\ 2x& 1& 0\\ 2x& 3x& 0\end{array}\right|+\left|\begin{array}{ccc}x-1& 5x& 7\\ x^2-1& x-1& 8\\ 2& 3& 0\end{array}\right|=3a{x}^2+2bx+c$
Putting $x=0$ we get
$c=\left|\begin{array}{ccc}1& 5& 0\\ -1& -1& 8\\ 0& 0& 0\end{array}\right|+\left|\begin{array}{ccc}-1& 0& 7\\ 0& 1& 0\\ 0& 0& 0\end{array}\right|+\left|\begin{array}{ccc}-1& 0& 7\\ -1& -1& 8\\ 2& 3& 0\end{array}\right|$
$\Rightarrow c=0+0+17=17$