Question

If $\Delta =\left|\begin{array}{ccc}x-2& 2x-3& 3x-4\\ 2x-3& 3x-4& 4x-5\\ 3x-5& 5x-8& 10x-17\end{array}\right|=A{x}^3+B{x}^2+Cx+D,$ then $B+C$ is equal to

• A. $1$
• B. $-1$
• C. $-3$
• D. $9$

Answer 1

The correct option is C $-3$

Given:
$\Delta =\left|\begin{array}{ccc}x-2& 2x-3& 3x-4\\ 2x-3& 3x-4& 4x-5\\ 3x-5& 5x-8& 10x-17\end{array}\right|$

$R_2\to R_2-R_1,R_3\to R_3-R_2$

$\Rightarrow \Delta =\left|\begin{array}{ccc}x-2& 2x-3& 3x-4\\ x-1& x-1& x-1\\ x-2& 2(x-2)& 6(x-2)\end{array}\right|$

$\Rightarrow \Delta =(x-1)(x-2)\left|\begin{array}{ccc}x-2& 2x-3& 3x-4\\ 1& 1& 1\\ 1& 2& 6\end{array}\right|$

$C_2\to C_2-C_1,C_3\to C_3-C_1$

$\Rightarrow \Delta =(x-1)(x-2)\left|\begin{array}{ccc}x-2& x-1& 2x-2\\ 1& 0& 0\\ 1& 1& 5\end{array}\right|$

$\Rightarrow \Delta =(x-1)(x-2)\left\{\right(-5(x-1)+(2x-2)\}$

$\Rightarrow \Delta =(x-1)(x-2)(-3x+3)$

$\Rightarrow \Delta =-3(x-1{)}^2(x-2)$

$\Rightarrow \Delta =-3x^3+12x^2-15x+6$

Also, $\Delta =A{x}^3+B{x}^2+Cx+D$
comparing the coefficient of same powers
$A=-3,B=12,C=-15$
$\Rightarrow B+C=12-15=-3$