Question

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

• A. $-1$
• B. 1
• C. -3
• D. 9

Answer 1

Answer: (C)

$\Delta =\left|\begin{array}{ccc}x-2& 2x-3& 3x-4\\ 2x-3& 3x-4& 4x-5\\ 3x-5& 5x-8& 10x-17\end{array}\right|=$
$=A{x}^3+B{x}^2+Cx+D$
$R_2\to R_2-R_1$
$R_3\to R_3-R_2$
$\Delta =\left|\begin{array}{ccc}x-2& 2x-3& 3x-4\\ x-1& x-1& x-1\\ x-2& 2\left(x-2\right)& 6\left(x-2\right)\end{array}\right|$
$=(x-1)(x-2)\left|\begin{array}{ccc}x-2& 2x-3& 3x-4\\ 1& 1& 1\\ 2& 2& 2\end{array}\right|$
$=-3(x-1{)}^2(x-2)=-3x^3+12x^2-15x+6$
$\therefore B+C=12-15=-3$