Question

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

• A. $-1$
• B. 1
• C. -3
• D. 9

Answer 1

Answer: (C)

$\Delta = \begin{vmatrix}x -2&2x-3&3x-4\\ 2x-3&3x-4&4x-5\\ 3x-5&5x-8&10x-17\end{vmatrix} =$
$= Ax ^{3}+ B x ^{2}+ Cx + D$
$R _{2} \rightarrow R _{2}- R _{1}$
$R _{3} \rightarrow R _{3}- R _{2}$
$\Delta = \begin{vmatrix}x -2&2x-3&3x-4\\ x-1&x-1&x-1\\ x-2&2\left(x-2\right)&6\left(x-2\right)\end{vmatrix}$
$=(x-1)(x-2) \begin{vmatrix}x -2&2x-3&3x-4\\ 1&1&1\\ 2&2&2\end{vmatrix}$
$=-3(x-1)^{2}(x-2)=-3 x^{3}+12 x^{2}-15 x+6$
$\therefore \,\,\,\, B+C=12-15=-3$