Question

The constant term of the polynomial
$\begin{vmatrix} {x+3}&{x} &{x+2}\\ {x}&{x+1}& {x-1} \\ {x+2}&{2x}&{3x+1}\\ \end{vmatrix} $ is ________

• A. -1
• B. 1
• C. 0
• D. 2

Answer 1

Answer: (A)

$\begin{vmatrix}x+3 & x & x+2 \\ x & x+1 & x-1 \\ x+2 & 2 x & 3 x+1\end{vmatrix}=f(x)$
Applying $R_{2} \rightarrow R_{2}-R_{1}$ and $R_{3} \rightarrow R_{3}-R_{1}$
$f(x)=\begin{vmatrix} x+3 & x & x+2 \\ -3 & 1 & -3 \\ -1 & x & 2 x-1 \end{vmatrix}$
Applying $C_{1} \rightarrow C_{1}-C_{3} \text { and } C_{2} \rightarrow C_{2}-C_{3}$
$f(x)= \begin{vmatrix} 1 & -2 & x+2 \\ 0 & 4 & -3 \\ -2 x & 1-x & 2 x-1 \end{vmatrix}$
Expand w.r.t. ' $C_{1}$ '
$f(x)=[4(2 x-1)+3(1-x)]$
$+(-2 x)[6-4(x+2)]$
$f(x)=[8 x-4+3-3 x]+[-2 x][-4 x-2]$
$f(x)=(5 x-1)+\left(8 x^{2}+4 x\right)$
$f(x)=8 x^{2}+9 x-1$
Hence, the constant term of quadratic equation is $-1$