Question

.Let $\Delta$ $\left(x\right)=$$\left|\begin{array}{ccc}x+a& x+b& x+a-c\\ x+b& x+c& x-1\\ x+c& x+d& x-b+d\end{array}\right|$ and ${\int }_0^2\Delta \left(x\right)dx=-16$, where $a$, b,c,d are in A.$P$. then the common difference of the A.$P$. is

• A. $\pm 1$
• B. $\pm 2$
• C. $\pm 3$
• D. $\pm 4$

Answer 1

The correct option is B $\pm 2$

Let $commondiff=m$
$R_1\to R_1-R_2$
$\Delta \left(x\right)=\left|\begin{array}{ccc}a-b& b-c& a-c+1\\ x+b& x+c& x-1\\ x+c& x+d& x-b+d\end{array}\right|$
$R_3\to R_3-R_2$
$\Delta \left(x\right)=\left|\begin{array}{ccc}a-b& b-c& a-c+1\\ x+b& x+c& x-1\\ c-b& d-c& d-b+1\end{array}\right|$
$=\left|\begin{array}{ccc}m& m& -2m+1\\ x+b& x+c& x-1\\ m& m& 2m+1\end{array}\right|$

Applying $C_2\to C_2-C_1$

=$\left|\begin{array}{ccc}m& 0& -2m+1\\ x+b& m& x-1\\ m& 0& 2m+1\end{array}\right|$

$=m(2m^2+m+2m^2-m)$
$=-4m^3$
${\int }_0^2\Delta =-m^3\left(2\right)=-16$
$m=\pm 2$