Question

If $A\left(x\right)=\left|\begin{array}{ccc}x+1& 2x+1& 3x+1\\ 2x+1& 3x+1& x+1\\ 3x+1& x+1& 2x+1\end{array}\right|$

then ${\int }_0^{1}A\left(x\right)dx=?$

• A. $-15$
• B. $-\frac{25}{3}$
• C. $-30$
• D. $-5$

Answer 1

Answer: (B)

$-\frac{25}{3}$

To solve this determinant

we can apply $R_1\to R_1-R_2$ and $R_2\to R_2-R_3$ ( Row Operations)

$\Rightarrow \left[\begin{array}{ccc}-x& -x& 2x\\ -x& 2x& -x\\ 3x+1& x+1& 2x+1\end{array}\right]$

On expanding above we get $A\left(x\right)=-(20x^3+10x^2)$

Now, ${\int }_0^{1}A\left(x\right)dx=-[\frac{20\times (1{)}^4}{4}+\frac{10\times (1{)}^3}{3}]=\frac{-25}{3}$

Therefore Answer is $B$