Question

Suppose $a\in R$. Let $f(x)=\left|\begin{array}{ccc}x+a& x& x\\ x& x+a& x\\ x& x& x+a\end{array}\right|$
Then $f(2x)-f(x)$ is equal to

• A. $3x{a}^2$
• B. $3x^{2}a$
• C. $x{a}^2$
• D. $a^{2}x$

Answer 1

Answer: (A)

Write $f(x)={\Delta }_1+x{\Delta }_2$ where
${\Delta }_1=\left|\begin{array}{ccc}a& x& x\\ 0& x+a& x\\ 0& x& x+a\end{array}\right|=a\left[(x+a{)}^2-x^2\right]$
$=2x{a}^2+a^3$
and ${\Delta }_2=\left|\begin{array}{ccc}1& x& x\\ 1& x+a& x\\ 1& x& x+a\end{array}\right|$
$=\left|\begin{array}{ccc}1& x& x\\ 0& a& 0\\ 0& 0& a\end{array}\right|=a^2$
Thus $f(x)=2x{a}^2+a^3+x{a}^2$
$=3x{a}^2+a^3$
$\therefore f(2x)-f(x)=3x{a}^2$