Question

If $f\left(x\right)=\left|\begin{array}{ccc}x+c_1& x+a& x+a\\ x+b& x+c_2& x+a\\ x+b& x+b& x+c_3\end{array}\right|$, then $f\left(x\right)$ is

• A. constant
• B. linear in x
• C. quadratic in x
• D. cubic in x

Answer 1

The correct option is B linear in x

Since $f\left(x\right)=\left|\begin{array}{ccc}x+c_1& x+a& x+a\\ x+b& x+c_2& x+a\\ x+b& x+b& x+c_3\end{array}\right|$ ---------$\left(1\right)$

$C_2\to C_2-C_1;C_3\to C_3-C_2$

$=\left|\begin{array}{ccc}x+c_1& a-c_1& 0\\ x+b& c_2-b& a-c_2\\ x+b& 0& c_3-b\end{array}\right|$

$f\left(x\right)=x\left|\begin{array}{ccc}1& a-c_1& 0\\ 1& c_2-b& a-c_2\\ 1& 0& c_3-b\end{array}\right|+\left|\begin{array}{ccc}{c}_1& a-c_1& 0\\ b& c_2-b& a-c_2\\ b& 0& c_3-b\end{array}\right|$
So, $f\left(x\right)$ is linear