Question

If $\begin{vmatrix}1 & a & a^{2} \\ 1 & x & x^{2} \\ b^{2} & a b & a^{2}\end{vmatrix}=0$ then, $x=$

• A. a
• B. $b$
• C. $\frac{a}{b}, a$
• D. $\frac{b}{a}, a$
• E. 0

Answer 1

Answer: (C)

$\begin{vmatrix}1 & a & a^{2} \\ 1 & x & x^{2} \\ b^{2} & a b & a^{2}\end{vmatrix}$
Applying, $R_{3} \rightarrow R_{3}-R_{1}$ and $R_{2} \rightarrow R _{2}-R_{1}$
We get,
$(x-a)(b-1)\begin{vmatrix}1 & a & a^{2} \\ 0 & 1 & x+a \\ b+1 & a & 0\end{vmatrix}=0$
Which gives $x=\frac{a}{b}, a$