Question

The equation $\left|\begin{array}{ccc}1& x& x^2\\ x^2& 1& x\\ x& x^2& 1\end{array}\right|=0$ has

• A. exactly two distinct roots
• B. one pair of equal real roots
• C. modulus of each root 1
• D. three pairs of equal roots

Answer 1

Answer: (C), (D)

The correct options are
C modulus of each root 1
D three pairs of equal roots

$\left|\begin{array}{ccc}1& x& x^2\\ x^2& 1& x\\ x& x^2& 1\end{array}\right|$

At, $R_1=R_1+R_2+R_3,$

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

$=(1+x+x^2)\left|\begin{array}{ccc}0& 0& 1\\ (x+1)(x-1)& -(x-1)& x\\ -x(x-1)& (x+1)(x-1)& 1\end{array}\right|$

$=(1+x+x^2)(x-1{)}^2\left|\begin{array}{ccc}0& 0& 1\\ (x+1)& -1& x\\ -x& (x+1)& 1\end{array}\right|$

$=(1+x+x^2)(x-1{)}^2(x^2+1+2x-x)$

$=(1+x+x^2{)}^2(x-1{)}^2$

$=(x^3-1{)}^2$

so, modulus of each root is $1$

It also has three pairs of equal roots