Question

The value of $a^3+b^3+c^3$ is -

• A. $p$
• B. $2p$
• C. $3p$
• D. None of these

Answer 1

The correct option is D None of these

$Q=\left[\begin{array}{ccc}a& b& c\\ c& a& b\\ b& c& a\end{array}\right]$ is an orthogoanal matrix.
Therefore according to given data,
$a^2+b^2+c^2=1$ and $ab+bc+ca=0$
$|Q|=\left|\begin{array}{ccc}a& b& c\\ c& a& b\\ b& c& a\end{array}\right|=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$
$\Rightarrow |Q|=a+b+c=\pm 1$ ($\because |Q{Q}^T|=1$)
$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$
$\Rightarrow a^3+b^3+c^3=\pm 1+3abc$
$\therefore a^3+b^3+c^3=\pm 1+3p$
Hence, option $D$.