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  4. The value of the determinant |ka&k2+a2&1 kb&k2+b2&1 kc&k2+c2&1| is

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Q. The value of the determinant $\begin{vmatrix}ka&k^{2}+a^{2}&1\\ kb&k^{2}+b^{2}&1\\ kc&k^{2}+c^{2}&1\end{vmatrix}$ is

Determinants

Solution:

We have, $\begin{vmatrix}ka&k^{2}+a^{2}&1\\ kb&k^{2}+b^{2}&1\\ kc&k^{2}+c^{2}&1\end{vmatrix}=\begin{vmatrix}ka&k^{2}&1\\ kb&k^{2}&1\\ kc&k^{2}&1\end{vmatrix}+\begin{vmatrix}ka&a^{2}&1\\ kb&b^{2}&1\\ kc&c^{2}&1\end{vmatrix}$
$=0+k \begin{vmatrix}a&a^{2}&1\\ b&b^{2}&1\\ c&c^{2}&1\end{vmatrix}$
$=k (a-b) (b-c) (c-a)$