JEE Advanced 2017 Paper 1Question 1
For a real number $\alpha$, if the system $$\begin{bmatrix}1&\alpha&\alpha^2\\\alpha&1&\alpha\\\alpha^2&\alpha&1\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}1\\-1\\1\end{bmatrix}$$ of linear equations, has infinitely many solutions, then $1+\alpha+\alpha^2=$
Determinant
Answer$1$
Solution
$\det=(1-\alpha^2)^2=0$ gives $\alpha=\pm1$. For $\alpha=1$ all rows equal but the right side is $(1,-1,1)$: inconsistent. For $\alpha=-1$: rows 1 and 3 coincide and row 2 is $-$row 1, consistent, infinitely many solutions. So $1+\alpha+\alpha^2=1-1+1=1$.