QJEEM24Ma3564: JEE Main 2024 Mathematics - Class 11 JEEM24Ma3564 Solution: If the sum of squares of all real values of , for which the…

$\begin{aligned}& 2 x-y+3=0 \\& 6 x+3 y+1=0 \\& \alpha x+2 y-2=0\end{aligned}$

Will not form a $\Delta$ if $\alpha x+2 y-2=0$ is concurrent with $2 x-y+3=0$ and $6 x+3 y+1=0$ or parallel to either of them so

Case-1: Concurrent lines

$\left|\begin{array}{ccc}2 & -1 & 3 \\6 & 3 & 1 \\\alpha & 2 & -2\end{array}\right|=0 \Rightarrow \alpha=\frac{4}{5}$

Case-2 : Parallel lines

$\begin{aligned}& -\frac{\alpha}{2}=\frac{-6}{3} \text { or }-\frac{\alpha}{2}=2 \\& \Rightarrow \alpha=4 \text { or } \alpha=-4 \\& P=16+16+\frac{16}{25} \\& {[P]=\left[32+\frac{16}{25}\right]=32}\end{aligned}$