JEE Main 2025 Apr 2 Shift 2, Mathematics Q8: Linear Equations And Cramer’s Rule

Infinitely many solutions require $\Delta =0$ and every ${\Delta }_i=0$.

From $\Delta =\left|\begin{array}{c}2\lambda 3\\ 32-1\\ 45\mu \end{array}\right|=0$, expanding along row 1:

$2(2\mu +5)-\lambda (3\mu +4)+3(15-8)=0\Rightarrow 4\mu -3\lambda \mu -4\lambda +31=0(\star )$.

Use ${\Delta }_3=\left|\begin{array}{c}2\lambda 5\\ 327\\ 459\end{array}\right|=0$:

$2(18-35)-\lambda (27-28)+5(15-8)=-34+\lambda +35=\lambda +1=0\Rightarrow \lambda =-1$.

Substitute $\lambda =-1$ in $(\star )$: $4\mu +3\mu +4+31=0\Rightarrow 7\mu =-35\Rightarrow \mu =-5$.

$\therefore {\lambda }^2+{\mu }^2=1+25=26$.