JEE Main 2025 Jan 24 Shift 1, Mathematics Q20: Linear Equations And Cramer’s Rule

For infinitely many solutions, the coefficient determinant and all numerator determinants (Cramer's ${\Delta }_x,{\Delta }_y,{\Delta }_z$) must vanish. Use $\Delta =0$ and one numerator determinant.

$\Delta =\left|\begin{array}{c}2-11\\ 5\lambda 3\\ 100-47\mu \end{array}\right|=2(\lambda \mu +141)+1\cdot (5\mu -300)+1\cdot (-235-100\lambda )$

$=2\lambda \mu +282+5\mu -300-235-100\lambda =2\lambda \mu +5\mu -100\lambda -253=0(1)$

${\Delta }_z=\left|\begin{array}{c}2-14\\ 5\lambda 12\\ 100-47212\end{array}\right|=0$. Expanding (using the same structure with $\mu \to 212/\text{etc.}$ in column 3) yields the simple relation

$6\lambda =-12\Rightarrow \lambda =-2$.

Substitute $\lambda =-2$ into (1):

$2(-2)\mu +5\mu -100(-2)-253=0\Rightarrow \mu +200-253=0\Rightarrow \mu =53$.

$\mu -2\lambda =53-(-4)=57$.

$\therefore$ Option (4).