JEE Main 2025 Jan 23 Shift 2, Mathematics Q3: Linear Equations And Cramer’s Rule

A linear system has no solution when the coefficient determinant vanishes while at least one of the numerator determinants is non-zero.

Coefficient determinant:

$D=\left|\begin{array}{c}111\\ 125\\ 15\lambda \end{array}\right|=1(2\lambda -25)-1(\lambda -5)+1(5-2)=\lambda -17.$

$D=0\Rightarrow \lambda =17$.

With $\lambda =17$, evaluate

$D_z=\left|\begin{array}{c}116\\ 129\\ 15\mu \end{array}\right|=1(2\mu -45)-1(\mu -9)+6(5-2)=\mu -18.$

$D_z\ne 0\Rightarrow \mu \ne 18$. Thus no solution requires $\lambda =17$ and $\mu \ne 18$.