NIMCET 2021 — Mathematics PYQ

The coefficient matrix is formed by the coefficients of $x,y,\text{ and }z$:

Step 2: Calculate the determinant.

Expanding along the first row ($R_1$):

Step 3: Determine the condition for "at least one solution".

If $\Delta \ne 0$, the system has a unique solution.

If $\Delta =0$, the system can have infinitely many solutions (consistent) or no solution (inconsistent).

Let's test the case where $\Delta =0$:

Step 4: Verify Consistency for $\lambda =-5$.

To ensure "at least one solution" exists when $\Delta =0$, we must check if ${\Delta }_x={\Delta }_y={\Delta }_z=0$.

Let's check ${\Delta }_x$:

Since $\Delta =0$ and ${\Delta }_x=0$ (and similarly ${\Delta }_y,{\Delta }_z$ will be $0$), the system has infinitely many solutions for $\lambda =-5$.

Final Answer:

The value of $\lambda$ is $-5$.