Matrices & Determinants
Determinants
nta_pyq_2025_jan
Grade 12
Question:
If the system $2x-y+z=4,\ 5x+\lambda y+3z=12,\ 100x-47y+\mu z=212$ has infinitely many solutions, then $\mu-2\lambda$ is equal to:
Step-by-Step Solution
Key Concept: $D_{z}=0$ pins down $\lambda$, then $D=0$ (with $\lambda$ substituted) pins down $\mu.$
$D_{z}=\begin{vmatrix}2&-1&4\\5&\lambda&12\\100&-47&212\end{vmatrix}=2(212\lambda+564)+(1060-1200)+4(-235-100\lambda).$
$=424\lambda+1128-140-940-400\lambda=24\lambda+48.$ Setting $=0$: $\lambda=-2.$
$D=\begin{vmatrix}2&-1&1\\5&-2&3\\100&-47&\mu\end{vmatrix}=2(-2\mu+141)+(5\mu-300)+(-235+200)=\mu-53=0\Rightarrow\mu=53.$
$\mu-2\lambda=53-2(-2)=57.$
Correct Answer: 1