Q. If the system of equations $2x-y+z=4$, $5x+\lambda y+3z=12$, $100x-47y+\mu z=212$ has infinitely many solutions, then $\mu-2\lambda$ is equal to
A) $57$
B) $59$
C) $55$
D) $56$
Explanation
[Solution by Dr. Abubacker] For infinitely many solutions, $E_3=aE_1+bE_2$. Matching $x$ and constant terms: $2a+5b=100$ and $4a+12b=212$, so $a+3b=53$. Hence $b=6$, $a=35$, and $\mu=a+3b=53$. Matching $y$: $-a+\lambda b=-47\Rightarrow-35+6\lambda=-47\Rightarrow\lambda=-2$. So $\mu-2\lambda=53+4=57$. Answer (1).
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.