JEE Main 2024 — 5 April, Shift 1 — Question 6

$11 \mathrm{x}+\mathrm{y}+\lambda \mathrm{z}=-5$

$2 x+3 y+5 z=3$ $8 x-19 y-39 z=\mu$

for infinite sol.

$\mathrm{D}=\left|\begin{array}{ccc}11 & 1 & \lambda\\ 2 & 3 & 5 \\8 & -19 & -39\end{array}\right|=0$

$\Rightarrow 11(-117+95)-1(-78-40)+\lambda(-38-24)$

$\Rightarrow 11(-22)+118-\lambda(62)=0$ $\Rightarrow 62 \lambda=118-242$

$\Rightarrow \lambda=\frac{-124}{62}=-2$

${{\text{D}}_{1}}=\left|\begin{array}{ccc}-5 & 1 & -2 \\ 3 & 3 & 5 \\ \mu & -19 & -39\end{array} \right|=0 \\{}$

$~\Rightarrow -5\left( -117+95 \right)-1\left( -117-5\mu \right)-2\left( -57-3\mu \right)=0 \\{}$

$~\Rightarrow -5\left( -22 \right)+117+5\mu +114+6\mu =0 \\{}$

$~\Rightarrow 11\mu =-110-231=-341 \\{} ~\Rightarrow \mu =-31 \\{}$

${{\lambda }^{4}}-\mu ={{(-2)}^{4}}-\left( -31 \right)=16+31=47$