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JEE Main 2-April-2025 Paper - Shift 2 — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEEJEE Main 2-April-2025 Paper - Shift 24 Marks
MCQ
If the system of equation
$2 x+\lambda y+3 z=5$
$3 x+2 y-z=7$
$4 x+5 y+\mu z=9$
has infinitely many solutions, then $\left(\lambda^{2}+\mu^{2}\right)$ is equal to :
  • A
    22
  • B
    18
  • C
    26
  • D
    30
✓

Answer

C. 26
$\Delta=0 \Rightarrow\left|\begin{array}{ccc}2 & \lambda & 3 \\ 3 & 2 & -1 \\ 4 & 5 & \mu\end{array}\right|=0$
$\Rightarrow 2(2 \mu+5)+\lambda(-4-3 \mu)+3(7)=0$
$\Rightarrow 4 \mu-3 \lambda \mu-4 \lambda+31=0 \ldots .$. (1)
$\Delta_{3}=0 \Rightarrow\left|\begin{array}{lll}2 & \lambda & 5 \\ 3 & 2 & 7 \\ 4 & 5 & 9\end{array}\right|=0$
$\Rightarrow 2(-17)+\lambda(1)+5(7)=0$
$\Rightarrow \lambda=-1$
from equation (1)
$4 \mu+3 \mu+4+31=0 \Rightarrow \mu=-5$
$\therefore \lambda^{2}+\mu^{2}=26$

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