જો સમીકરણ સંહતિ

$2 x+y-z=5$

$2 x-5 y+\lambda z=\mu$

$x+2 y-5 z=7$

ને અસંખ્ય ઉકેલો હોય,તો

$(\lambda+\mu)^2+(\lambda-\mu)^2=........$

Question

જો સમીકરણ સંહતિ

$2 x+y-z=5$

$2 x-5 y+\lambda z=\mu$

$x+2 y-5 z=7$

ને અસંખ્ય ઉકેલો હોય,તો

$(\lambda+\mu)^2+(\lambda-\mu)^2=........$

JEE MAIN 2023, Difficult

Download our app for free and get started

Solution

$\left|\begin{array}{ccc}2 & 1 & -1 \\ 2 & -5 & \lambda \\ 1 & 2 & -5\end{array}\right|=0$

$2(25-2 \lambda)-(-10-\lambda)-(4+5)=0$

$50-4 \lambda+10+\lambda-9=0$

$51=3 \lambda \Rightarrow \lambda=17$

$\left|\begin{array}{ccc}2 & 1 & 5 \\ 2 & -5 & \mu \\ 1 & 2 & 7\end{array}\right|=0$

$2(-35-2 \mu)-(14-\mu)+5(4+5)=0$

$-70-4 \mu-14+\mu+45=0$

$-3 \mu=39$

$-\mu=13$

$(\lambda+\mu)^2+(\lambda-\mu)^2=2\left(\lambda^2+\mu^2\right)$

$=2\left(17^2+13^2\right)=916$

Download our app
and get started for free