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3 and 4 .Determinants and Matrices
hard
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
$(\lambda+\mu)^2+(\lambda-\mu)^2=........$
A
$916$
B
$912$
C
$920$
D
$904$
(JEE MAIN-2023)
Solution
$\left|\begin{array}{ccc}2 & 1 & -1 \\ 2 & -5 & \lambda \\ 1 & 2 & -5\end{array}\right|=0$
$2(25-2 \lambda)-(-10-\lambda)-(4+5)=0$
$50-4 \lambda+10+\lambda-9=0$
$51=3 \lambda \Rightarrow \lambda=17$
$\left|\begin{array}{ccc}2 & 1 & 5 \\ 2 & -5 & \mu \\ 1 & 2 & 7\end{array}\right|=0$
$2(-35-2 \mu)-(14-\mu)+5(4+5)=0$
$-70-4 \mu-14+\mu+45=0$
$-3 \mu=39$
$-\mu=13$
$(\lambda+\mu)^2+(\lambda-\mu)^2=2\left(\lambda^2+\mu^2\right)$
$=2\left(17^2+13^2\right)=916$
Standard 12
Mathematics
