If system of equation 2 x+λ y+3 z=5 , 3 x+2 y-z=7 , 4 x+5 y+μ z=9 has infinitely many solutions, t

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3 and 4 .Determinants and Matrices

hard

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$30$

(JEE MAIN-2025)

Solution

$\Delta=0 \Rightarrow\left|\begin{array}{ccc}2 & \lambda & 3 \\ 3 & 2 & -1 \\ 4 & 5 & \mu\end{array}\right|=0$
$\Rightarrow 2(2 \mu+5)+\lambda(-4-3 \mu)+3(7)=0$
$\Rightarrow 4 \mu-3 \lambda \mu-4 \lambda+31=0 \ldots \ldots .(1)$
$\Delta_3=0 \Rightarrow\left|\begin{array}{lll}2 & \lambda & 5 \\ 3 & 2 & 7 \\ 4 & 5 & 9\end{array}\right|=0$
$\Rightarrow 2(-17)+\lambda(1)+5(7)=0$
$\Rightarrow \lambda=-1$
from equation $(1)$
$4 \mu+3 \mu+4+31=0 \Rightarrow \mu=-5$
$\therefore \lambda^2+\mu^2=26$

Standard 12

Mathematics

The system of equations : $2x\, \cos^2\theta + y\, \sin2\theta – 2\sin\theta = 0$ $x\, \sin2\theta + 2y\, \sin^2\theta = – 2\, \cos\theta$ $x\, \sin\theta – y \cos\theta = 0$ , for all values of $\theta$ , can

normal

The value of the determinant $\left| {\begin{array}{*{20}{c}}{{a^2}}&a&1\\{\cos \,(nx)}&{\cos \,(n\, + \,1)\,x}&{\cos \,(n\, + \,2)\,x}\\{\sin \,(nx)}&{\sin \,(n\, + \,1)\,x}&{\sin \,(n\, + \,2)\,x}\end{array}} \right|$ is independent of :

normal

Let $m$ and $M$ be respectively the minimum and maximum values of

$\left|\begin{array}{ccc}\cos ^{2} x & 1+\sin ^{2} x & \sin 2 x \\ 1+\cos ^{2} x & \sin ^{2} x & \sin 2 x \\ \cos ^{2} x & \sin ^{2} x & 1+\sin 2 x\end{array}\right|$.

Then the ordered pair $( m , M )$ is equal to

hard

(JEE MAIN-2020)

$\left| {\,\begin{array}{*{20}{c}}1&5&\pi \\{{{\log }_e}e}&5&{\sqrt 5 }\\{{{\log }_{10}}10}&5&e\end{array}\,} \right| = $

medium

The following system of linear equations $2 x+3 y+2 z=9$ ; $3 x+2 y+2 z=9$ ;$x-y+4 z=8$

medium

(JEE MAIN-2021)