Let system of linear equations 4 x+λ y+2 z=0 ; 2 x-y+z=0 ; μ x +2 y +3 z =0, λ, μ ?

English

Gujarati

Hindi

3 and 4 .Determinants and Matrices

hard

Let the system of linear equations $4 x+\lambda y+2 z=0$ ; $2 x-y+z=0$ ; $\mu x +2 y +3 z =0, \lambda, \mu \in R$ has a non-trivial solution. Then which of the following is true?

A

$\mu=6, \lambda \in R$

B

$\lambda=2, \mu \in R$

C

$\lambda=3, \mu \in R$

D

$\mu=-6, \lambda \in R$

(JEE MAIN-2021)

Solution

For non-trivial solution

$\left|\begin{array}{ccc} 4 & \lambda & 2 \\ 2 & -1 & 1 \\ \mu & 2 & 3 \end{array}\right|=0$

$\Rightarrow 2 \mu-6 \lambda+\lambda \mu=12$

when $\mu=6, \quad 12-6 \lambda+6 \lambda=12$

which is satisfied by all $\lambda$

Standard 12

Mathematics

Share

Similar Questions

If $x, y, z$ are in arithmetic progression with common difference $d , x \neq 3 d ,$ and the
determinant of the matrix $\left[\begin{array}{ccc}3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z\end{array}\right]$ is zero, then the value of $k ^{2}$ is ….. .

hard

(JEE MAIN-2021)

If $\left| {\,\begin{array}{*{20}{c}}a&b&{a\alpha – b}\\b&c&{b\alpha – c}\\2&1&0\end{array}\,} \right| = 0$ and $\alpha \ne \frac{1}{2},$ then

medium

$\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&{{\omega ^2}}&\omega \\1&\omega &{{\omega ^2}}\end{array}\,} \right| = $

easy

Let $M$ and $m$ respectively be the maximum and the minimum values of $f(x)=\left|\begin{array}{ccc}1+\sin ^2 x & \cos ^2 x & 4 \sin 4 x \\ \sin ^2 x & 1+\cos ^2 x & 4 \sin 4 x \\ \sin ^2 x & \cos ^2 x & 1+4 \sin 4 x\end{array}\right|, x \in R$ Then $M ^4- m ^4$ is equal to :____

hard

(JEE MAIN-2025)

If the lines $x + 2ay + a = 0$, $x + 3by + b = 0$ and $x + 4cy + c = 0$ are concurrent, then $a$, $b$ and $c$ are in

normal