1. Tardigrade
  2. Question
  3. Mathematics
  4. Let θ ∈(0, (π/2)). If the system of linear equations (1+ cos 2 θ) x+ sin 2 θ y+4 sin 3 θ z=0 cos 2 θ x+(1+ sin 2 θ) y+4 sin 3 θ z=0 cos 2 θ x+ sin 2 θ y+(1+4 sin 3 θ) z=0 has a non-trivial solution, then the value of θ is :

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. Let $\theta \in\left(0, \frac{\pi}{2}\right)$. If the system of linear equations
$\left(1+\cos ^{2} \theta\right) x+\sin ^{2} \theta y+4 \sin 3\, \theta\, z=0 $
$\cos ^{2} \theta\, x+\left(1+\sin ^{2} \theta\right) y+4 \sin 3 \,\theta \,z=0 $
$\cos ^{2} \theta \,x+\sin ^{2} \theta\, y+(1+4 \sin 3\, \theta) z=0$
has a non-trivial solution, then the value of $\theta$ is :

JEE MainJEE Main 2021Determinants

Solution:

Case-I
$\begin{vmatrix} 1+\cos ^{2} \theta & \sin ^{2} \theta & 4 \sin 3 \theta \\ \cos ^{2} \theta & 1+\sin ^{2} \theta & 4 \sin 3 \theta \\ \cos ^{2} \theta & \sin ^{2} \theta & 1+4 \sin 3 \theta \end{vmatrix}=0$
$C _{1} \rightarrow C _{1}+ C _{2} $
$\begin{vmatrix}2 & \sin ^{2} \theta & 4 \sin 3 \theta \\ 2 & 1+\sin ^{2} \theta & 4 \sin 3 \theta \\1 & \sin ^{2} \theta & 1+4 \sin 3 \theta\end{vmatrix}=0$
$ R _{1} \rightarrow R _{1}- R _{2}, R _{2} \rightarrow R _{2}- R _{3}$
$\begin{vmatrix}0 & -1 & 0 \\1 & 1 & -1 \\1 & \sin ^{2} \theta & 1+4 \sin ^{3} \theta \end{vmatrix}=0$
or $4 \sin 3 \theta=-2$
$\sin 3 \theta=-\frac{1}{2}$
$\theta=\frac{7 \pi}{18}$