# The number of values of

Question:

The number of values of $\theta \in(0, \pi)$ for which the system of linear equations

$x+3 y+7 z=0$

$-x+4 y+7 z=0$

$(\sin 3 \theta) x+(\cos 2 \theta) y+2 z=0$

has a non-trivial solution, is :

1. One

2. Three

3. Four

4. Two

Correct Option: , 4

Solution:

$\left|\begin{array}{ccc}1 & 3 & 7 \\ -1 & 4 & 7 \\ \sin 3 \theta & \cos 2 \theta & 2\end{array}\right|=0$

$(8-7 \cos 2 \theta)-3(-2-7 \sin 3 \theta)$

$+7(-\cos 2 \theta-4 \sin 3 \theta)=0$

$14-7 \cos 2 \theta+21 \sin 3 \theta-7 \cos 2 \theta$

$-28 \sin 3 \theta=0$

$14-7 \sin 3 \theta-14 \cos 2 \theta=0$

$14-7\left(3 \sin \theta-4 \sin ^{3} \theta\right)-14\left(1-2 \sin ^{2} \theta\right)=0$

$-21 \sin \theta+28 \sin ^{3} \theta+28 \sin ^{2} \theta=0$

$7 \sin \theta\left[-3+4 \sin ^{2} \theta+4 \sin \theta\right]=0$

$\sin \theta$

$4 \sin ^{2} \theta+6 \sin \theta-2 \sin \theta-3=0$

$2 \sin \theta(2 \sin \theta+3)-1(2 \sin \theta+3)=0$

Hence, 2 solutions in $(0, \pi)$

Option (4)

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