KVPY Sample Paper KVPY Stream-SX Model Paper-29

The number of values of \[\theta \in (0,\pi )\] for which the system of linear equations \[x+3y+7z=0\], \[-x+4y+7z=0\], \[(sin3\theta )x+(cos2\theta )y+2z=0\] has a non-trivial solution, is:

A) three

B) two

C) four

D) one

Correct Answer: B

Solution :

\[\left| \begin{matrix} \sin 3\theta & -1 & 1 \\ \cos 2\theta & 4 & 3 \\ 2 & 7 & 7 \\ \end{matrix} \right|=0\]

\[7\sin 3\theta +14\cos 2\theta -14=0\]

\[\sin 3\theta +2\cos 2\theta -2=0,\sin \theta =\frac{1}{2}.\]