Q91 / 160 · WBJEE 2009 · Mathematics

If $5 \cos 2 \theta+2 \cos ^2 \frac{\theta}{2}+1=0$, when $(0 < \theta < \pi)$, then the values of $\theta$ are :

  • A
    $\frac{\pi}{3} \pm \pi$
  • B
    $\frac{\pi}{3}, \cos ^{-1}\left(\frac{3}{5}\right)$
  • C
    $\cos ^{-1}\left(\frac{3}{5}\right) \pm \pi$
  • Correct answer
    D
    $\frac{\pi}{3}, \pi-\cos ^{-1}\left(\frac{3}{5}\right)$

Solution

Hints : $5 \cos 2 \theta+1+\cos \theta+1=0$ $$ \begin{aligned} & 5\left(2 \cos ^2 \theta-1\right)+\cos \theta+2=0 \\ & 10 \cos ^2 \theta+\cos \theta-3=0 \\ & (5 \cos \theta+3)(2 \cos \theta-1)=0 \\ & \cos \theta=\frac{1}{2} \\ & \theta=\frac{\pi}{3} \end{aligned} $$ $$ \mid \begin{aligned} & \cos \theta=-\frac{3}{5} \\ & \theta=\cos ^{-1}\left(-\frac{3}{5}\right) \\ & =\pi-\cos ^{-1}\left(\frac{3}{5}\right) \end{aligned} $$

Hence, the correct answer is option: 'D'.

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