Q73 / 170 · Manipal (MET) 2016 · Mathematics

$\left|\begin{array}{ccc}\sin \theta & \cos \theta & \sin 2 \theta \\ \sin \left(\theta+\frac{2 \pi}{3}\right) & \cos \left(\theta+\frac{2 \pi}{3}\right) & \sin \left(2 \theta+\frac{4 \pi}{3}\right) \\ \sin \left(\theta-\frac{2 \pi}{3}\right) & \cos \left(\theta-\frac{2 \pi}{3}\right) & \sin \left(2 \theta-\frac{4 \pi}{3}\right)\end{array}\right|$ equals

  • A
    $3 \sin \theta$
  • B
    $\sin ^3 \theta$
  • Correct answer
    C
    $0$
  • D
    None of these

Solution

$\Delta=\left|\begin{array}{ccc}\sin \theta & \cos \theta & \sin 2 \theta \\ \sin \left(\theta+\frac{2 \pi}{3}\right) & \cos \left(\theta+\frac{2 \pi}{3}\right) & \sin \left(2 \theta+\frac{4 \pi}{3}\right) \\ \sin \left(\theta-\frac{2 \pi}{3}\right) & \cos \left(\theta-\frac{2 \pi}{3}\right) & \sin \left(2 \theta-\frac{4 \pi}{3}\right)\end{array}\right|$ $=\left|\begin{array}{ccc}\sin \theta & \cos \theta & \sin 2 \theta \\ 2 \sin \theta \cos \frac{2 \pi}{3} & 2 \cos \theta \cos \frac{2 \pi}{3} & 2 \sin 2 \theta \cos \frac{4 \pi}{3} \\ \sin \left(\theta-\frac{2 \pi}{3}\right) & \cos \left(\theta-\frac{2 \pi}{3}\right) & \sin \left(2 \theta-\frac{4 \pi}{3}\right)\end{array}\right|$ Applying $R_2 \rightarrow R_2+R_3$, $=-\left|\begin{array}{ccc}\sin \theta & \cos \theta & \sin 2 \theta \\ \sin \theta & \cos \theta & \sin 2 \theta \\ \sin \left(\theta-\frac{2 \pi}{3}\right) & \cos \left(\theta-\frac{2 \pi}{3}\right) & \sin \left(2 \theta-\frac{4 \pi}{3}\right)\end{array}\right|=0$ as $R_1=R_2$

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

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