Q198 / 270 · NDA 2018 (Phase 1) · Mathematics

Suppose $\cos A$ is given. If only one value of $\cos \left(\frac{A}{2}\right)$ is possible, then A must be

  • A
    An odd multiple of $90^{\circ}$
  • B
    A multiple of $90^{\circ}$
  • Correct answer
    C
    An odd multiple of $180^{\circ}$
  • D
    A multiple of $180^{\circ}$

Solution

Given, $\cos \frac{\mathrm{A}}{2}$ has only one value. We know, $\cos \mathrm{A}=2 \cos ^2 \frac{\mathrm{~A}}{2}-1$ $\Rightarrow 2 \cos ^2 \frac{\mathrm{~A}}{2}=\cos \mathrm{A}+1 \Rightarrow \cos \frac{\mathrm{~A}}{2}=\sqrt{\frac{\cos \mathrm{A}+1}{2}}$ Since, $\cos \frac{A}{2}$ is single value, $\frac{\cos A+1}{2}=0$ $\Rightarrow \cos \mathrm{A}=-1$ So, A is an odd multiple of $180^{\circ}$.

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

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