Q187 / 270 · NDA 2022 (Phase 1) · Mathematics

If $3 \sin ^{-1} x+\cos ^{-1} x=\pi$, then what is $x$ equal to?

  • A
    0
  • B
    $\frac{1}{2}$
  • Correct answer
    C
    $\frac{1}{\sqrt{2}}$
  • D
    $\frac{1}{\sqrt{3}}$

Solution

$\begin{aligned} & 3 \sin ^{-1} x+\cos ^{-1} x=\pi \\ \Rightarrow \quad & 2 \sin ^{-1} x+\sin ^{-1} x+\cos ^{-1} x=\pi\end{aligned}$ $\begin{aligned} & \Rightarrow \quad 2 \sin ^{-1} x+\frac{\pi}{2}=\pi \\ & \Rightarrow \quad 2 \sin ^{-1} x=\frac{\pi}{2} \quad \Rightarrow \quad \sin ^{-1} x=\frac{\pi}{4} \\ & \Rightarrow \quad x=\sin \frac{\pi}{4}=\frac{1}{\sqrt{2}}\end{aligned}$

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

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