Q51 / 100 · BITSAT 2024 (Memory Based Paper 1) · Mathematics
If $\cos \cot ^{-1}\left(\frac{1}{2}\right)=\cot \left(\cos ^{-1} x\right)$, then the value of $x$ is
- Correct answerA$\frac{1}{\sqrt{6}}$
- B$\frac{-1}{\sqrt{12}}$
- C$\frac{2}{\sqrt{6}}$
- D$\frac{-2}{\sqrt{6}}$
Solution
We have
$\cos \left(\cot ^{-1}\left(\frac{1}{2}\right)\right)=\cot \left(\cos ^{-1} x\right)$
Let $\cot ^{-1}\left(\frac{1}{2}\right)=\alpha \Rightarrow \cot \alpha=\frac{1}{2} \Rightarrow \cos ^{-1} \frac{1}{\sqrt{5}}$
$\Rightarrow \cos \left(\cos ^{-1} \frac{1}{\sqrt{5}}\right)=\cot \left(\cot ^{-1} \frac{x}{\sqrt{1-x^2}}\right)$
$\Rightarrow \frac{1}{\sqrt{5}}=\frac{x}{\sqrt{1-x^2}} \Rightarrow \sqrt{1-x^2}=\sqrt{5} x$
On squaring both sides, we get,
$1-x^2=5 x^2 \Rightarrow 1=6 x^2$
$\Rightarrow x= \pm \frac{1}{\sqrt{6}}$
$\begin{array}{ll}& x=\frac{1}{\sqrt{6}} \quad \text { [neglecting - ve sign] }\end{array}$
Hence, the correct answer is option: 'A'.