Question

All possible values of p and q for which

$\cos^{- 1}\sqrt{p} + \cos^{- 1}\sqrt{1 - p} + \cos^{- 1}\sqrt{1 - q} = \frac{3\pi}{4}$ holds, is

• A. $p = - 1,q = \frac{1}{2}$
• B. $q > 1,p = \frac{1}{2}$
• C. $0 \leq p \leq 1,q = \frac{1}{2}$
• D. None of these

Answer 1

Answer: (C)

$0 \leq p \leq 1,q = \frac{1}{2}$

Answer 2

$\cos^{- 1}\sqrt{p} + \cos^{- 1}\sqrt{1 - p} = \frac{3\pi}{4} - \cos^{- 1}\sqrt{1 - q}$

$\Rightarrow$ $\cos^{- 1}\sqrt{p} + \cos^{- 1}\sqrt{1 - p} = \cos^{- 1}\left( \frac{- 1}{\sqrt{2}} \right) - \cos^{- 1}\sqrt{1 - q}$

$\Rightarrow$ $\sqrt{p}\sqrt{1 - p} - \sqrt{1 - p}\sqrt{p} = - \left\lbrack \frac{1}{\sqrt{2}}.\sqrt{1 - q} - \frac{1}{\sqrt{2}}.\sqrt{q} \right\rbrack$

$\Rightarrow 0 = \sqrt{1 - q} - \sqrt{q}$ $\Rightarrow$ $q = \frac{1}{2}$