First slide

Properties of ITF

Question

If $\cos^{- 1} \sqrt{p} + \cos^{- 1} \sqrt{1 - p} + \cos^{- 1} \sqrt{1 - q} = \frac{3 π}{2}$ , then the value of q is

Moderate

A

1
B

$\frac{1}{\sqrt{2}}$
C

$\frac{1}{3}$
D

$\frac{1}{2}$

Solution

Let $α = \cos^{- 1} \sqrt{p}, β = \cos^{- 1} \sqrt{1 - p}$
and $γ = \cos^{- 1} \sqrt{1 - q} or \cos α = \sqrt{p}, \cos β = \sqrt{1 - p}$
and $\cos γ = \sqrt{1 - q}$
Therefore, $\sin α = \sqrt{1 - p}, \sin β = \sqrt{p}, \sin γ = \sqrt{q}$
The given equation may be written as $α + β + γ = \frac{3 π}{4}$
or $α + β = \frac{3 π}{4} - γ or \cos (α + β) = \cos (\frac{3 π}{4} - γ)$
$\Rightarrow \cos αcos β - \sin γsin β = \cos [π - (π / 4 + γ)] = - \cos (\frac{π}{4} + γ)$
$\Rightarrow \sqrt{p} \sqrt{1 - p} - \sqrt{1 - p} \sqrt{p} = - (\frac{1}{\sqrt{2}} \sqrt{1 - q} - \frac{1}{\sqrt{2}} \sqrt{q}) \Rightarrow 0 = \sqrt{1 - q} - \sqrt{q} \Rightarrow 1 - q = q \Rightarrow q = \frac{1}{2}$

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