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Inverse Trigonometric Functions

Multiple Choice Questions

111.

If $\cos^{- 1} (p) + \cos^{- 1} (q) + \cos^{- 1} (r) = π$ , then p² + q² + r² + 2pqr is equal to

112.

If $\sin^{- 1} (\frac{x}{5}) + \csc^{- 1} (\frac{5}{4}) = \frac{π}{2}$ , then x is equal to

113.

$\sin (\frac{1}{2} \cos^{- 1} (\frac{4}{5}))$

$- \frac{1}{\sqrt{10}}$
$\frac{1}{\sqrt{10}}$
$- \frac{1}{10}$
$\frac{1}{10}$

114.

If $\cos^{- 1} (x) + \cos^{- 1} (y) + \cos^{- 1} (z) = 3 π$ , then xy + yz + zx is equal to

115.

If $\tan^{- 1} (\frac{x + 1}{x - 1}) + \tan^{- 1} (\frac{x - 1}{x}) = \tan^{- 1} (- 7)$ , then the value of x is

zero
- 2
1
2

116.

If $\cos^{- 1} (\sqrt{p}) + \cos^{- 1} (\sqrt{1 - p}) + \cos^{- 1} (\sqrt{1 - q}) = \frac{3 π}{4}$ , then the value of q is

$\frac{2}{\sqrt{2}}$
1
1/2
1/3

117.

The soluton of $\sin^{- 1} (x) - \sin^{- 1} (2 x) = \pm \frac{π}{3}$ is

$\pm \frac{1}{3}$
$\pm \frac{1}{4}$
$\pm \frac{\sqrt{3}}{2}$
$\pm \frac{1}{2}$

118.

If $\cos^{- 1} (x) = α, (0 < x < 1)$ and $\sin^{- 1} (2 x \sqrt{1 - x^{2}})$ + $\sec^{- 1} (\frac{1}{2 x^{2} - 1}) = \frac{2 π}{3}$ , then $\tan^{- 1} (2 x)$ equals :

$\frac{π}{6}$
$\frac{π}{4}$
$\frac{π}{3}$
$\frac{π}{2}$

119.

If a > b > 0, then the value of $\tan^{- 1} (\frac{a}{b}) + \tan^{- 1} (\frac{a + b}{a - b})$ depends on :

both a and b
b and not a
a and not b
neither a nor b

120.
The solution of $\tan^{- 1} (x) + 2 \cot^{- 1} (x) = \frac{2 π}{3}$ is

$- \frac{1}{\sqrt{3}}$

$\frac{1}{\sqrt{3}}$

$- \sqrt{3}$

$\sqrt{3}$

$\sqrt{3}$

$We have, \tan^{- 1} (x) + 2 \cot^{- 1} (x) = \frac{2 π}{3} \Rightarrow \tan^{- 1} (x) + 2 \tan^{- 1} (\frac{1}{x}) = \frac{2 π}{3} \Rightarrow \tan^{- 1} (x) + \tan^{- 1} (\frac{2 (\frac{1}{x})}{1 - {(\frac{1}{x})}^{2}}) = \frac{2 π}{3} [∵ 2 \tan^{- 1} (x) = \tan^{- 1} (\frac{2 x}{1 - x^{2}})] \Rightarrow \tan^{- 1} (\frac{x + \frac{2 x}{x^{2} - 1}}{1 - \frac{2 x}{x^{2} - 1}}) = \tan (\frac{2 π}{3}) \Rightarrow \frac{x^{3} - x + 2 x}{x^{2} - 1 - 2 x^{2}} = \tan (\frac{2 π}{3}) \Rightarrow \frac{x^{3} + x}{- 1 - x^{2}} = \tan (\frac{2 π}{3}) \Rightarrow \frac{x (x^{2} + 1)}{- 1 (x^{2} + 1)} = - \sqrt{3} \Rightarrow x = \sqrt{3}$

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Multiple Choice Questions

120. The solution of tan-1x + 2cot-1x = 2π3 is - 13 13 - 3 3

120.
The solution of $\tan^{- 1} (x) + 2 \cot^{- 1} (x) = \frac{2 π}{3}$ is

$- \frac{1}{\sqrt{3}}$

$\frac{1}{\sqrt{3}}$

$- \sqrt{3}$

$\sqrt{3}$