Untitled Document

Question Bank No: 1

1. $\cos^{- 1}$ (xy + $\sqrt{1 - x^{2}} \sqrt{1 - y^{2}}$ ) is equal to

a)		$\cos^{- 1}$ x - $\cos^{- 1}$ y
b)		$\cos^{- 1}$ y - $\cos^{- 1}$ x
c)		$\cos^{- 1}$ x + $\cos^{- 1}$ y
d)		None of these

2. It x $>$ 1 then 2 $\tan^{- 1}$ x cannot be equal to

a)		$\cos^{- 1} (\frac{1 - x^{2}}{1 + x^{2}})$
b)		$π$ - $\sin^{- 1} (\frac{2 x}{1 + x^{2}})$
c)		$π + \tan^{- 1} (\frac{2 x}{1 - x^{2}})$
d)		$π - \cos^{- 1} (\frac{1 - x^{2}}{1 + x^{2}})$

3. If x $\in$ $(\frac{1}{2})$ , then $\sin^{- 1}$ (2x $\sqrt{1 - x^{2}})$ is equal to

a)		2 $\sin^{- 1}$ x
b)		2 $\cos^{- 1}$ x
c)		π - 2 $\sin^{- 1}$ x
d)		None of these

4. If minimum value of $(\sin^{- 1} x)^{2}$ + $(\cos^{- 1} x)^{2}$ is $\frac{π^{2}}{k}$ , then value of k is

a)		4
b)		6
c)		8
d)		None of these

5. $\tan^{- 1} (\sqrt{1 + x^{2}}$ -x) is equal to

a)		$\frac{π}{4} + \frac{1}{2} \tan^{- 1} \frac{x}{2}$
b)		$\frac{π}{4} - \frac{1}{2} \tan^{- 1} \frac{x}{2}$
c)		$\frac{π}{2} - \frac{1}{2} \tan^{- 1} \frac{x}{2}$
d)		None of these

6. $\cos^{- 1}$ $\frac{3}{5}$ - $\cos^{- 1}$ $\frac{5}{13}$ is equal to

a)		$\cos^{- 1}$ $(\frac{63}{65})$
b)		- $\cos^{- 1}$ $(\frac{63}{65})$
c)		$π$ – $\cos^{- 1}$ $(\frac{63}{65})$
d)		None of these

7. $\tan^{- 1}$ $\frac{2}{3}$ + $\tan^{- 1}$ $\frac{3}{4}$ is equal to

a)		$\tan^{- 1}$ $\frac{17}{6}$
b)		$\tan^{- 1}$ $\frac{6}{17}$
c)		$π$ - $\tan^{- 1}$ $\frac{17}{6}$
d)		None of these

8. For 0 $\leq$ $\cos^{- 1}$ x $\leq$ $π$ and - $\frac{π}{2}$ $\leq$ $\sin^{- 1}$ x $\leq$ $\frac{π}{2}$ , the value of cos ( $\sin^{- 1}$ x + 2 $\cos^{- 1}$ x) at x = $\frac{1}{5}$

a)		$\frac{- 2 \sqrt{6}}{5}$
b)		$- \frac{\sqrt{6}}{5}$
c)		$\frac{2 \sqrt{6}}{5}$
d)		$\frac{\sqrt{6}}{5}$

9. If $\tan^{- 1}$ 2, $\tan^{- 1}$ 3 are two angles of a triangle, then the third angle is

a)		$30^{0}$
b)		$45^{0}$
c)		$60^{0}$
d)		$75^{0}$

10. If $\sin^{- 1}$ x + $\sin^{- 1}$ y + $\sin^{- 1}$ z = 3 $π$ /2, the value of $x^{100} + y^{100} + z^{100}$ - $\frac{9}{x^{101} + y^{101} + z^{101}}$ is

a)		0
b)		1
c)		2
d)		3

11. Two angles of a triangle are $\cot^{- 1}$ 2 and $\cot^{- 1}$ 3. Then the third angle is

a)		$π$ /4
b)		3 $π$ /4
c)		$π$ /6
d)		$π$ /3

12. If x + 1/x = 2, the principle value of $\sin^{- 1}$ x is

a)		$π$ /4
b)		π $π$ /2
c)		$π$
d)		3 $π$ /2

13. If $\sin^{- 1}$ (1 - x) - 2 $\sin^{- 1}$ x = $π$ /2, then x equals

a)		0, -1/2
b)		0, 12
c)		0
d)		None of these

14. The value of $\tan^{- 1}$ 1 + $\tan^{- 1}$ 2 + $\tan^{- 1}$ 3 is

a)		0
b)		1
c)		$π$
d)		- $π$

15. If $π$ $\leq$ x $\leq$ 2 $π$ , then $\cos^{- 1}$ (cos x) is equal to

a)		x
b)		-x
c)		2π + x
d)		2π - x

16. If A = $\tan^{- 1} (\frac{x \sqrt{3}}{2 k - x})$ and B = $\tan^{- 1} (\frac{2 x - k}{k \sqrt{3}})$ , then the value of A - B is

a)		$0^{0}$
b)		$45^{0}$
c)		$60^{0}$
d)		$30^{0}$

17. The value of sin ( $\cot^{- 1}$ x) is

a)		$\frac{1}{1 + x^{2}}$
b)		$\frac{1}{\sqrt{1 + x^{2}}}$
c)		$\frac{x}{\sqrt{1 + x^{2}}}$
d)		$\frac{x}{1 + x^{2}}$

18. The principal value of sin-1( sin $\frac{2 π}{3})$ is

a)		2 $π$ /3
b)		- 2 $π$ /3
c)		$π$ /3
d)		4 $π$ /3

19. If xy + yz + zx = 1, then $\tan^{- 1}$ x + $\tan^{- 1}$ y + $\tan^{- 1}$ z is equal to

a)		$π$
b)		$π$ /2
c)		1
d)		None of these

20. $\cos^{- 1}$ $\frac{1}{2}$ + 2 $\sin^{- 1}$ $\frac{1}{2}$ is equal to

a)		$π$ /4
b)		$π$ /6
c)		$π$ /3
d)		2 $π$ /3

21. A solution of the equation $\tan^{- 1} (1 + x) + \tan^{- 1} (1 - x) = \frac{π}{2}$ is

a)		x=1
b)		x= -1
c)		x=0
d)		x= $π$

22. Domain of $\sin^{- 1} (x)$ is

a)		(- $π, π)$
b)		$[- 1, 1]$
c)		(0,2 $π)$
d)		(- $\propto, \propto)$

23. If $\tan^{- 1} (x + 1) + \tan^{- 1} (x - 1) = \tan^{- 1} (\frac{8}{31})$ then x is

a)		1
b)		$\frac{1}{2}$
c)		- $\frac{1}{2}$
d)		$\frac{1}{4}$

24. The principal value of $\sin^{- 1} (\frac{- \sqrt{3}}{2})$ is

a)		$\frac{- 2 π}{3}$
b)		$\frac{- π}{3}$
c)		$\frac{4 π}{3}$
d)		$\frac{5 π}{3}$

25. If x = $\sin^{- 1} K$ y = $\cos^{- 1} K; - 1 \leq K \leq 1$ then the correct relationship is

a)		x+y=2
b)		x-y=2
c)		x+y= $\frac{π}{2}$
d)		x-y= $\frac{π}{2}$

26. Considering only the principal values of tan $(\cos^{- 1} x)$ = sin $(\cot^{- 1} \frac{1}{2})$ then x equals

a)		$\frac{1}{\sqrt{5}}$
b)		$\frac{2}{\sqrt{5}}$
c)		$\frac{3}{\sqrt{5}}$
d)		$\frac{\sqrt{5}}{3}$

27. $\cot^{- 1} \frac{ab + 1}{a - b} + \cot^{- 1} \frac{bc + 1}{b - c} + \cot^{- 1} \frac{ca + 1}{c - a}$ is equal

a)		0
b)		1
c)		$\frac{π}{4}$
d)		none of these

28. If A = $\tan^{- 1} x$ then the value of sin 2A is

a)		$\frac{2 x}{\sqrt{1 - x^{2}}}$
b)		$\frac{x}{\sqrt{1 - x^{2}}}$
c)		$\frac{2 x}{1 + x^{2}}$
d)		none of these

29. $\sin (\sin^{- 1} \frac{1}{2} + \cos^{- 1} \frac{1}{2})$ is

a)		0
b)		1
c)		$\frac{1}{2}$
d)		$\frac{1}{\sqrt{2}}$

30. tan $(\frac{π}{4} + \frac{1}{2} \cot^{- 1} x) + \tan (\frac{π}{4} - \frac{1}{2} \cot^{- 1} x)$ where (x $\neq 0)$ is

a)		x
b)		2x
c)		$\frac{2}{x}$
d)		none of these

31. If $\cos^{- 1} x - \sin^{- 1} x = 0$ then x is equal to

a)		$\pm \frac{1}{\sqrt{2}}$
b)		1
c)		$\sqrt{2}$
d)		$\frac{1}{\sqrt{2}}$

32. If $\tan^{- 1} 2, \tan^{- 1} 3$ are two angles of a triangle, then the third angle is

a)		$\frac{π}{4}$
b)		$\frac{3 π}{4}$
c)		$\frac{π}{2}$
d)		none of these

33. The principal value of $\sin^{- 1} [\cos (\sin^{- 1} \frac{\sqrt{3}}{2})]$ is

a)		$\frac{π}{6}$
b)		$\frac{π}{3}$
c)		$\frac{- π}{3}$
d)		none of these

34. $\tan^{- 1} \tan (\frac{3 π}{4})$ is equals to

a)		$\frac{π}{4}$
b)		$\frac{- π}{4}$
c)		$\frac{3 π}{4}$
d)		none of these

35. The principal value of $\sin^{- 1} (\sin \frac{2 π}{3})$ is

a)		$\frac{- 2 π}{3}$
b)		$\frac{2 π}{3}$
c)		$\frac{4 π}{3}$
d)		$\frac{π}{3}$

36. $\cos^{- 1} (\cos (\frac{7 π}{6}))$ is equals to

a)		$\frac{7 π}{6}$
b)		$\frac{5 π}{6}$
c)		$\frac{π}{3}$
d)		$\frac{π}{6}$

37. sin $[\frac{1}{2} \cot^{- 1} (- \frac{3}{4})]$ is equal to

a)		$\frac{1}{\sqrt{5}}$
b)		$\frac{- 2}{\sqrt{5}}$
c)		$\frac{2}{\sqrt{5}}$
d)		$\frac{- 1}{\sqrt{5}}$

38. If we consider only the principal value of the inverse trigometric functions then the value of tan $(\cos^{- 1} \frac{1}{5 \sqrt{2}} - \sin^{- 1} \frac{4}{\sqrt{17}})$ is

a)		$\frac{\sqrt{29}}{3}$
b)		$\frac{29}{3}$
c)		$\frac{\sqrt{3}}{29}$
d)		$\frac{3}{29}$

39. The value of tan $[\cos^{- 1} (\frac{4}{5}) + \tan^{- 1} (\frac{2}{3})]$ is

a)		$\frac{6}{17}$
b)		$\frac{17}{6}$
c)		$\frac{16}{7}$
d)		None of these

40. $\cos^{- 1} (\frac{1}{2}) + 2 \sin^{- 1} (\frac{1}{2})$ is equal to

a)		$\frac{π}{4}$
b)		$\frac{π}{6}$
c)		$\frac{π}{3}$
d)		$\frac{2 π}{3}$