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#### Inverse Trigonometric Functions Important Questions

12th Standard EM

Reg.No. :
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Maths

Time : 01:00:00 Hrs
Total Marks : 50
5 x 1 = 5
1. The value of sin-1 (cos x),0$\le x\le\pi$ is

(a)

$\pi-x$

(b)

$x-\frac{\pi}{2}$

(c)

$\frac{\pi}{2}-x$

(d)

$\pi-x$

2. If sin-1 x+sin-1 y=$\frac{2\pi}{3};$then cos-1x+cos-1 y is equal to

(a)

$\frac{2\pi}{3}$

(b)

$\frac{\pi}{3}$

(c)

$\frac{\pi}{6}$

(d)

$\pi$

3. ${ sin }^{ -1 }\frac { 3 }{ 5 } -{ cos }^{ -1 }\frac { 12 }{ 13 } +{ sec }^{ -1 }\frac { 5 }{ 3 } { -cosec }^{ 1- }\frac { 13 }{ 2 }$is equal to

(a)

2$\pi$

(b)

$\pi$

(c)

0

(d)

tan-1$\frac{12}{65}$

4. If $\alpha ={ tan }^{ -1 }\left( tan\cfrac { 5\pi }{ 4 } \right)$ and $\beta ={ tan }^{ -1 }\left( -tan\cfrac { 2\pi }{ 3 } \right)$ then

(a)

$4\alpha =3\beta \quad$

(b)

$3\alpha =4\beta$

(c)

$\alpha -\beta =\cfrac { 7\pi }{ 12 }$

(d)

none

5. The number of real solutions of the equation $\sqrt { 1+cos2x } ={ 2sin }^{ -1 }\left( sinx \right) ,-\pi <x<\pi$ is

(a)

0

(b)

1

(c)

2

(d)

infinte

6. 5 x 1 = 5
7. Amplitude of sine function

8. (1)

3cos-1x

9. sin-1(3x-4x3)

10. (2)

cosec-1x

11. cos-1(4x3-3x)

12. (3)

1

13. $sin^{ -1 }\left( \cfrac { 1 }{ x } \right)$

14. (4)

3sin-1x

15. cos-1(-x)

16. (5)

$\pi -{ cos }^{ -1 }x$

5 x 2 = 10
17. Find the period and amplitude of
y=sin 7x

18. Sketch the graph of y= sin$(\frac{1}{3}x)$for 0$\le x <6\pi$.

19. Prove that ${ tan }^{ -1 }\left( \cfrac { 1 }{ 7 } \right) +{ tan }^{ -1 }\left( \cfrac { 1 }{ 13 } \right) ={ tan }^{ -1 }\left( \cfrac { 2 }{ 9 } \right)$

20. Ecalute $sin\left( { cos }^{ -1 }\left( \cfrac { 3 }{ 5 } \right) \right)$

21. 5 x 3 = 15
22. Find the domain of cos-1$(\frac{2+sinx}{3})$

23. Find the value of
$tan\left( { cos }^{ -1 }\left( \frac { 1 }{ 2 } \right) -{ sin }^{ -1 }\left( -\frac { 1 }{ 2 } \right) \right)$

24. Evaluate $sin\left[ { sin }^{ -1 }\left( \frac { 3 }{ 5 } \right) +{ sec }^{ -1 }\left( \frac { 5 }{ 4 } \right) \right]$

25. Prove that
tan-1$\frac{1}{2}+tan^{-1}\frac{1}{3}=\frac{\pi}{4}$

26. Solve: ${ tan }^{ -1 }\left( \cfrac { x-1 }{ x-2 } \right) +{ tan }^{ -1 }\left( \cfrac { x+1 }{ x+2 } \right) =\cfrac { \pi }{ 4 }$

27. 3 x 5 = 15
28. If a1, a2, a3, ... an is an arithmetic progression with common difference d, prove that tan $x = {-b \pm \sqrt{b^2-4ac} \over 2a}\quad \left[ tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 1 }{ a }_{ 2 } } \right) +tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 2 }{ a }_{ 3 } } \right) +....tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ n }{ a }_{ n-1 } } \right) \right] =\frac { { a }_{ n }-{ a }_{ 1 } }{ 1+{ a }_{ 1 }{ a }_{ n } }$

29. Solve $tan^{ -1 }\left( \frac { x-1 }{ x-2 } \right) +tan^{ -1 }\left( \frac { x+1 }{ x+2 } \right) =\frac { \pi }{ 4 }$

30. Simplify ${ sin }^{ -1 }\left( \cfrac { sinx+cosx }{ \sqrt { 2 } } \right) ,\cfrac { \pi }{ 4 } <x<\cfrac { \pi }{ 4 }$