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#### Inverse Trigonometric Functions Model Question Paper

12th Standard EM

Reg.No. :
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Maths

Time : 01:00:00 Hrs
Total Marks : 50
5 x 1 = 5
1. 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)

$\pi$

2. If cot−1x=$\frac{2\pi}{5}$ for some x$\in$R, the value of tan-1 x is

(a)

$\frac{-\pi}{10}$

(b)

$\frac{\pi}{5}$

(c)

$\frac{\pi}{10}$

(d)

$-\frac{\pi}{5}$

3. If the function f(x)sin-1(x2-3), then x belongs to

(a)

[-1,1]

(b)

[$\sqrt2$,2]

(c)

$\\ \\ \\ \left[ -2,-\sqrt { 2 } \right] \cup \left[ \sqrt { 2 } ,2 \right]$

(d)

$\left[ -2,-\sqrt { 2 } \right] \cap \left[ \sqrt { 2 } ,2 \right]$

4. The value of ${ cos }^{ -1 }\left( \cfrac { cos5\pi }{ 3 } \right) +sin^{ -1 }\left( \cfrac { sin5\pi }{ 3 } \right)$ is

(a)

$\cfrac { \pi }{ 2 }$

(b)

$\cfrac { 5\pi }{ 3 }$

(c)

$\cfrac { 10\pi }{ 3 }$

(d)

0

5. If ${ tan }^{ -1 }\left( \cfrac { x+1 }{ x-1 } \right) +{ tan }^{ -1 }\left( \cfrac { x-1 }{ x } \right) ={ tan }^{ -1 }\left( -7 \right)$ then x Is

(a)

0

(b)

-2

(c)

1

(d)

2

6. 5 x 2 = 10
7. For what value of x does sinx=sin−1x?

8. Find the value of sin-1$\left( sin\frac { 5\pi }{ 9 } cos\frac { \pi }{ 9 } +cos\frac { 5\pi }{ 9 } sin\frac { \pi }{ 9 } \right)$.

9. Find all values of x such that
6$\pi\le x \le 6\pi$ and cos x -0

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

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

12. 5 x 3 = 15
13. For what value of x , the inequality$\cfrac { \pi }{ 2 } <{ cos }^{ -1 }(3x-1)<\pi$

14. Find the value of
$cos\left( { cos }^{ -1 }\left( \frac { 4 }{ 5 } \right) +{ sin }^{ -1 }\left( \frac { 4 }{ 5 } \right) \right)$

15. Prove that tan(sin-1x)=$\frac{x}{\sqrt1-x^2},-1<x<1.$

16. Solve ${ tan }^{ -1 }\left( \cfrac { 2x }{ 1-{ x }^{ 2 } } \right) +{ cot }^{ -1 }\left( \cfrac { 1-{ x }^{ 2 } }{ 2x } \right) =\cfrac { \pi }{ 3 } ,x>0$

17. If $sin\left( { sin }^{ -1 }\cfrac { 1 }{ 5 } +{ cos }^{ -1 }x \right) =1$ then find the value ofx.

18. 4 x 5 = 20
19. Find the principal value of cos−1$\left( \frac { \sqrt { 3 } }{ 3 } \right)$

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

21. Write thefunction$f(x)={ tan }^{ -1 }\sqrt { \cfrac { a-x }{ a+x } } -a<x<a$ in the simplest form

22. Provethat ${ tan }^{ -1 }\left( \cfrac { 1-x }{ 1+x } \right) -{ tan }^{ -1 }\left( \cfrac { 1-y }{ 1+y } \right) ={ sin }^{ -1 }\left( \cfrac { y-x }{ \sqrt { 1+{ x }^{ 2 } } .\sqrt { 1+{ y }^{ 2 } } } \right) \\$