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#### Inverse Trigonometric Functions Model Question Paper 1

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
5 x 1 = 5
1. sin−1(cos x)$=\frac{\pi}{2}-x$ is valid for

(a)

$-\pi \le x\le 0$

(b)

$0\pi \le x\le 0$

(c)

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

(d)

$-\frac { \pi }{ 4 } \le x\le \frac { 3\pi }{ 4 }$

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. The number of solutions of the equation ${ tan }^{ -1 }2x+{ tan }^{ -1 }3x=\cfrac { \pi }{ 4 }$

(a)

2

(b)

3

(c)

1

(d)

none

4. ${ tan }^{ -1 }\left( \cfrac { 1 }{ 4 } \right) +{ tan }^{ -1 }\left( \cfrac { 2 }{ 11 } \right)$ =

(a)

0

(b)

$\cfrac { 1 }{ 2 }$

(c)

-1

(d)

none

5. If ${ cos }^{ -1 }x>x>{ sin }^{ -1 }x$ then

(a)

$\cfrac { 1 }{ \sqrt { 2 } } <x\le 1$

(b)

$0\le x<\cfrac { 1 }{ \sqrt { 2 } }$

(c)

$-1\le x<\cfrac { 1 }{ \sqrt { 2 } }$

(d)

x>0

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. Show that cot−1$\left( \frac { 1 }{ \sqrt { { x }^{ 2 }-1 } } \right) ={ sec }^{ -1 }x,|x|>1$

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

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. Solve: cos(tan-1x) = $sin\left( { cot }^{ -1 }\cfrac { 3 }{ 4 } \right)$

18. 4 x 5 = 20
19. Solve $tan^{ -1 }\left( \frac { x-1 }{ x-2 } \right) +tan^{ -1 }\left( \frac { x+1 }{ x+2 } \right) =\frac { \pi }{ 4 }$

20. Solve $cos\left( sin^{ -1 }\left( \frac { x }{ \sqrt { 1+{ x }^{ 2 } } } \right) \right) =sin\left\{ cot^{ -1 }\left( \frac { 3 }{ 4 } \right) \right\}$

21. If ${ tan }^{ -1 }\left( \cfrac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right) =a$ than prove that x2=sin2a

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