#### 12th Standard Maths English Medium Inverse Trigonometric Functions Reduced Syllabus Important Questions 2021

12th Standard

Reg.No. :
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Maths

Time : 01:00:00 Hrs
Total Marks : 100

Multiple Choice Questions

15 x 1 = 15
1. 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)

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

2. The domain of the function defined by f(x)=sin−1$\sqrt{x-1}$ is

(a)

[1,2]

(b)

[-1,1]

(c)

[0,1]

(d)

[-1,0]

3. If |x|$\le$1, then 2tan-1 x-sin-1 $\frac{2x}{1+x^2}$ is equal to

(a)

tan-1x

(b)

sin-1x

(c)

0

(d)

$\pi$

4. The equation tan-1 x-cot-1 x=tan-1$\left( \frac { 1 }{ \sqrt { 3 } } \right)$has

(a)

no solution

(b)

unique solution

(c)

two solutions

(d)

infinite number of solutions

5. If ${ tan }^{ -1 }\left\{ \cfrac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right\} =\alpha$ then x2 =

(a)

$sin2\alpha$

(b)

$sin\alpha$

(c)

$cos2\alpha$

(d)

$cos\alpha$

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

(a)

2

(b)

3

(c)

1

(d)

none

7. ·If $\alpha ={ tan }^{ -1 }\left( \cfrac { \sqrt { 3 } }{ 2y-x } \right) ,\beta ={ tan }^{ -1 }\left( \cfrac { 2x-y }{ \sqrt { 3y } } \right)$ then $\alpha -\beta$

(a)

$\cfrac { \pi }{ 6 }$

(b)

$\cfrac { \pi }{ 3 }$

(c)

$\cfrac { \pi }{ 2 }$

(d)

$\cfrac { -\pi }{ 3 }$

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

(a)

0

(b)

$\cfrac { 1 }{ 2 }$

(c)

-1

(d)

none

9. 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

10. 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

11. The domain of cos-1(x2 - 4) is______

(a)

[3, 5]

(b)

[-1, 1]

(c)

$\left[ -\sqrt { 5 } ,-\sqrt { 3 } \right] \cup \left[ \sqrt { 3 } ,\sqrt { 5 } \right]$

(d)

[0, 1]

12. The value of tan $\left( { cos }^{ -1 }\cfrac { 3 }{ 5 } +{ tan }^{ -1 }\cfrac { 1 }{ 4 } \right)$ is ______

(a)

$\cfrac { 19 }{ 8 }$

(b)

$\cfrac { 8 }{ 19 }$

(c)

$\cfrac { 19 }{ 12 }$

(d)

$\cfrac { 3 }{ 4 }$

13. The value of ${ sin }^{ -1 }\left( cos\cfrac { 33\pi }{ 5 } \right)$ is________

(a)

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

(b)

$\cfrac { -\pi }{ 10 }$

(c)

$\cfrac { \pi }{ 10 }$

(d)

$\cfrac { 7\pi }{ 5 }$

14. If x < 0, y < 0 such that xy = 1, then tan--1(x) + tan-l(y) =_____

(a)

$\cfrac { \pi }{ 2 }$

(b)

$\cfrac { -\pi }{ 2 }$

(c)

$-\pi$

(d)

none

15. The pricipal value of ${ sin }^{ -1 }\left( \cfrac { -1 }{ 2 } \right)$ is _________

(a)

$\cfrac { \pi }{ 6 }$

(b)

$\cfrac { -\pi }{ 6 }$

(c)

$\cfrac { \pi }{ 3 }$

(d)

$\cfrac { -\pi }{ 3 }$

1. 2 Marks

10 x 2 = 20
16. Find the principal value of sin-1 $\left( -\frac { 1 }{ 2 } \right)$(in radians and degrees).

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

18. State the reason for cos-1$[cos(-\frac{\pi}{6})]\neq \frac{\pi}{6}.$

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

20. Find the value of
i) $sin\left[ \frac { \pi }{ 3 } -{ sin }^{ 2 }\left( -\frac { 1 }{ 2 } \right) \right]$

21. Find all values of x such that
-5$\pi\le x \le 5\pi$ and cos x =1

22. Find the principal value of sin-1(-l).

23. If ${ cot }^{ -1 }\left( \cfrac { 1 }{ 7 } \right) =\theta$ find the value of cos $\theta$

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

25. Prove that $2{ tan }^{ -1 }\left( \cfrac { 2 }{ 3 } \right) ={ tan }^{ -1 }\left( \cfrac { 12 }{ 5 } \right)$

1. 3 Marks

10 x 3 = 30
26. Find the domain of sin−1(2−3x2)

27. Find the domain of cos-1$(\frac{2+sinx}{3})$

28. Find
i)  tan−1($-\sqrt3$)
ii)  tan−1$(tan\frac{3\pi}{5})$
iii) tan(tan-1(2019))

29. Find the value of tan−1(−1 )+cos-1$(\frac{1}{2})+sin^-1(-\frac{1}{2})$

30. Solve
${ sin }^{ -1 }\frac { 5 }{ x } +{ sin }^{ -1 }\frac { 12 }{ x } =\frac { \pi }{ 2 }$

31. Find the number of solution of the equation tan-1(x-1) tan-1x+tan-1(x+1)tan-1(3x)

32. Find the domain of
g(x)=sin−1x+cos−1x

33. Find the value of
$cot\left( { sin }^{ -1 }\frac { 3 }{ 5 } +{ sin }^{ -1 }\frac { 4 }{ 5 } \right)$

34. Prove that ${ cos }^{ -1 }\left( \cfrac { 4 }{ 5 } \right) +{ tan }^{ -1 }\left( \cfrac { 3 }{ 5 } \right) ={ tan }^{ -1 }\left( \cfrac { 27 }{ 11 } \right)$

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

1. 5 Marks

7 x 5 = 35
36. Find the principal value of cos−1$\left( \frac { \sqrt { 3 } }{ 2 } \right)$

37. Find the principal value of
cosec−1(−1)

38. Find the principal value of
sec−1(−2).

39. Find the domain of the following functions
(i) f(x) = sin-1(2x - 3)
(ii) f(x) = sin-1x + cos x

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

41. 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

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