12th Standard Maths English Medium Inverse Trigonometric Functions Reduced Syllabus Important Questions With Answer Key 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. 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 x=$\frac{1}{5}$, the valur of cos (cos-1x+2sin-1x) is

(a)

$-\sqrt { \frac { 24 }{ 25 } }$

(b)

$\sqrt { \frac { 24 }{ 25 } }$

(c)

$\frac{1}{5}$

(d)

-$\frac{1}{5}$

3. If sin-1 x+cot-1$(\frac{1}{2})=\frac{\pi}{2}$, then x is equal to

(a)

$\frac{1}{2}$

(b)

$\frac{1}{\sqrt{5}}$

(c)

$\frac{2}{\sqrt{5}}$

(d)

$\frac{\sqrt3}{2}$

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

(a)

2

(b)

3

(c)

1

(d)

none

5. 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

6. 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

7. 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

8. $sin\left\{ 2{ cos }^{ -1 }\left( \cfrac { -3 }{ 5 } \right) \right\} =$

(a)

$\cfrac { 6 }{ 15 }$

(b)

$\cfrac { 24 }{ 25 }$

(c)

$\cfrac { 4 }{ 5 }$

(d)

$\cfrac { -24 }{ 25 }$

9. $cot\left( \cfrac { \pi }{ 4 } -{ cot }^{ -1 }3 \right)$

(a)

7

(b)

6

(c)

5

(d)

none

10. If tan-1(cot$\theta$) = 2$\theta$, then$\theta$ = _____________

(a)

$\pm 3$

(b)

$\pm \cfrac { \pi }{ 4 }$

(c)

$\pm \cfrac { \pi }{ 6 }$

(d)

none

11. The value of sin(2(tan-1 0.75) is___________

(a)

0.75

(b)

1.5

(c)

0.96

(d)

sin-1(1.5)

12. If $\theta ={ sin }^{ -1 }\left( sin(-{ 60 }^{ 0 }) \right)$ then one of the possible values of $\theta$ is _________

(a)

$\cfrac { \pi }{ 3 }$

(b)

$\cfrac { \pi }{ 2 }$

(c)

$\cfrac { 2\pi }{ 3 }$

(d)

$\cfrac { -2\pi }{ 3 }$

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. ${ tan }^{ -1 }\left( tan\cfrac { 9\pi }{ 8 } \right)$

(a)

$\cfrac { 9\pi }{ 8 }$

(b)

$\cfrac { 9\pi }{ 8 }$

(c)

$\cfrac { \pi }{ 8 }$

(d)

$\cfrac { -\pi }{ 8 }$

1. 2 Marks

10 x 2 = 20
16. Find the value of sin-1$\left( sin\frac { 5\pi }{ 9 } cos\frac { \pi }{ 9 } +cos\frac { 5\pi }{ 9 } sin\frac { \pi }{ 9 } \right)$.

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

18. Prove that tan $\left( { sin }^{ -1 }x \right) =\frac { x }{ \sqrt { 1-{ x }^{ 2 } } } for|x|<1$

19. Find the period and amplitude of
y=4sin(−2x)

20. Find the value of
i) ${ sin }^{ -1 }\left( sin\left( \frac { 5\pi }{ 4 } \right) \right)$
ii) ${ sin }^{ -1 }\left( sin\left( \frac { 2\pi }{ 3 } \right) \right)$

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

22. Find the principal value of ${ cos }^{ -1 }\left( \cfrac { -1 }{ 2 } \right)$

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

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

25. Evaluate $sin\left( { cos }^{ -1 }\left( \cfrac { 1 }{ 2 } \right) \right)$

1. 3 Marks

10 x 3 = 30
26. Find the domain of the following
$f\left( x \right) { =sin }^{ -1 }\left( \frac { { x }^{ 2 }+1 }{ 2x } \right)$

27. For what value of x , the inequality$\cfrac { \pi }{ 2 } <{ cos }^{ -1 }(3x-1)<\pi$ holds?

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 x > cos-1x

31. Prove that ${ tan }^{ -1 }x+{ tan }^{ -1 }\frac { 2x }{ 1-{ x }^{ 2 } } ={ tan }^{ -1 }\frac { 3x-{ x }^{ 2 } }{ 1-{ 3x }^{ 2 } } ,|x|<\frac { 1 }{ \sqrt { 3 } }$

32. Find the value of
${ cos }^{ -1 }\left( cos\left( \frac { 4\pi }{ 3 } \right) \right) +{ cos }^{ -1 }\left( cos\left( \frac { 5\pi }{ 4 } \right) \right) .$

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

34. Find the real solutions of the equation
${ tan }^{ -1 }\sqrt { x(x+1) } +{ sin }^{ -1 }\sqrt { { x }^{ 2 }+x+1 } =\cfrac { \pi }{ 2 }$

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 (i) cos-1 $(-\frac{1}{\sqrt2})$
ii) cos-1$(cos(-\frac{\pi}{3}))$
iii) cos-1$(cos(-\frac{7\pi}{6}))$

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

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

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