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12th Standard Maths English Medium Inverse Trigonometric Functions Reduced Syllabus Important Questions With Answer Key 2021

12th Standard

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

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