Inverse Trigonometric Functions Important Questions

12th Standard EM

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Maths

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

    \(\pi\)

  3. \({ sin }^{ -1 }\frac { 3 }{ 5 } -{ cos }^{ -1 }\frac { 12 }{ 13 } +{ sec }^{ -1 }\frac { 5 }{ 3 } { -cosec }^{ 1- }\frac { 13 }{ 2 } \)is equal to

    (a)

    2\(\pi\)

    (b)

    \(\pi\)

    (c)

    0

    (d)

    tan-1\(\frac{12}{65}\)

  4. 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

  5. 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

  6. 5 x 1 = 5
  7. Amplitude of sine function

  8. (1)

    \(\pi -{ cos }^{ -1 }x\)

  9. sin-1(3x-4x3)

  10. (2)

    cosec-1x

  11. cos-1(4x3-3x)

  12. (3)

    3cos-1x

  13. \(sin^{ -1 }\left( \cfrac { 1 }{ x } \right) \)

  14. (4)

    1

  15. cos-1(-x)

  16. (5)

    3sin-1x

    5 x 2 = 10
  17. Find the period and amplitude of
    y=sin 7x

  18. Sketch the graph of y= sin\((\frac{1}{3}x)\)for 0\(\le x <6\pi\).

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

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

  21. 5 x 3 = 15
  22. Find the domain of cos-1\((\frac{2+sinx}{3})\)

  23. Find the value of
    \(tan\left( { cos }^{ -1 }\left( \frac { 1 }{ 2 } \right) -{ sin }^{ -1 }\left( -\frac { 1 }{ 2 } \right) \right) \)

  24. Evaluate \(sin\left[ { sin }^{ -1 }\left( \frac { 3 }{ 5 } \right) +{ sec }^{ -1 }\left( \frac { 5 }{ 4 } \right) \right] \)

  25. Prove that
     tan-1\(\frac{1}{2}+tan^{-1}\frac{1}{3}=\frac{\pi}{4}\)

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

  27. 3 x 5 = 15
  28. If a1, a2, a3, ... an is an arithmetic progression with common difference d, prove that tan \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\quad \left[ tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 1 }{ a }_{ 2 } } \right) +tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 2 }{ a }_{ 3 } } \right) +....tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ n }{ a }_{ n-1 } } \right) \right] =\frac { { a }_{ n }-{ a }_{ 1 } }{ 1+{ a }_{ 1 }{ a }_{ n } } \)

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

  30. Simplify \({ sin }^{ -1 }\left( \cfrac { sinx+cosx }{ \sqrt { 2 } } \right) ,\cfrac { \pi }{ 4 } <x<\cfrac { \pi }{ 4 } \)
     

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