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

12th Standard

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

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