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Inverse Trigonometric Functions Model Question Paper 1

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
    5 x 1 = 5
  1. \(\sin ^{-1}(\cos x)=\frac{\pi}{2}-x\) is valid for

    (a)

    \(-\pi \le x\le 0\)

    (b)

    \(0 \le x\le \pi\)

    (c)

    \(-\frac { \pi }{ 2 } \le x\le \frac { \pi }{ 2 } \)

    (d)

    \(-\frac { \pi }{ 4 } \le x\le \frac { 3\pi }{ 4 } \)

  2. If \(\cot ^{-1} x=\frac{2 \pi}{5}\) 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)

    \(-\frac{\pi}{5}\)

  3. The number of solutions of the equation \({ tan }^{ -1 }2x+{ tan }^{ -1 }3x=\frac { \pi }{ 4 } \) ____________

    (a)

    2

    (b)

    3

    (c)

    1

    (d)

    none

  4. \({ tan }^{ -1 }\left( \frac { 1 }{ 4 } \right) +{ tan }^{ -1 }\left( \frac { 2 }{ 11 } \right) \) = ____________

    (a)

    0

    (b)

    \(\frac { 1 }{ 2 } \)

    (c)

    -1

    (d)

    none

  5. If \({ cos }^{ -1 }x>x>{ sin }^{ -1 }x\) then _________

    (a)

    \(\cfrac { 1 }{ \sqrt { 2 } }

    (b)

    \(0\le x<\frac { 1 }{ \sqrt { 2 } } \)

    (c)

    \(-1\le x<\frac { 1 }{ \sqrt { 2 } } \)

    (d)

    x>0

  6. 5 x 2 = 10
  7. For what value of x does sinx = sin−1x?

  8. Find the value of sin-1\(\left( sin\frac { 5\pi }{ 9 } cos\frac { \pi }{ 9 } +cos\frac { 5\pi }{ 9 } sin\frac { \pi }{ 9 } \right) \).

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

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

  11. Evaluate \(sin\left( { cos }^{ -1 }\left( \frac { 3 }{ 5 } \right) \right) \)

  12. 5 x 3 = 15
  13. For what value of x, the inequality \(\frac { \pi }{ 2 } <{ cos }^{ -1 }(3x-1)<\pi \) holds?

  14. Find the value of
     \(cos\left( { cos }^{ -1 }\left( \frac { 4 }{ 5 } \right) +{ sin }^{ -1 }\left( \frac { 4 }{ 5 } \right) \right) \)

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

  16. Solve \({ tan }^{ -1 }\left( \frac { 2x }{ 1-{ x }^{ 2 } } \right) +{ cot }^{ -1 }\left( \frac { 1-{ x }^{ 2 } }{ 2x } \right) =\frac { \pi }{ 3 } ,x>0\)

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

  18. 4 x 5 = 20
  19. Solve \(tan^{ -1 }\left( \frac { x-1 }{ x-2 } \right) +tan^{ -1 }\left( \frac { x+1 }{ x+2 } \right) =\frac { \pi }{ 4 } \)

  20. Solve \(cos\left( sin^{ -1 }\left( \frac { x }{ \sqrt { 1+{ x }^{ 2 } } } \right) \right) =sin\left\{ cot^{ -1 }\left( \frac { 3 }{ 4 } \right) \right\} \)

  21. If \({ tan }^{ -1 }\left( \frac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right) =a\) than prove that x= sin 2a

  22. Prove that \({ tan }^{ -1 }\left( \frac { 1-x }{ 1+x } \right) -{ tan }^{ -1 }\left( \frac { 1-y }{ 1+y } \right) ={ sin }^{ -1 }\left( \frac { y-x }{ \sqrt { 1+{ x }^{ 2 } } .\sqrt { 1+{ y }^{ 2 } } } \right)\)

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