Mathematics - Inverse Circular Functions

Question - 1

\(x\epsilon\left({3\pi\over 2}, 2\pi\right)\)then the value of the expression sin-1 [cos {cos-1 (cos x)} + sin-1 (sin x)], is

  • A \(-{\pi\over 2}\)
  • B \({\pi\over 2}\)
  • C 0
  • D \(\pi\)

Question - 2

The sum of the infinite terms of the series \(tan^{-1}\left(1\over 3\right)+tan^{-1}\left(2\over 9\right)+tan^{-1}\left(4\over 33\right)+......\) is equal to

  • A \(\pi\over 6\)
  • B \(\pi\over 4\)
  • C \(\pi\over 3\)
  • D \(\pi\over 2\)

Question - 3

If \(sin^{-1}\sqrt{(x^2+2x+1)}+sec^{c-1}\sqrt{(x^2+2x+1)}={\pi\over 2}\)x ≠ 0, then the value of \(2sec^{-1}\left(x\over 2\right)+sin^{-1}\left(x\over 2\right)\) is equal to

  • A \(-{3\pi\over 2}\)
  • B \({3\pi\over 2}\)
  • C \(-{\pi\over 2}\)
  • D \({\pi\over 2}\)

Question - 4

The greatest of tan1, tan-1 1, sin-11, sin1, cos 1, is

  • A sin 1
  • B tan 1
  • C tan-1 1
  • D none of these

Question - 5

The value of 'a' for which a x2 + sin-1 (x2 - 2x + 2) + cos-1 (x2 - 2x + 2) = 0 has a real solution, is

  • A \(-{2\over\pi}\)
  • B \({2\over\pi}\)
  • C \(-{\pi\over2}\)
  • D \({\pi\over2}\)

Question - 6

If \(sin^{-1}x+sin^{-1}y+sin^{-1}z={3\pi\over 2}\) and f(1)=2 f(p + q) = f(p)· f(q), ∀ p, q ∈ R, \(X^{f(1)} + y^{f(2)} + z^{f(3)}-{(x+y+z)\over X^{f(1)} + y^{f(2)} + z^{f(3)}}\)

  • A 0
  • B 1
  • C 2
  • D 3

Question - 7

If the mapping f(x)=ax+b,a>0 maps [-1,1] onto [0, 2], then cot [cot-1 7 + cot-1 8 + cot-1 18] is equal to

  • A f( - 1)
  • B f(0)
  • C f(1)
  • D f(2)

Question - 8

The sum of the infinite series \(sin^{-1}\left(1\over \sqrt2\right)+sin^{-1}\left(\sqrt2-1\over \sqrt6\right)+sin^{-1}\left(\sqrt3-\sqrt2\over \sqrt{12}\right)+.......+........+sin^{-1}\left(\sqrt n-\sqrt{(n-1)}\over \sqrt\{n(n+1)\}\right)+.........\)is

  • A \(\pi\over 8\)
  • B \(\pi\over 4\)
  • C \(\pi\over 2\)
  • D \(\pi\)

Question - 9

If [sin-1 cos-1 sin-1 tan-1 x] = 1, where [.] denotes the greatest integer function, then x belongs to the interval

  • A [tan sin cos 1, tan sin cos sin 1]
  • B (tan sin cos 1, tan sin cos sin 1)
  • C [- 1, 1]
  • D [sin cos tan 1, sin cos sin tan 1]

Question - 10

If [cot-1x]+ [cos-1 x] = 0, where x is a non-negative real number and [.] denotes the greatest integer function, then complete set of values of x is

  • A (cos 1,1]
  • B (cot 1,1)
  • C (cos 1, cot 1)
  • D none of these
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