JEE Maths - Inverse Trigonometric Functions

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Question - 1

If \( a x+b\left(\sec \left(\tan ^{-1} x\right)\right)=c and a y+b\left(\sec \left(\tan ^{-1} y\right)\right)=c, then \frac{x+y}{1-x y}= \) _____________ .

  • A \(\frac{a c}{a^{2}+c^{2}} \)
  • B \(\frac{2 a c}{a-c} \)
  • C \(\frac{2 a c}{a^{2}-c^{2}} \)
  • D \(\frac{a+c}{1-a c}\)

Question - 2

The range of the function \(f(x)=\sin ^{-1}\left[x^{2}+\frac{1}{2}\right]+\cos ^{-1}\left[x^{2}-\frac{1}{2}\right], where [.] is the greatest integer function, is \) ____________ .

  • A \(\left\{\frac{\pi}{2}, \pi\right\} \)
  • B \(\left\{0,-\frac{1}{2}\right\} \)
  • C \(\{\pi\} \)
  • D \(\left(0, \frac{\pi}{2}\right)\)

Question - 3

If \(\sum_{r=1}^{k} \cos ^{-1} \beta_{r}=\frac{k \pi}{2} for any k \geq 1 where $\beta_{r} \geq 0 . \forall r and A=\sum_{r=1}^{k}\left(\beta_{r}\right)^{r}. Then \lim _{x \rightarrow A} \frac{\left(1+x^{2}\right)^{1 / 3}-(1-2 x)^{1 / 4}}{x+x^{2}}=\) _____________ .

  • A \(\frac{1}{2} \)
  • B \(0 \)
  • C \(\frac{A}{2} \)
  • D \(\frac{\pi}{2}\)

Question - 4

The range of the function \(f(x)=\sin ^{-1}(\log [x])+\log \left(\sin ^{-1}[x]\right) ; \) (where [.] denotes the greatest integer function) is ___________ .

  • A \(\mathrm{R} \)
  • B \([1,2) \)
  • C \(\left\{\log \frac{\pi}{2}\right\} \)
  • D \(\{-\sin 1\}\)

Question - 5

If \( x_{1}, x_{2}, x_{3}, x_{4} \) are roots of the equation  \(x^{4}-x^{3} \sin 2 \beta+x^{2} \cos 2 \beta-x \cos \beta-\sin \beta=0, then \sum_{i=1}^{4} \tan ^{-1} x_{i}\)  is equal to _____________ .

  • A \(\pi-\beta \)
  • B \(\pi-2 \beta \)
  • C \(\frac{\pi}{2}-\beta \)
  • D \(\frac{\pi}{2}-2 \beta\)

Question - 6

The maximum value of f(x) \(=\tan ^{-1}\left(\frac{(\sqrt{12}-2) x^{2}}{x^{4}+2 x^{2}+3}\right) is \) ____________ .

  • A \(18^{\circ} \)
  • B \(36^{\circ} \)
  • C \(22.5^{\circ} \)
  • D \(15^{\circ}\)

Question - 7

The number of solutions of the equation \(\left|\tan ^{-1}\right| x||=\sqrt{\left(x^{2}+1\right)^{2}-4 x^{2}} \text { is }\) _____________ . 

  • A 2
  • B 3
  • C 4
  • D none of these

Question - 8

\(S=\tan ^{-1}\left(\frac{1}{n^{2}+n+1}\right)+\tan ^{-1}\left(\frac{1}{n^{2}+3 n+3}\right)+\ldots+\tan ^{-1}\left(\frac{1}{1+(n+19)(n+20)}\right), then \tan S^{3} s equal to : \)

  • A \(\frac{20}{401+20 n} \)
  • B \(\frac{n}{n^{2}+20 n+1} \)
  • C \(\frac{20}{n^{2}+20 n+1} \)
  • D \(\frac{n}{401+20 n}\)

Question - 9

The number of roots of the equation \(\sin ^{-1} x-\frac{1}{\sin ^{-1} x}=\cos ^{-1} x-\frac{1}{\cos ^{-1} x} \text { is }\)______________ .

  • A 0
  • B 1
  • C 2
  • D 3

Question - 10

The minimum integral value of α for which the quadratic equation \(\left(\cot ^{-1} \alpha\right) x^{2}-\left(\tan ^{-1} \alpha\right)^{3 / 2} x+2\left(\cot ^{-1} \alpha\right)^{2}=0\) has both positive roots _____________ .

  • A 1
  • B 2
  • C 3
  • D 4
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