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Yukti Vault

Trigonometry

Inverse Trigonometric Functions: JEE Main & Advanced PYQs

143 previous year questions on Inverse Trigonometric Functions from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 101-143 of 143

  1. $\tan^{-1}\frac{1+\sqrt3}{3+\sqrt3}+\sec^{-1}\sqrt{\frac{8+4\sqrt3}{6+3\sqrt3}}=$
  2. Let $S=\left\{x\in\mathbb R:0<x<1\text{ and }2\tan^{-1}\left(\frac{1-x}{1+x}\right)=\cos^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)\right\}$. If $n(S)$ denotes the number …
  3. Considering only the principal values of inverse trigonometric functions, the number of positive real values of $x$ satisfying $\tan^{-1}(x)+\tan^{-1}(2x)=\frac\pi4$ is :
  4. If $\cos^{-1}x-\cos^{-1}\frac y2=\alpha$, where $-1\le x\le1$, $-2\le y\le2$, $x\le\frac y2$, then for all $x,y$, $4x^{2}-4xy\cos\alpha+y^{2}$ is equal to :
  5. For non-negative integers $n$, let …
  6. If we consider only the principal values of the inverse trigonometric functions, then the value of …
  7. The principal value of $\sin^{-1}\left(\sin\dfrac{2\pi}{3}\right)$ is
  8. The numerical value of $\tan\left\{2\tan^{-1}\left(\dfrac15\right)-\dfrac{\pi}{4}\right\}$ is equal to ______.
  9. The greater of the two angles $A=2\tan^{-1}\left(2\sqrt2-1\right)$ and $B=3\sin^{-1}(1/3)+\sin^{-1}(3/5)$ is ______.
  10. If $\sin^{-1}\left(x-\dfrac{x^{2}}{2}+\dfrac{x^{3}}{4}-\cdots\right)+\cos^{-1}\left(x^{2}-\dfrac{x^{4}}{2}+\dfrac{x^{6}}{4}-\cdots\right)=\dfrac\pi2$ for $0<|x|<\sqrt2$ …
  11. The number of real solutions of $\tan^{-1}\sqrt{x(x+1)}+\sin^{-1}\sqrt{x^{2}+x+1}=\pi/2$ is
  12. The value of $x$ for which $\sin\left(\cot^{-1}(1+x)\right)=\cos\left(\tan^{-1}x\right)$ is
  13. Let the maximum value of $\left(\sin^{-1}x\right)^2+\left(\cos^{-1}x\right)^2$ for $x\in\left[-\frac{\sqrt3}{2},\frac{1}{\sqrt2}\right]$ be $\frac mn\pi^2$, where …
  14. The number of solutions of $\tan^{-1}4x+\tan^{-1}6x=\frac\pi6$, where $-\frac{1}{2\sqrt6}<x<\frac{1}{2\sqrt6}$, is equal to :
  15. Considering the principal values of inverse trigonometric functions, the value of the expression …
  16. If $k=\tan\left(\frac\pi4+\frac12\cos^{-1}\left(\frac23\right)\right)+\tan\left(\frac12\sin^{-1}\left(\frac23\right)\right)$, then the number of solutions of the …
  17. Let $0<\alpha<1$, $\beta=\frac{1}{3\alpha}$ and $\tan^{-1}(1-\alpha)+\tan^{-1}(1-\beta)=\frac{\pi}{4}$. Then $6(\alpha+\beta)$ is equal to :
  18. If the domain of the function $f(x)=\cos^{-1}\left(\frac{2x-5}{11-3x}\right)+\sin^{-1}\left(2x^2-3x+1\right)$ is the interval $[\alpha,\beta]$, then $\alpha+2\beta$ is …
  19. Statement I: The equation $(\sin^{-1}x)^{3}+(\cos^{-1}x)^{3}-a\pi^{3}=0$ has a solution for all $a\ge\frac1{32}$. Statement II: For any $x\in R$ …
  20. If $0<x<1$, then $\sqrt{1+x^2}\left[\{x\cos(\cot^{-1}x)+\sin(\cot^{-1}x)\}^2-1\right]^{\frac12}=$
  21. Let the function $g:(-\infty,\infty)\to\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ be given by $g(u)=2\tan^{-1}(e^{u})-\frac{\pi}{2}$. Then, $g$ is
  22. Let $f(\theta)=\sin\left(\tan^{-1}\left(\frac{\sin\theta}{\sqrt{\cos2\theta}}\right)\right)$, where $-\frac\pi4<\theta<\frac\pi4$. Then the value of …
  23. Match List I with List II and select the correct answer using the code given below the lists: List I: P. …
  24. Let $f:[0,4\pi]\to[0,\pi]$ be defined by $f(x)=\cos^{-1}(\cos x)$. The number of points $x\in[0,4\pi]$ satisfying the equation $$f(x)=\frac{10-x}{10}$$ Is
  25. For any positive integer $n$, let $S_n:(0,\infty)\to\mathbb{R}$ be defined by $$S_n(x)=\sum_{k=1}^{n}\cot^{-1}\left(\frac{1+k(k+1)x^2}{x}\right),$$ where for any …
  26. The value of $\cot\left(\sum_{n=1}^{23}\cot^{-1}\left(1+\sum_{k=1}^{n}2k\right)\right)$ is
  27. If $\alpha=3\sin^{-1}\left(\frac{6}{11}\right)$ and $\beta=3\cos^{-1}\left(\frac{4}{9}\right)$, where the inverse trigonometric functions take only the principal values …
  28. The number of real solutions of the equation …
  29. For any positive integer $n$, define $f_n:(0,\infty)\to\mathbb{R}$ as $$f_n(x)=\sum_{j=1}^{n}\tan^{-1}\left(\frac{1}{1+(x+j)(x+j-1)}\right)\text{ for all …
  30. $\cos\left(\sin^{-1}\frac{3}{5}+\sin^{-1}\frac{5}{13}+\sin^{-1}\frac{33}{65}\right)$ is equal to :
  31. The value of $\cot\left(\operatorname{cosec}^{-1}\frac53+\tan^{-1}\frac23\right)$ is :
  32. Match the following: (i) $\displaystyle\sum_{i=1}^{\infty}\tan^{-1}\left(\dfrac{1}{2i^2}\right)=t$ then $\tan t=$ (ii) Sides $a,b,c$ of a triangle $ABC$ are in AP and …
  33. Let $(x,y)$ be such that $\sin^{-1}(ax)+\cos^{-1}(y)+\cos^{-1}(bxy)=\frac{\pi}{2}$. Match the statements in Column I with statements in Column II and indicate your …
  34. Using mathematical induction, prove that $$\tan^{-1}(1/3)+\tan^{-1}(1/7)+\cdots+\tan^{-1}\left\{1/\left(n^{2}+n+1\right)\right\}=\tan^{-1}\left\{n/(n+2)\right\}$$
  35. Prove that $\cos\tan^{-1}\sin\cot^{-1}x=\sqrt{\dfrac{x^{2}+1}{x^{2}+2}}$.
  36. Let $S_k=\displaystyle\sum_{r=1}^{k}\tan^{-1}\left(\dfrac{6^{r}}{2^{2r+1}+3^{2r+1}}\right)$. Then $\displaystyle\lim_{k\to\infty}S_k$ is equal to :
  37. Given that the inverse trigonometric function assumes principal values only. Let $x,y$ be any two real numbers in $[-1,1]$ such that $\cos^{-1}x-\sin^{-1}y=\alpha$ …
  38. Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of …
  39. If $\dfrac{\pi}{2}\le x\le\dfrac{3\pi}{4}$, then $\cos^{-1}\left(\dfrac{12}{13}\cos x+\dfrac{5}{13}\sin x\right)$ is equal to
  40. If $\alpha>\beta>\gamma>0$, then the expression …
  41. Let $[x]$ denote the greatest integer less than or equal to $x$. Then the domain of $f(x)=\sec^{-1}(2[x]+1)$ is:
  42. The number of solutions of the equation, $\sin^{-1}x=2\tan^{-1}x$ (in principal values) is :
  43. The value of $\tan\left(2\tan^{-1}\left(\dfrac{3}{5}\right)+\sin^{-1}\left(\dfrac{5}{13}\right)\right)$ is equal to :

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