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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 1-50 of 143

  1. Let $f(x)=\cos\left(2\tan^{-1}\sin\left(\cot^{-1}\sqrt{\frac{1-x}{x}}\right)\right)$, $0<x<1$. Then :
  2. A possible value of $\tan\left(\frac{1}{4}\sin^{-1}\frac{\sqrt{63}}{8}\right)$ is :
  3. The number of solutions of the equation $\sin^{-1}\left[x^{2}+\frac{1}{3}\right]+\cos^{-1}\left[x^{2}-\frac{2}{3}\right]=x^{2}$, for $x\in[-1,1]$, and $[x]$ denotes the …
  4. If $\sin\left(\tan^{-1}(x\sqrt{2})\right)=\cot\left(\sin^{-1}\sqrt{1-x^{2}}\right)$, $x\in(0,1)$, then the value of $x$ is :
  5. The sum of possible values of $x$ for $\tan^{-1}(x+1)+\cot^{-1}\left(\frac{1}{x-1}\right)=\tan^{-1}\left(\frac{8}{31}\right)$ is :
  6. If $0<a,b<1$, and $\tan^{-1}a+\tan^{-1}b=\frac{\pi}{4}$, then the value of …
  7. Given that the inverse trigonometric functions take principal values only. Then, the number of real values of $x$ which satisfy …
  8. If $\cot^{-1}(\alpha)=\cot^{-1}2+\cot^{-1}8+\cot^{-1}18+\cot^{-1}32+\ldots$ upto $100$ terms, then $\alpha$ is :
  9. If $\frac{\sin^{-1}x}{a}=\frac{\cos^{-1}x}{b}=\frac{\tan^{-1}y}{c}$; $0<x<1$, then the value of $\cos\left(\frac{\pi c}{a+b}\right)$ is :
  10. Let $\alpha=3\sin^{-1}\left(\frac{6}{11}\right)$ and $\beta=3\cos^{-1}\left(\frac{4}{9}\right)$, where inverse trigonometric functions take only the principal values. …
  11. $\operatorname{cosec}\left[2\cot^{-1}(5)+\cos^{-1}\left(\frac{4}{5}\right)\right]$ is equal to :
  12. Let $g(x)=f(x)+f(1-x)$ and $f''(x)>0$, $x\in(0,1)$. If $g$ is decreasing in the interval $(0,\alpha)$ and increasing in the interval $(\alpha,1)$, then …
  13. For $n\in\mathbb{N}$, if $\cot^{-1}3+\cot^{-1}4+\cot^{-1}5+\cot^{-1}n=\frac{\pi}{4}$, then $n$ is equal to $\_\_\_\_\_\_$ .
  14. $\tan\left(2\tan^{-1}\frac15+\sec^{-1}\frac{\sqrt5}{2}+2\tan^{-1}\frac18\right)$ is equal to :
  15. If $S=\left\{x\in\mathbb{R}:\sin^{-1}\left(\frac{x+1}{\sqrt{x^{2}+2x+2}}\right)-\sin^{-1}\left(\frac{x}{\sqrt{x^{2}+1}}\right)=\frac{\pi}{4}\right\}$, then …
  16. For $x\in(-1,1]$, the number of solutions of the equation $\sin^{-1}x=2\tan^{-1}x$ is equal to $\_\_\_\_\_\_$ .
  17. Considering the principal values of the inverse trigonometric functions, $\sin^{-1}\left(\frac{\sqrt{3}}{2}x+\frac{1}{2}\sqrt{1-x^{2}}\right)$ …
  18. The domain of the function $\operatorname{cosec}^{-1}\left(\frac{1+x}{x}\right)$ is :
  19. Let $x=\sin(2\tan^{-1}\alpha)$ and $y=\sin\left(\frac12\tan^{-1}\frac43\right)$. If $S=\{\alpha\in\mathbb{R}:y^2=1-x\}$, then $\sum_{\alpha\in S}16\alpha^3$ is equal to …
  20. For $k\in\mathbb{R}$, let the solutions of the equation $\cos\left(\sin^{-1}\left(x\cot\left(\tan^{-1}\left(\cos\left(\sin^{-1}x\right)\right)\right)\right)\right)=k$ …
  21. Let $M$ and $m$ respectively be the maximum and minimum values of the function $f(x)=\tan^{-1}(\sin x+\cos x)$ in $\left[0,\frac{\pi}{2}\right]$. Then the value of …
  22. Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $2\sin^{-1}x+3\cos^{-1}x=\frac{2\pi}{5}$, is $\_\_\_\_\_\_$ .
  23. If $\tan A=\frac{1}{\sqrt{x\left(x^{2}+x+1\right)}}$, $\tan B=\frac{\sqrt{x}}{\sqrt{x^{2}+x+1}}$ and $\tan C=\left(x^{-3}+x^{-2}+x^{-1}\right)^{1/2}$ …
  24. The domain of the function $f(x)=\sin^{-1}\left(\frac{|x|+5}{x^2+1}\right)$ is $(-\infty,-a]\cup[a,\infty)$. Then $a$ is equal to :
  25. The value of $\lim_{n\to\infty}6\tan\left\{\sum_{r=1}^{n}\tan^{-1}\left(\frac{1}{r^{2}+3r+3}\right)\right\}$ is equal to :
  26. The value of $\tan^{-1}\left(\frac{\cos\left(\frac{15\pi}{4}\right)-1}{\sin\left(\frac{\pi}{4}\right)}\right)$ is equal to :
  27. The set of all values of $k$ for which $(\tan^{-1}x)^{3}+(\cot^{-1}x)^{3}=k\pi^{3}$, $x\in\mathbf{R}$, is the interval :
  28. The sum of the infinite series …
  29. If $y=\cos\left(\frac{\pi}{3}+\cos^{-1}\frac{x}{2}\right)$, then $(x-y)^{2}+3y^{2}$ is equal to $\_\_\_\_\_\_$ .
  30. If $a=\sin^{-1}(\sin(5))$ and $b=\cos^{-1}(\cos(5))$, then $a^{2}+b^{2}$ is equal to
  31. Considering only the principal values of the inverse trigonometric functions, the domain of the function $f(x)=\cos^{-1}\left(\frac{x^2-4x+2}{x^2+3}\right)$ is :
  32. Let $f(x)=2\cos^{-1}x+4\cot^{-1}x-3x^{2}-2x+10$, $x\in[-1,1]$. If $[a,b]$ is the range of the function $f$, then $4a-b$ is equal to :
  33. The range of $f(x)=4\sin^{-1}\left(\frac{x^{2}}{x^{2}+1}\right)$ is
  34. Let $f(x)=\left(\sin\left(\tan^{-1}x\right)+\sin\left(\cot^{-1}x\right)\right)^2-1$, $|x|>1$. If $\frac{dy}{dx}=\frac12\frac{d}{dx}\left(\sin^{-1}(f(x))\right)$ and …
  35. The inverse function of $f(x)=\frac{8^{2x}-8^{-2x}}{8^{2x}+8^{-2x}}$, $x\in(-1,1)$, is
  36. If $y=\sum_{k=1}^{6}k\cos^{-1}\left\{\frac35\cos kx-\frac45\sin kx\right\}$, then $\frac{dy}{dx}$ at $x=0$ is ________.
  37. The domain of the function $\cos^{-1}\left(\frac{2\sin^{-1}\left(\frac{1}{4x^{2}-1}\right)}{\pi}\right)$ is :
  38. $50\tan\left(3\tan^{-1}\left(\frac{1}{2}\right)+2\cos^{-1}\left(\frac{1}{\sqrt{5}}\right)\right)+4\sqrt{2}\tan\left(\frac{1}{2}\tan^{-1}(2\sqrt{2})\right)$ is equal to …
  39. If $S$ is the sum of the first $10$ terms of the series …
  40. If $y(x)=\cot^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right)$, $x\in\left(\frac{\pi}{2},\pi\right)$, then $\frac{dy}{dx}$ at …
  41. Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation $\cos^{-1}(x)-2\sin^{-1}(x)=\cos^{-1}(2x)$ is equal …
  42. For $\alpha,\beta,\gamma\neq0$, if $\sin^{-1}\alpha+\sin^{-1}\beta+\sin^{-1}\gamma=\pi$ and $(\alpha+\beta+\gamma)(\alpha-\gamma+\beta)=3\alpha\beta$, then $\gamma$ …
  43. The value of $\cot\left(\sum_{n=1}^{50}\tan^{-1}\left(\frac{1}{1+n+n^{2}}\right)\right)$ is
  44. If $\sum_{r=1}^{50}\tan^{-1}\frac{1}{2r^{2}}=p$, then the value of $\tan p$ is :
  45. Let $S=\left\{\theta\in[0,2\pi]-\left\{\frac\pi2,\frac{3\pi}{2}\right\}:\sin^{-1}(\sin\theta)+\cos^{-1}(\cos\theta)+\tan^{-1}(\tan\theta)=\frac{4\pi}{5}\right\}$. Then …
  46. If $0<x<\frac{1}{\sqrt2}$ and $\frac{\sin^{-1}x}{\alpha}=\frac{\cos^{-1}x}{\beta}$, then a value of $\sin\left(\frac{2\pi\alpha}{\alpha+\beta}\right)$ is
  47. Let $\alpha=\tan\left(\frac{5\pi}{16}\sin\left(2\cos^{-1}\left(\frac{1}{\sqrt{5}}\right)\right)\right)$ and …
  48. The domain of the function $f(x)=\sin^{-1}\left(\frac{x^2-3x+2}{x^2+2x+7}\right)$ is :
  49. The value of …
  50. If the domain of the function $f(x)=\sec^{-1}\left(\frac{2x}{5x+3}\right)$ is $[\alpha,\beta)\cup(\gamma,\delta]$, then $|3\alpha+10(\beta+\gamma)+21\delta|$ is equal to …

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