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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 51-100 of 143

  1. If $\left(\sin^{-1}x\right)^{2}-\left(\cos^{-1}x\right)^{2}=a$; $0<x<1$, $a\neq0$, then the value of $2x^{2}-1$ is :
  2. $2\pi-\left(\sin^{-1}\frac45+\sin^{-1}\frac5{13}+\sin^{-1}\frac{16}{65}\right)$ is equal to :
  3. $\lim_{n\to\infty}\tan\left\{\sum_{r=1}^{n}\tan^{-1}\left(\frac{1}{1+r+r^{2}}\right)\right\}$ is equal to $\_\_\_\_\_\_$ .
  4. If the inverse trignometric functions take principal values, then …
  5. $\sin^{-1}\left(\sin\frac{2\pi}{3}\right)+\cos^{-1}\left(\cos\frac{7\pi}{6}\right)+\tan^{-1}\left(\tan\frac{3\pi}{4}\right)$ is equal to :
  6. The number of real roots of the equation $\tan^{-1}\sqrt{x(x+1)}+\sin^{-1}\sqrt{x^{2}+x+1}=\frac\pi4$ is :
  7. $\cos^{-1}(\cos(-5))+\sin^{-1}(\sin(6))-\tan^{-1}(\tan(12))$ is equal to : (The inverse trigonometric functions take the principal values)
  8. The value of $\tan^{-1}\left[\frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}\right]$, $|x|<\frac12$, $x\neq0$, is equal to :
  9. If $x,y,z$ are in A.P. and $\tan^{-1}x$, $\tan^{-1}y$ and $\tan^{-1}z$ are also in A.P., then :
  10. Let $x\in(0,1)$. The set of all $x$ such that $\sin^{-1}x>\cos^{-1}x$, is the interval :
  11. The domain of the function $f(x)=\frac{\sin^{-1}(x-3)}{\sqrt{9-x^{2}}}$ is
  12. The value of $\cot\left(\sum_{n=1}^{19}\cot^{-1}\left(1+\sum_{p=1}^{n}2p\right)\right)$ is :
  13. The domain of $\sin^{-1}\left[\log_{3}\left(\frac x3\right)\right]$ is
  14. A value of $x$ satisfying the equation $\sin\left[\cot^{-1}(1+x)\right]=\cos\left[\tan^{-1}x\right]$, is :
  15. All $x$ satisfying the inequality $\left(\cot^{-1}x\right)^{2}-7\left(\cot^{-1}x\right)+10>0$, lie in the interval :
  16. Let $f:(-1,1)\to B$, be a function defined by $f(x)=\tan^{-1}\frac{2x}{1-x^{2}}$, then $f$ is both one-one and onto when $B$ is the interval
  17. $\cot^{-1}\left(\sqrt{\cos\alpha}\right)-\tan^{-1}\left(\sqrt{\cos\alpha}\right)=x$, then $\sin x$ is equal to :
  18. If $\cos^{-1}x-\cos^{-1}\frac y2=\alpha$, then $4x^{2}-4xy\cos\alpha+y^{2}$ is equal to :
  19. A value of $x$ for which $\sin\left(\cot^{-1}(1+x)\right)=\cos\left(\tan^{-1}x\right)$, is :
  20. If $f(x)=2\tan^{-1}x+\sin^{-1}\left(\frac{2x}{1+x^2}\right)$, $x>1$, then $f(5)$ is equal to
  21. If $\alpha=\cos^{-1}\frac35$, $\beta=\tan^{-1}\frac13$, where $0<\alpha,\beta<\frac\pi2$, then $\alpha-\beta$ is equal to
  22. Let $\tan^{-1}y=\tan^{-1}x+\tan^{-1}\left(\frac{2x}{1-x^{2}}\right)$, where $|x|<\frac{1}{\sqrt3}$. Then a value of $y$ is :
  23. If $\sin^{-1}\left(\frac x5\right)+\operatorname{cosec}^{-1}\left(\frac54\right)=\frac\pi2$, then the value of $x$ is :
  24. The principal value of $\tan^{-1}\left(\cot\frac{43\pi}{4}\right)$ is
  25. The trigonometric equation $\sin^{-1}x=2\sin^{-1}a$ has a solution for :
  26. $S=\tan^{-1}\left(\frac{1}{n^{2}+n+1}\right)+\tan^{-1}\left(\frac{1}{n^{2}+3n+3}\right)+\ldots+\tan^{-1}\left(\frac{1}{1+(n+19)(n+20)}\right)$, then $\tan S$ is equal to …
  27. Considering only the principal values of the inverse trigonometric functions, the value of …
  28. If the sum of all the solutions of $\tan^{-1}\left(\frac{2x}{1-x^{2}}\right)+\cot^{-1}\left(\frac{1-x^{2}}{2x}\right)=\frac\pi3,\ -1<x<1,\ x\neq0$, is …
  29. Let $\tan^{-1}(x)\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$, for $x\in\mathbb{R}$. Then the number of real solutions of the equation …
  30. Considering only the principal values of the inverse trigonometric functions, the value of …
  31. If for some $\alpha,\beta$; $\alpha\le\beta$, $\alpha+\beta=8$ and $\sec^{2}\left(\tan^{-1}\alpha\right)+\operatorname{cosec}^{2}\left(\cot^{-1}\beta\right)=36$, then …
  32. Let $x=\frac mn$ ($m,n$ are co-prime natural numbers) be a solution of the equation $\cos\left(2\sin^{-1}x\right)=\frac19$ and let $\alpha,\beta$ ($\alpha>\beta$) be the …
  33. The total number of real solutions of the equation $$\theta=\tan^{-1}(2\tan\theta)-\frac{1}{2}\sin^{-1}\left(\frac{6\tan\theta}{9+\tan^2\theta}\right)$$ is (Here, the …
  34. The value of $\tan\left[\cos^{-1}\left(\dfrac45\right)+\tan^{-1}\left(\dfrac23\right)\right]$ is
  35. Let $S=\left\{x:\cos^{-1}x=\pi+\sin^{-1}x+\sin^{-1}(2x+1)\right\}$. Then $\sum_{x\in S}(2x-1)^2$ is equal to ______.
  36. The derivative of $\tan^{-1}\left(\frac{\sin x-\cos x}{\sin x+\cos x}\right)$ with respect to $\frac x2$, where $x\in\left(0,\frac\pi2\right)$, is
  37. Considering only the principal values of inverse functions, the set $A=\left\{x\ge0:\tan^{-1}(2x)+\tan^{-1}(3x)=\frac\pi4\right\}$
  38. Let $y=f(x)$ represent a parabola with focus $\left(-\frac12,0\right)$ and directrix $y=-\frac12$. Then …
  39. For any $y\in\mathbb{R}$, let $\cot^{-1}(y)\in(0,\pi)$ and $\tan^{-1}(y)\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. Then the sum of all the solutions of the equation …
  40. The value of $$\sec^{-1}\left(\frac14\sum_{k=0}^{10}\sec\left(\frac{7\pi}{12}+\frac{k\pi}{2}\right)\sec\left(\frac{7\pi}{12}+\frac{(k+1)\pi}{2}\right)\right)$$ in the …
  41. Let $S$ be the set of all solutions of the equation $\cos^{-1}2x-2\cos^{-1}\sqrt{1-x^2}=\pi$, $x\in\left[-\frac12,\frac12\right]$. Then …
  42. Find the value of $\cos\left(2\cos^{-1}x+\sin^{-1}x\right)$ at $x=\dfrac15$, where $0\le\cos^{-1}x\le\pi$ and $-\pi/2\le\sin^{-1}x\le\pi/2$.
  43. Considering only the principal values of the inverse trigonometric functions, the value of …
  44. If $\frac{\pi}{4}+\sum_{p=1}^{11}\tan^{-1}\left(\frac{2^{p-1}}{1+2^{2p-1}}\right)=\alpha$, then $\tan\alpha$ is equal to ________.
  45. Let $a,b,c$ be positive real numbers. Let $$\theta=\tan^{-1}\sqrt{\frac{a(a+b+c)}{bc}}+\tan^{-1}\sqrt{\frac{b(a+b+c)}{ca}}+\tan^{-1}\sqrt{\frac{c(a+b+c)}{ab}}$$ Then …
  46. If $x=\sin^{-1}(\sin10)$ and $y=\cos^{-1}(\cos10)$, then $y-x$ is equal to :
  47. If $\sin^{-1}\frac{\alpha}{17}+\cos^{-1}\frac45-\tan^{-1}\frac{77}{36}=0$, $0<\alpha<13$, then $\sin^{-1}(\sin\alpha)+\cos^{-1}(\cos\alpha)$ is equal to
  48. The value of $\sin^{-1}\frac{12}{13}-\sin^{-1}\frac35$ is equal to:
  49. Let $(a,b)\subset(0,2\pi)$ be the largest interval for which $\sin^{-1}(\sin\theta)-\cos^{-1}(\sin\theta)>0$, $\theta\in(0,2\pi)$, holds. If …
  50. If $\cos^{-1}\frac2{3x}+\cos^{-1}\frac3{4x}=\frac\pi2$ $\left(x>\frac34\right)$, then $x$ is equal to :

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