JEE Trigonometry PYQs with Solutions PDF, Download Now

Let the maximum value of $$\left(\sin^{-1}x\right)^2+\left(\cos^{-1}x\right)^2$$ for $$x\epsilon \left[-\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}\right]$$ be $$\frac{m}{n}\pi^{2}$$, where gcd
(m, n) = l. Then m + n is equal to ____________

If the domain of the function $$f(x)=\cos^{-1}\left(\frac{2x-5}{11-3x}\right)+\sin^{-1}(2x^{2}-3x+1)$$ is the interval $$[\alpha, \beta]$$, then $$\alpha+2\beta$$ is equal to:

The number of solutions of $$ \tan^{-1}4x + \tan^{-1}6x = \frac{\pi}{6} $$, where $$ -\frac{1}{2\sqrt{6}}<x<\frac{1}{2\sqrt{6}}, $$ is equal to

If the domain of the function f(x) = $$\sin^{-1}\frac{1}{x^{2}-2x-2}$$, is $$\left[-\infty, \alpha\right] \cup \left[\beta,\gamma\right]\cup \left[\delta,\infty\right],$$ then $$\alpha+\beta+\gamma+\delta$$ is equal to

If $$K=\tan\left(\frac{\pi}{4}+\frac{1}{2}\cos^{-1}\left(\frac{2}{3}\right)\right)+\tan\left(\frac{1}{2}\sin^{-1}\left(\frac{2}{3}\right)\right)$$, then the number of solutions of the equation $$\sin^{-1}(kx-1)=\sin^{-1} x-\cos^{-1} x$$ is______.

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $$16\left(\left(\sec^{-1}x\right)^{2}\left(\cosec^{-1}x\right)^{2}\right) \text{is :} $$

$$ \text{If for some } \alpha,\beta;\; \alpha\le\beta,\; \alpha+\beta=8$$ and  $$\sec^2(\tan^{-1}\alpha)+\cosec^2(\cot^{-1}\beta)=36,$$ $$\alpha^2+\beta^2$$  is:_______

$$\text{If } \alpha > \beta > \gamma > 0,\text{ then the expression}\cot^{-1}\!\left\{\beta+\frac{(1+\beta^2)}{(\alpha-\beta)}\right\} + \cot^{-1}\!\left\{\gamma+\frac{(1+\gamma^2)}{(\beta-\gamma)}\right\} + \cot^{-1}\!\left\{\alpha+\frac{(1+\alpha^2)}{(\gamma-\alpha)}\right\}\text{ is equal to:}$$

$$\cos \left(\sin^{-1}\frac{3}{5}+\sin^{-1}\frac{5}{13}+\sin^{-1}\frac{33}{65}\right)$$ is equal to:

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_________.

If $$\frac{\pi}{2}\leq x\leq \frac{3\pi}{4}$$, then $$\cos^{-1}\left(\frac{12}{13}\cos x+\frac{5}{13}\sin x\right)$$ is equal to

Let $$[x]$$ denote the greatest integer less than or equal to $$x$$. Then the domain of $$f(x)=sec^{-1}(2[x]+1)$$ is:

Let $$\{x\}$$ denote the fractional part of $$x$$ and $$f(x) = \frac{\cos^{-1}(1-\{x\}^2)\sin^{-1}(1-\{x\})}{\{x\} - \{x\}^3}$$, $$x \neq 0$$. If $$L$$ and $$R$$ respectively denotes the left hand limit and the right hand limit of $$f(x)$$ at $$x = 0$$, then $$\frac{32}{\pi^2}(L^2 + R^2)$$ is equal to:

Considering only the principal values of inverse trigonometric functions, the number of positive real values of $$x$$ satisfying $$\tan^{-1}(x) + \tan^{-1}(2x) = \frac{\pi}{4}$$ is :

For $$\alpha, \beta, \gamma \neq 0$$. If $$\sin^{-1}\alpha + \sin^{-1}\beta + \sin^{-1}\gamma = \pi$$ and $$(\alpha + \beta + \gamma)(\alpha - \gamma + \beta) = 3\alpha\beta$$, then $$\gamma$$ equal to

If $$a = \sin^{-1}(\sin 5)$$ and $$b = \cos^{-1}(\cos 5)$$, then $$a^2 + b^2$$ is equal to

Given that the inverse trigonometric function assumes principal values only. Let $$x, y$$ be any two real numbers in $$[-1, 1]$$ such that $$\cos^{-1} x - \sin^{-1} y = \alpha$$, $$\frac{-\pi}{2} \leq \alpha \leq \pi$$. Then, the minimum value of $$x^2 + y^2 + 2xy \sin \alpha$$ is

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 _____

If the domain of the function $$f(x) = \sin^{-1}\left(\frac{x-1}{2x+3}\right)$$ is $$\mathbb{R} - (\alpha, \beta)$$, then $$12\alpha\beta$$ is equal to :

Let $$\lim_{n \to \infty} \left(\frac{n}{\sqrt{n^4+1}} - \frac{2n}{(n^2+1)\sqrt{n^4+1}} + \frac{n}{\sqrt{n^4+16}} - \frac{8n}{(n^2+4)\sqrt{n^4+16}} + \ldots + \frac{n}{\sqrt{n^4+n^4}} - \frac{2n \cdot n^2}{(n^2+n^2)\sqrt{n^4+n^4}}\right)$$ be $$\frac{\pi}{k}$$, using only the principal values of the inverse trigonometric functions. Then $$k^2$$ is equal to ________

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 ________

Let $$S$$ be the set of all solutions of the equation $$\cos^{-1}(2x) - 2\cos^{-1}(\sqrt{1-x^2}) = \pi, x \in \left[-\frac{1}{2}, \frac{1}{2}\right]$$. Then $$\sum_{x \in S} \left(2\sin^{-1}(x^2) - 1\right)$$ is equal to

Let $$f(x) = 2x + \tan^{-1}(x)$$ and $$g(x) = \log_e(\sqrt{1+x^2} + x), \quad x \in [0, 3]$$. Then

Let $$S = \left\{x \in R : 0 \lt x \lt 1 \text{ and } 2\tan^{-1}\left(\frac{1-x}{1+x}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right\}$$. If $$n(S)$$ denotes the number of elements in $$S$$ then :

$$\tan^{-1}\frac{1+\sqrt{3}}{3+\sqrt{3}} + \sec^{-1}\sqrt{\frac{8+4\sqrt{3}}{6+3\sqrt{3}}} =$$

The value of $$\operatorname{cosec}10°-\sqrt{3}\sec10°$$ is equal to :

Let $$\cos(\alpha+\beta)= -\frac{1}{10} \text{and} \sin (\alpha -\beta)= \frac{3}{8}$$, where $$0<\alpha<\frac{\pi}{3}$$ and $$0<\beta<\frac{\pi}{4}$$. If $$\tan 2\alpha = \frac{3(1-r\sqrt{5})}{\sqrt{11}(s+\sqrt{5})}, r,s\in N$$, then r + s is equal to __________.

Number of solutions of $$\sqrt{3}\cos2\theta+8\cos\theta+3\sqrt{3}=0,\theta\epsilon[-3\pi,2\pi]$$ is:

Let $$\alpha$$ and $$\beta$$ respectively be the maximum and the minimum values of the function $$f(\theta)=4\left(\sin^4\left(\frac{7\pi}{2}-\theta\right)+\sin^4(11\pi+\theta)\right)-2\left(\sin^6\left(\frac{3\pi}{2}-\theta\right)+\sin^6(9\pi-\theta)\right),\ \ \theta\in\ R$$. Then $$\alpha+2\beta$$ is equal to:

If $$\dfrac{\cos^{2}48^{o}-\sin^{2}12^{o}}{\sin^{2}24^{o}-\sin^{2}6^{o}}=\dfrac{\alpha+\beta\sqrt{5}}{2}$$, where $$\alpha, \beta \text{ }\epsilon \text{ }N$$, then $$\alpha + \beta $$ is equal to ________

The value of $$\dfrac{\sqrt{3}  \operatorname{cosec} 20^{\circ}-\sec20^{\circ}}{\cos20^{\circ}\cos40^{\circ}\cos60^{\circ}\cos80^{\circ}}$$ is equal to:

If $$\cot x=\frac{5}{12}$$ for some $$x\in \left(\pi,\frac{3\pi}{2}\right)$$, then $$\sin 7x \left(\cos \frac{13x}{2}+\sin \frac{13x}{2}\right)+\cos 7x\left(\cos \frac{13x}{2}-\sin \frac{13x}{2}\right)$$ is equal to

The number of elements in the set $$\left\{x \in [0,180^{\circ}]:\tan (x+100^{\circ}) = \tan (x+50^{\circ}) \tan x \tan(x-50^{\circ})\right\}$$ is ___________.

If $$\frac{\tan (A-B)}{\tan A}+\frac{\sin^{2}C}{\sin^{2}A}=1,A,B,C \in \left(0,\frac{\pi}{2}\right)$$, Then

Let $$ \frac{\pi}{2} < \theta < \pi $$ and $$\cot\theta=-\frac{1}{2\sqrt{2}}.$$ Then the value of $$\sin\left( \frac{150}{2}\right)\left(\cos 80 + \sin 80\right)+\cos\left( \frac{150}{2}\right)\left(\cos 80 - \sin 80\right)$$ is equal to

The least value of $$(\cos^{2} \theta- 6\sin \theta \cos \theta + 3\sin^{2} \theta +2)$$ is

Considering the principal values of inverse trigonometric functions, the value of the expression $$ \tan\left( 2\sin^{-1} \left( \frac{2}{\sqrt{13}}-2\cos ^{-1}\left( \frac{3}{\sqrt{10}}\right)\right)\right) $$
is equal to:

The sum of all values of $$\theta \in [0,2\pi]$$ satisfying $$2\sin^{2}\theta =\cos2\theta \text{ and }2\cos^{2}\theta =3\sin\theta$$ is

The value of $$\left(\sin 70^{\circ}\right)\left(\cot 10^{\circ}\cot 70^{\circ}-1\right)$$ is

If $$\sin x + \sin^2 x = 1$$, $$x \in (0, \tfrac{\pi}{2})$$ then $$(\cos^{12}x+\tan^{12}x)+3(\cos^{10}x+\tan^{10}x+\cos^{8}x+\tan^{8} x)+(\cos^{6}x+\tan^{6}x)$$
is equal to :

If $$\sum_{r=1}^{13}\left\{\frac{1}{\sin(\frac{\pi}{4}+(r-1)\frac{\pi}{6})\sin(\frac{\pi}{4}+\frac{r\pi}{6})}\right\}=a\sqrt{3}+b,a,b \in Z$$ then $$a^{2}+b^{2}$$ is equal to:

If $$\tan A = \frac{1}{\sqrt{xx^2+x+1}}$$, $$\tan B = \frac{\sqrt{x}}{\sqrt{x^2+x+1}}$$ and $$\tan C=x^{-3}+x^{-2}+x^{-11/2}$$, $$0 < A, B, C < \frac{\pi}{2}$$, then $$A + B$$ is equal to:

The number of solutions of the equation $$4\sin^2 x - 4\cos^3 x + 9 - 4\cos x = 0$$; $$x \in [-2\pi, 2\pi]$$ is:

Let the set of all $$a \in \mathbb{R}$$ such that the equation $$\cos 2x + a \sin x = 2a - 7$$ has a solution be $$[p, q]$$ and $$r = \tan 9° - \tan 27° - \frac{1}{\cot 63°} + \tan 81°$$, then $$pqr$$ is equal to _______.

If $$2\tan^2\theta - 5\sec\theta = 1$$ has exactly 7 solutions in the interval $$\left[0, \frac{n\pi}{2}\right]$$, for the least value of $$n \in \mathbb{N}$$ then $$\sum_{k=1}^{n} \frac{k}{2^k}$$ is equal to :

If $$\alpha$$, $$-\frac{\pi}{2} < \alpha < \frac{\pi}{2}$$ is the solution of $$4\cos\theta + 5\sin\theta = 1$$, then the value of $$\tan\alpha$$ is

The sum of the solutions $$x \in R$$ of the equation $$\frac{3\cos 2x + \cos^3 2x}{\cos^6 x - \sin^6 x} = x^3 - x^2 + 6$$ is

If $$2\sin^3 x + \sin 2x \cos x + 4\sin x - 4 = 0$$ has exactly $$3$$ solutions in the interval $$\left[0, \frac{n\pi}{2}\right]$$, $$n \in \mathbb{N}$$, then the roots of the equation $$x^2 + nx + (n - 3) = 0$$ belong to :

For $$\alpha, \beta \in \left(0, \frac{\pi}{2}\right)$$, let $$3\sin(\alpha + \beta) = 2\sin(\alpha - \beta)$$ and a real number $$k$$ be such that $$\tan\alpha = k\tan\beta$$. Then the value of $$k$$ is equal to

Let $$S = \{\sin^2 2\theta : (\sin^4 \theta + \cos^4 \theta)x^2 + (\sin 2\theta)x + (\sin^6 \theta + \cos^6 \theta) = 0$$ has real roots$$\}$$. If $$\alpha$$ and $$\beta$$ be the smallest and largest elements of the set $$S$$, respectively, then $$3((\alpha - 2)^2 + (\beta - 1)^2)$$ equals _____