Trigonometric Functions and Equations

Let $[\cdot]$ denote the greatest integer function. If the domain of the function $f(x)=\sin^{-1}\left(\dfrac{x+[x]}3\right)$ is $[\alpha,\beta)$, then $\alpha^2+\beta^2$ is equal to:

Let $0<\alpha<1$, $\beta=\dfrac1{3\alpha}$ and $\tan^{-1}(1-\alpha)+\tan^{-1}(1-\beta)=\dfrac\pi4$. Then $6(\alpha+\beta)$ is equal to:

Let $S=\left\{\theta\in(-2\pi,2\pi):\cos\theta+1=\sqrt3\sin\theta\right\}$. Then $\displaystyle\sum_{\theta\in S}\theta$ is equal to:

The sum of all possible values of $\theta\in[0,2\pi]$, for which the system of equations :
$$x\cos3\theta-8y-12z=0,\qquad x\cos2\theta+3y+3z=0,\qquad x+y+3z=0$$
has a non-trivial solution, is equal to :

If $\sin\left(\tan^{-1}\left(x\sqrt2\right)\right)=\cot\left(\sin^{-1}\sqrt{1-x^2}\right)$, $x\in(0,1)$, then the value of $x$ is :

Let $\alpha=3+4+8+9+13+14+\dots$ upto 40 terms. If $(\tan\beta)^{\frac\alpha{1020}}$ is a root of the equation $x^2+x-2=0$, $\beta\in\left(0,\dfrac\pi2\right)$, then $\sin^2\beta+3\cos^2\beta$ is equal to :

Let $\alpha=3\sin^{-1}\left(\dfrac6{11}\right)$ and $\beta=3\cos^{-1}\left(\dfrac49\right)$, where inverse trigonometric functions take only the principal values.

Given below are two statements :

Statement I : $\cos(\alpha+\beta)>0$.

Statement II : $\cos(\alpha)<0$.

In the light of the above statements, choose the correct answer from the options given below :

If $\displaystyle\int_{\pi/6}^{\pi/4}\left(\cot\left(x-\frac\pi3\right)\cot\left(x+\frac\pi3\right)+1\right)dx=\alpha\log_e\left(\sqrt3-1\right)$, then $9\alpha^2$ is equal to ________.

The total number of real solutions of the equation

$$\theta = \tan^{-1}(2\tan\theta) - \frac12\sin^{-1}\left(\frac{6\tan\theta}{9 + \tan^2\theta}\right)$$

is

(Here, the inverse trigonometric functions $\sin^{-1}x$ and $\tan^{-1}x$ assume values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, respectively.)

Let

$$\alpha = \frac{1}{\sin60^\circ\sin61^\circ} + \frac{1}{\sin62^\circ\sin63^\circ} + \cdots + \frac{1}{\sin118^\circ\sin119^\circ}.$$

Then the value of

$$\left(\frac{\operatorname{cosec}1^\circ}{\alpha}\right)^2$$

is _____.

If

$$\alpha = \int_{1/2}^{2}\frac{\tan^{-1}x}{2x^2 - 3x + 2}\,dx,$$

then the value of $\sqrt7\tan\left(\dfrac{2\alpha\sqrt7}{\pi}\right)$ is _____.

(Here, the inverse trigonometric function $\tan^{-1}x$ assumes values in $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$.)

Let $\dfrac{\pi}{2} < x < \pi$ be such that $\cot x = \dfrac{-5}{\sqrt{11}}$. Then

$$\left(\sin\frac{11x}{2}\right)(\sin 6x - \cos 6x) + \left(\cos\frac{11x}{2}\right)(\sin 6x + \cos 6x)$$

is equal to

Considering only the principal values of the inverse trigonometric functions, the value of

$$\tan\left(\sin^{-1}\left(\frac{3}{5}\right) - 2\cos^{-1}\left(\frac{2}{\sqrt5}\right)\right)$$

is

Let $f : \mathbb{R} \to \mathbb{R}$ be a function defined by

$$f(x) = \begin{cases} x^2\sin\left(\dfrac{\pi}{x^2}\right), & \text{if } x \neq 0, \\ 0, & \text{if } x = 0. \end{cases}$$

Then which of the following statements is TRUE?

Let $\tan^{-1}(x) \in \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$, for $x \in \mathbb{R}$. Then the number of real solutions of the equation

$$\sqrt{1 + \cos(2x)} = \sqrt2\tan^{-1}(\tan x)$$

in the set $\left(-\dfrac{3\pi}{2}, -\dfrac{\pi}{2}\right)\cup\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)\cup\left(\dfrac{\pi}{2}, \dfrac{3\pi}{2}\right)$ is equal to _____.

Let $A = \left\{\dfrac{1967 + 1686i\sin\theta}{7 - 3i\cos\theta} : \theta \in \mathbb{R}\right\}$. If $A$ contains exactly one positive integer $n$, then the value of $n$ is _____.

For any $y \in \mathbb{R}$, let $\cot^{-1}(y) \in (0,\pi)$ and $\tan^{-1}(y) \in \left(-\dfrac\pi2, \dfrac\pi2\right)$. Then the sum of all the solutions of the equation $\tan^{-1}\left(\dfrac{6y}{9-y^2}\right) + \cot^{-1}\left(\dfrac{9-y^2}{6y}\right) = \dfrac{2\pi}{3}$ for $0 < |y| < 3$, is equal to

For $x \in \mathbb{R}$, let $\tan^{-1}(x) \in \left(-\dfrac\pi2, \dfrac\pi2\right)$. Then the minimum value of the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \displaystyle\int_0^{x\tan^{-1}x} \frac{e^{(t-\cos t)}}{1+t^{2023}}\,dt$ is _____.

Considering only the principal values of the inverse trigonometric functions, the value of
$$\frac32\cos^{-1}\sqrt{\frac{2}{2+\pi^2}} + \frac14\sin^{-1}\frac{2\sqrt2\,\pi}{2+\pi^2} + \tan^{-1}\frac{\sqrt2}{\pi}$$
is _____.

Let $|M|$ denote the determinant of a square matrix $M$. Let $g : \left[0, \dfrac\pi2\right] \to \mathbb{R}$ be the function defined by
$$g(\theta) = \sqrt{f(\theta) - 1} + \sqrt{f\left(\frac\pi2 - \theta\right) - 1}$$
where
$$f(\theta) = \frac12\begin{vmatrix}1 & \sin\theta & 1\\ -\sin\theta & 1 & \sin\theta\\ -1 & -\sin\theta & 1\end{vmatrix} + \begin{vmatrix}\sin\pi & \cos\left(\theta + \frac\pi4\right) & \tan\left(\theta - \frac\pi4\right)\\ \sin\left(\theta - \frac\pi4\right) & -\cos\frac\pi2 & \log_e\left(\frac4\pi\right)\\ \cot\left(\theta + \frac\pi4\right) & \log_e\left(\frac\pi4\right) & \tan\pi\end{vmatrix}.$$
Let $p(x)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g(\theta)$, and $p(2) = 2 - \sqrt2$. Then, which of the following is/are TRUE?