Trigonometric Equations — JEE Advanced PYQs

Match each entry in List-I to the correct entry in List-II and choose the correct option.
| (Q) …

Consider the curve ${\mathrm{C}}_1$ given by

and the curve ${\mathrm{C}}_2$ given by

L …

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

Then which of the following statements is TRUE ?

Consider the following lists:
| (II) | …

Let $f:[0,2]\to \mathbb{R}$ be the function defined by

If $\alpha ,\beta \in [0,2]$ are such that $\{x\in [0,2]:f(x)\ge 0\}=[\alpha ,\beta ]$, then the value of …

Answer the following by appropriately matching the lists based on the information given in the paragraph
Let $f(x)=\sin (\pi \cos x)$ and $g(x)=\cos (2\pi \sin x)$ be two functions defined for $x>0$. Define the following sets whose elements are written in the increasing or …

Let $a,b,c$ be three non-zero real numbers such that the equation $\sqrt{3}a\cos x+2b\sin x=c$, $x\in \left[-\frac{\pi }{2},\frac{\pi }{2}\right]$, has two distinct real roots $\alpha$ and $\beta$ with $\alpha +\beta =\frac{\pi }{3}$. Then, the value of …

Let $S=\left\{x\in (-\pi ,\pi ):x\ne 0,\pm \frac{\pi }{2}\right\}$. The sum of all distinct solutions of the equation $\sqrt{3}\sec x+\mathrm{cosec}x+2(\tan x-\cot x)=0$ in the set $S$ is equal to

The number of distinct solutions of the equation

in the interval $[0,2\pi ]$ is

For $x\in (0,\pi )$, the equation $\sin x+2\sin 2x-\sin 3x=3$ has

Let $\theta ,\phi \in [0,2\pi ]$ be such that
$2\cos \theta (1-\sin \phi )={\sin }^2\theta \left(\tan \frac{\theta }{2}+\cot \frac{\theta }{2}\right)\cos \phi -1$,
$\tan (2\pi -\theta )>0$ and $-1<\sin \theta <-\frac{\sqrt{3}}{2}$.
Then $\phi$ cannot satisfy

Let $P=\{\theta :\sin \theta -\cos \theta =\sqrt{2}\cos \theta \}$ and $Q=\{\theta :\sin \theta +\cos \theta =\sqrt{2}\sin \theta \}$ be two sets. Then

The positive integer value of $n>3$ satisfying the equation

is

The number of values of $\theta$ in the interval $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ such that $\theta \ne \frac{n\pi }{5}$ for $n=0,\pm 1,\pm 2$ and $\tan \theta =\cot 5\theta$ as well as $\sin 2\theta =\cos 4\theta$ is

The number of all possible values of $\theta$, where $0<\theta <\pi$, for which the system of equations …

The number of solutions of the pair of equations

in the interval $[0,2\pi ]$ is

If $0<\theta <2\pi$, then the intervals of values of $\theta$ for which $2{\sin }^2\theta -5\sin \theta +2>0$, is