Embibe Experts Solutions for Chapter: Trigonometric Equations and Inequalities, Exercise 1: JEE Advanced Paper 1 - 2016

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

Let $f:\left[0, 2\right]\to ℝ$ be the function defined by

$f\left(x\right)=\left(3-\sin \left(2\pi x\right)\right)\sin \left(\pi x-\frac{\pi }{4}\right)-\sin \left(3\pi x+\frac{\pi }{4}\right)$.

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

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 $\frac{a}{b}$ is

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