JEE Main 2024 — Trigonometry

In triangle ABC if a=5, b=4 and cos(A-B)=31/32 then c is:

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

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

Let $|\cos \theta \cos (60-\theta) \cos (60+\theta)| \leq \frac{1}{8}, \theta \epsilon[0,2 \pi]$. Then, the sum of all $\theta \epsilon[0,2 \pi]$, where $\cos 3 \theta$ attains its maximum value, is :

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 $x^2+y^2+2 x y \sin \alpha$ is

If $\sin x=-\frac{3}{5}$, where $\pi < x < \frac{3 \pi}{2}$, then $80\left(\tan ^2 x-\cos x\right)$ is equal to

Suppose $\theta \epsilon\left[0, \frac{\pi}{4}\right]$ is a solution of $4 \cos \theta-3 \sin \theta=1$. Then $\cos \theta$ is equal to :

The number of solutions of $\sin ^2 x+\left(2+2 x-x^2\right) \sin x-3(x-1)^2=0$, where $-\pi \leq x \leq \pi$, is________

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

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

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

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

The integral $\int_{1 / 4}^{3 / 4} \cos \left(2 \cot ^{-1} \sqrt{\frac{1-x}{1+x}}\right) d x$ is equal to

If the value of $\frac{3 \cos 36^{\circ}+5 \sin 18^{\circ}}{5 \cos 36^{\circ}-3 \sin 18^{\circ}}$ is $\frac{a \sqrt{5}-b}{c}$, where $a, b, c$ are natural numbers and $\operatorname{gcd}(a, c)=1$, then $a+b+c$ is equal to :

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

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 $x=\frac{m}{n}(m,n$ are co-prime natural numbers) be a solution of the equation $\mathrm{cos}(2{\mathrm{sin}}^{-1}x)=\frac{1}{9}$ and let $\alpha ,\beta (\alpha >\beta )$ be the roots of the equation $m{x}^{2}-nx-m+n=0$. Then the point $(\alpha ,\beta )$ lies on the line

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

For $n \in \mathrm{N}$, if $\cot ^{-1} 3+\cot ^{-1} 4+\cot ^{-1} 5+\cot ^{-1} n=\frac{\pi}{4}$, then $n$ is equal to_____

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

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

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

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