Trigonometric Functions
140 JEE Maths previous year questions on Trigonometric Functions — options free on every question; 14 include the answer & explanation free, the rest unlock with PYQ Pass.
Evaluate: \(\sum _{r=1}^{13}\frac{1}{\sin \left[\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right]\sin \left[\frac{\pi }{4}+\frac{r\pi }{6}\right]}\)
\(2\sqrt{3}-2\)
\(\sum _{r=1}^{13}\frac{1}{\sin \left[\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right]\sin \left[\frac{\pi }{4}+\frac{r\pi }{6}\right]}\\ \sum _{r=1}^{13}\frac{\sin \left[\frac{\pi }{4}+r\frac{\pi }{6}\right]-\left[\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right]}{\left(\sin \frac{\pi }{6}\right)\times \sin \left(\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right)\sin \left(\frac{\pi }{4}+\frac{r\pi }{6}\right)}\\ \sum _{r=1}^{13}2\left(\cot \left(\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right)-\cot \left(\frac{\pi }{4}+\frac{r\pi }{6}\right)\right)\\ 2\left[\cot \frac{\pi }{4}-\cot \left(\frac{\pi }{4}+\frac{13\pi }{6}\right)\right]\\ 2\left(1-\cot \frac{5\pi }{12}\right)\\ 2(1-2+\sqrt{3})\Rightarrow (2\sqrt{3}-2)\)
If \(2 x^2+(\cos \theta) x-1=0, \theta \in[0,2 \pi]\) has roots \(\alpha\) and \(\beta\). Then the sum of maximum and minimum value of \(\alpha^4+\beta^4\) is
[JEE Main 2025]
\(\frac{25}{16}\)
\(\begin{aligned}& 2 x^2+(\cos \theta) x-1=0 \\& \alpha+\beta=\frac{-\cos \theta}{2}; \quad \alpha \beta=-\frac{1}{2} \\& \alpha^2+\beta^2=(\alpha+\beta)^2-2 \alpha \beta =\frac{\cos ^2 \theta}{4}+1 \\& \alpha^4+\beta^4=\left(\alpha^2+\beta^2\right)^2-2 \alpha^2 \beta^2=\left(\frac{\cos ^2 \theta}{4}+1\right)^2-\frac{2}{4} \\& \alpha^4+\beta^4=\left(\frac{\cos ^2 \theta}{4}+1\right)^2-\frac{1}{2}\end{aligned}\)
Maximum when \(\cos \theta=1\)
\(\begin{aligned}& M=\left(\frac{1}{4}+1\right)^2-\frac{1}{2} \\& M=\frac{17}{16}\end{aligned}\)
Minimum when \(\cos \theta=0\)
\(m=1-\frac{1}{2}=\frac{1}{2}\)
\(\left.\therefore \quad 16(M+m)=16 (\frac{17}{16}+\frac{1}{2}\right)=25\)
Let \(S={\theta \in (-2\pi ,2\pi ):\cos \theta +1=\sqrt{3}\sin \theta }\). Then \(\sum _{\theta \in S}\theta\) is equal to:
[JEE Main 2026, 6 Apr (Shift 1)]
\(-\frac{4\pi }{3}\)
Given, \(\cos \theta+1=\sqrt{3} \sin \theta\)
\(2 \cos ^2 \frac{\theta}{2}=\sqrt{3} \cdot 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}\)
\(2 \cos \frac{\theta}{2}\left(\cos \frac{\theta}{2}-\sqrt{3} \sin \frac{\theta}{2}\right)=0\)
If \(\cos \frac{\theta}{2}=0\)
Now in the interval \((-2 \pi, 2 \pi)\), the possible values are.
\(\theta=-\pi, \pi\)
If \(\cos \frac{\theta}{2}-\sqrt{3} \sin \frac{\theta}{2}=0\)
\(\tan \frac{\theta}{2}=\frac{1}{\sqrt{3}}\)
Now in the interval \((-2 \pi, 2 \pi)\), the possible values are:
\(\theta=-\frac{5 \pi}{3}, \frac{\pi}{3}\)
Therefore,
\(S=\left\{-\pi, \pi,-\frac{5 \pi}{3}, \frac{\pi}{3}\right\}\)
Hence,
\(\sum_{\theta \in S} \theta=-\frac{4 \pi}{3}\)
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 :
[JEE Main 2024, 08 Apr (Shift 2)]
52
\(\frac{3\cos 36^\circ +5\sin 18^\circ }{5\cos 36^\circ -3\sin 18^\circ }\\ =\frac{\frac{3(\sqrt{5}+1)}{4}+5\left(\frac{\sqrt{5}-1}{4}\right)}{5\left(\frac{\sqrt{5}-1}{4}\right)-3\left(\frac{\sqrt{5}-1}{4}\right)}\)
\(=\frac{8\sqrt{5}-2}{2\sqrt{5}+8}\)
\(=\frac{4\sqrt{5}-1}{\sqrt{5}+4}\times \frac{\sqrt{5}-4}{\sqrt{5}-4}\)
\(=\frac{20-16\sqrt{5}-\sqrt{5}+4}{-11}\)
\(=\frac{17\sqrt{5}-24}{11}\\ \Rightarrow a=17,b=24,c=11\)
\(a+b+c=52\)
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 :
[JEE Main 2024, 08 Apr (Shift 2)]
52
\(\frac{3\cos 36^\circ +5\sin 18^\circ }{5\cos 36^\circ -3\sin 18^\circ }\\ =\frac{\frac{3(\sqrt{5}+1)}{4}+5\left(\frac{\sqrt{5}-1}{4}\right)}{5\left(\frac{\sqrt{5}-1}{4}\right)-3\left(\frac{\sqrt{5}-1}{4}\right)}\)
\(=\frac{8\sqrt{5}-2}{2\sqrt{5}+8}\)
\(=\frac{4\sqrt{5}-1}{\sqrt{5}+4}\times \frac{\sqrt{5}-4}{\sqrt{5}-4}\)
\(=\frac{20-16\sqrt{5}-\sqrt{5}+4}{-11}\)
\(=\frac{17\sqrt{5}-24}{11}\\ \Rightarrow a=17,b=24,c=11\)
\(a+b+c=52\)
If \(\sin x+\sin ^2 x=1, x \in\left(0, \frac{\pi}{2}\right)\), then \(\left(\cos ^{12} x+\tan ^{12} x\right)+3\left(\cos ^{10} x+\tan ^{10} x+\cos ^8 x+\tan ^8 x\right)+\left(\cos ^6 x+\tan ^6 x\right)\) is equal to:
[JEE Main 2025]
2
\(\sin x+\sin ^2 x=1\)
\(\Rightarrow \sin x=\cos ^2 x\) and \(\tan x=\cos x\)
Given expression becomes
\(=2 \cos ^{12} x+6\left[\cos ^{10} x+\cos ^8 x\right]+2 \cos ^6 x\)
\(=2\left[\sin ^6 x+3 \sin ^5 x+3 \sin ^4 x+\sin ^3 x\right]\)
\(=2 \sin ^3 x\left[(\sin x+1)^3\right]\)
\(=2\left[\sin ^2 x+\sin x\right]^3\)
\(=2\)
The number of solutions of the equation \(2\mathrm{x}+3\mathrm{tanx}=\pi ,\mathrm{x}\in [-2\pi ,2\pi ]-\left\{\pm \frac{\pi }{2},\pm \frac{3\pi }{2}\right\}\) is
[JEE Main 2025, 3 Apr (Shift 1)]
\(5\)
\(f(x)=2x+3\tan x-\pi\)
\(f'(x)=2+3\sec^2x\)
\(\sec^2x>0\),
\(f'(x)>0\)
for every point in the domain.
Therefore \(f(x)\) is strictly increasing on each interval
\(\left(-2\pi,-\frac{3\pi}{2}\right)\),
\(\left(-\frac{3\pi}{2},-\frac{\pi}{2}\right)\),
\(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\),
\(\left(\frac{\pi}{2},\frac{3\pi}{2}\right)\),
\(\left(\frac{3\pi}{2},2\pi\right)\).
Now check sign changes.
\(\left(-2\pi,-\frac{3\pi}{2}\right)\),
\(f(-2\pi)=-5\pi<0\),
\(f(x)\to+\infty\) as \(x\to\left(-\frac{3\pi}{2}\right)^-\).
Exactly one solution.
\(\left(-\frac{3\pi}{2},-\frac{\pi}{2}\right)\),
\(f(x)\to-\infty\) as \(x\to\left(-\frac{3\pi}{2}\right)^+\),
\(f(x)\to+\infty\) as \(x\to\left(-\frac{\pi}{2}\right)^-\).
Exactly one solution.
\(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\),
\(f(x)\to-\infty\) as \(x\to\left(-\frac{\pi}{2}\right)^+\),
\(f(x)\to+\infty\) as \(x\to\left(\frac{\pi}{2}\right)^-\).
Exactly one solution.
\(\left(\frac{\pi}{2},\frac{3\pi}{2}\right)\),
\(f(x)\to-\infty\) as \(x\to\left(\frac{\pi}{2}\right)^+\),
\(f(x)\to+\infty\) as \(x\to\left(\frac{3\pi}{2}\right)^-\).
Exactly one solution.
\(\left(\frac{3\pi}{2},2\pi\right)\),
\(f(x)\to-\infty\) as \(x\to\left(\frac{3\pi}{2}\right)^+\),
\(f(2\pi)=3\pi>0\).
Exactly one solution.
Total number of solutions
\(=1+1+1+1+1=5\).
If for \(\theta \in \left[-\frac{\pi }{3},0\right]\), the points \((\mathrm{x},\mathrm{y})=\left(3\tan \left(\theta +\frac{\pi }{3}\right),2\tan \left(\theta +\frac{\pi }{6}\right)\right)\) lie on \(xy+\alpha x+\beta y+\gamma =0\), then \({\alpha }^{2}+{\beta }^{2}+{\gamma }^{2}\) is equal to :
[JEE Main 2025, 7 Apr (Shift 1)]
\(75\)
Given \((\mathrm{x},\mathrm{y})=\left(3\tan \left(\theta +\frac{\pi }{3}\right),2\tan \left(\theta +\frac{\pi }{6}\right)\right)\)
\( \mathrm{x}=3\left(\frac{\tan \theta+\sqrt{3}}{1-\sqrt{3} \tan \theta}\right) \)
\( \mathrm{x}-\sqrt{3} \tan \theta=3 \tan \theta+3 \sqrt{3} \)
\( \tan \theta=\frac{x-3 \sqrt{3}}{3+\sqrt{3} x} \quad \ldots(1)\)
\( 2\left(\frac{\tan \theta+\frac{1}{\sqrt{3}}}{1-\frac{\tan \theta}{\sqrt{3}}}=y\right) \)
\( 2(\sqrt{3} \tan \theta+1)=y(\sqrt{3}-\tan \theta) \ldots(2) \)
using (1) and (2)
\( 2\left(\frac{x-3 \sqrt{3}}{\sqrt{3}+x}+1\right)=y\left(\sqrt{3}-\frac{(x-3 \sqrt{3})}{\sqrt{3}(\sqrt{3}+x)}\right) \)
\( 2 \sqrt{3}(x-3 \sqrt{3}+x+\sqrt{3})=y(3(\sqrt{3}+x)-x+3 \sqrt{3})\)
\( 4 \sqrt{3} x-12=y(2 x+6 \sqrt{3}) \)
\(x y-2 \sqrt{3} x+3 \sqrt{3} y-6=0 \)
\( \Rightarrow \alpha=-2 \sqrt{3}, \beta=3 \sqrt{3}, \gamma=-6\)
\( \alpha^2+\beta^2+\gamma^2=12+27+36=75\)
If \(2 x^2+(\cos \theta) x-1=0, \theta \in[0,2 \pi]\) has roots \(\alpha\) and \(\beta\). Then the sum of maximum and minimum value of \(\alpha^4+\beta^4\) is
[JEE Main 2025]
\(\frac{25}{16}\)
\(\begin{aligned}& 2 x^2+(\cos \theta) x-1=0 \\& \alpha+\beta=\frac{-\cos \theta}{2}; \quad \alpha \beta=-\frac{1}{2} \\& \alpha^2+\beta^2=(\alpha+\beta)^2-2 \alpha \beta =\frac{\cos ^2 \theta}{4}+1 \\& \alpha^4+\beta^4=\left(\alpha^2+\beta^2\right)^2-2 \alpha^2 \beta^2=\left(\frac{\cos ^2 \theta}{4}+1\right)^2-\frac{2}{4} \\& \alpha^4+\beta^4=\left(\frac{\cos ^2 \theta}{4}+1\right)^2-\frac{1}{2}\end{aligned}\)
Maximum when \(\cos \theta=1\)
\(\begin{aligned}& M=\left(\frac{1}{4}+1\right)^2-\frac{1}{2} \\& M=\frac{17}{16}\end{aligned}\)
Minimum when \(\cos \theta=0\)
\(m=1-\frac{1}{2}=\frac{1}{2}\)
\(\left.\therefore \quad 16(M+m)=16 (\frac{17}{16}+\frac{1}{2}\right)=25\)
If\(\sum _{r=1}^{13}\left\{\frac{1}{\sin \left(\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right)\sin \left(\frac{\pi }{4}+\frac{r\pi }{6}\right)}\right\}=a\sqrt{3}+b,a,b\in Z\), then \(a^2+b^2\) is equal to :
[JEE Main 2025, 28 Jan (Shift 2)]
8
Let \(S=\sum _{r=1}^{13}\frac{1}{\sin \left(\frac{\pi }{4}+\frac{r\pi }{6}\right)\sin \left(\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right)}\)
\(=2\sum _{r=1}^{13}\frac{\sin \left(\frac{\pi }{4}+\frac{r\pi }{6}\right)-\left(\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right)}{\sin \left(\frac{\pi }{4}+\frac{\pi }{6}\right)\sin \left(\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right)}\)
\(=2\sum _{r=1}^{13}\left(\cot \left(\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right)-\cot \left(\frac{\pi }{4}+\frac{\mathrm{r}\pi }{6}\right)\right)\)
\(=2\left[\cot \left(\frac{\pi }{4}\right)-\cot \left(\frac{\pi }{4}+\frac{13\pi }{6}\right)\right]\)
\(=2\left[1-\cot \left(\frac{\pi }{4}+2\pi +\frac{\pi }{6}\right)\right]\)
\(=2\left[1-\cot \left(\frac{5\pi }{12}\right)\right]=2\left(1-2+\sqrt{3}\right)\)
\(=2\sqrt{3}-2=a\sqrt{3}+b\)
\({a}^{2}+{b}^{2}=8\)
Let \(\alpha\) and \(\beta\) respectively be the maximum and the minimum values of the function
\( f(\theta)=4\left(\sin ^4\left(\frac{7 \pi}{2}-\theta\right)+\sin ^4(11 \pi+\theta)\right) -2\left(\sin ^6\left(\frac{3 \pi}{2}-\theta\right)+\sin ^6(9 \pi-\theta)\right)\). Then \(\alpha +2\beta\) is equal to:
[JEE Main 2026, 23 Jan (Shift 2)]
\(5\)
Given:
\( f(\theta)=4\left(\sin ^4\left(\frac{7 \pi}{2}-\theta\right)+\sin ^4(11 \pi+\theta)\right) -2\left(\sin ^6\left(\frac{3 \pi}{2}-\theta\right)+\sin ^6(9 \pi-\theta)\right)\)
\(\Rightarrow f(\theta)=4\left(\cos ^4(\theta)+\sin ^4(\theta)\right)-2\left(\cos ^6 \theta+\sin ^6 \theta\right)\)
\(\Rightarrow f(\theta)=4\left(1-2 \sin ^2 \theta \cos ^2 \theta\right)-2\left(1-3 \sin ^2 \theta \cos ^2 \theta\right)\)
\(\Rightarrow f(\theta)=2-2 \sin ^2 \theta \cos ^2 \theta\)
\(\Rightarrow f(\theta)=2-\frac{\sin ^2(2 \theta)}{2}\)
So, \(\alpha=f(\theta)_{\max }=2\),
\(\beta=f(\theta)_{\min }=\frac{3}{2}\)
\(\Rightarrow \alpha+2 \beta=5\)
If \(\sin x=-\frac{3}{5}\), where \(\pi [JEE Main 2024, 8 Apr (Shift 1)]
109
\(\sin x=\frac{-3}{5},\pi \(\Rightarrow \tan x=\frac{3}{4}\) and \(\cos x=-\frac{4}{5}\) \(80\left({\tan }^{2}x-\cos x\right)=80\left(\frac{9}{16}+\frac{4}{5}\right)=45+64=109\)
If \(10{\sin }^{4}\theta +15{\cos }^{4}\theta =6\), then the value of \(\frac{27{\csc }^{6}\theta +8{\sec }^{6}\theta }{16{\sec }^{8}\theta }\) is:
[JEE Main 2025, 4 Apr (Shift 1)]
\(\frac{2}{5}\)
\(10{\sin }^{4}\theta +15{\cos }^{4}\theta =6\)
\(\Rightarrow 10\left[1-2 \sin ^2 \theta \cos ^2 \theta\right]+5 \cos ^4 \theta=6\)
\(\Rightarrow 25 \cos ^4 \theta-20 \cos ^2 \theta+4=0\)
\(\Rightarrow\left(5 \cos ^2 \theta-2\right)^2=0\)
\(\Rightarrow {\cos }^{2}\theta =\frac{2}{5}\) and \({\sin }^{2}\theta =\frac{3}{5}\)
\(\Rightarrow \frac{27{\csc }^{6}\theta +8{\sec }^{6}\theta }{16{\sec }^{8}\theta }\\ =\frac{27\cdot {\left(\frac{5}{3}\right)}^{3}+8\cdot {\left(\frac{5}{2}\right)}^{3}}{16\cdot {\left(\frac{5}{4}\right)}^{4}}\\ =\frac{2}{5}\)
If \(\theta \in \left[-\frac{7\pi }{6},\frac{4\pi }{3}\right]\), then the number of solutions of \(\sqrt{3}{\csc }^{2}\theta -2(\sqrt{3}-1)\csc \theta -4=0\), is equal to
[JEE Main 2025, 2 Apr (Shift 2)]
\(6\)
\(\sqrt{3}{\csc }^{2}\theta −2(\sqrt{3}−1)\csc \theta −4=0\)
Let \(x=\csc \theta\).
\(\sqrt{3}{x}^{2}−2(\sqrt{3}−1)x−4=0\)
\(\sqrt{3}x(x−2)+2(x−2)=0\)
\((\sqrt{3}x+2)(x−2)=0\)
So, the solutions for \(x\) are: \(x=2 \) or \(x=-\frac{2}{\sqrt{3}}\)
Substituting \(x=\csc \theta\) back:
\(\csc \theta =2⟹\sin \theta =\frac{1}{2}\)
\(\csc \theta =−\frac{2}{\sqrt{3}}⟹\sin \theta =−\frac{\sqrt{3}}{2}\)
The solutions are: \(−\frac{7\pi }{6},−\frac{2\pi }{3},−\frac{\pi }{3},\frac{\pi }{6},\frac{5\pi }{6},\frac{4\pi }{3}\).
Therefore, the number of solutions is 6.
If \(\theta \in \left[-\frac{7\pi }{6},\frac{4\pi }{3}\right]\), then the number of solutions of \(\sqrt{3}{\csc }^{2}\theta -2(\sqrt{3}-1)\csc \theta -4=0\), is equal to
[JEE Main 2025, 2 Apr (Shift 2)]
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Number of solutions of \(\sqrt{3}\cos 2\theta +8\cos \theta +3\sqrt{3}=0,\theta \in [-3\pi ,2\pi ]\) is:
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Let \(\alpha\) and \(\beta\) respectively be the maximum and the minimum values of the function
\( f(\theta)=4\left(\sin ^4\left(\frac{7 \pi}{2}-\theta\right)+\sin ^4(11 \pi+\theta)\right) -2\left(\sin ^6\left(\frac{3 \pi}{2}-\theta\right)+\sin ^6(9 \pi-\theta)\right)\). Then \(\alpha +2\beta\) is equal to:
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Let \(S={x\in [-\pi ,\pi ]:\sin x(\sin x+\cos x)=a,a\in Z}\). Then \(n\left(S\right)\) is equal to:
[JEE Main 2026, 2 Apr (Shift 1)]
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If \(K=\sin \left(\frac{\pi }{18}\right)\sin \left(\frac{5\pi }{18}\right)\sin \left(\frac{7\pi }{18}\right)\), then the value of \(\sin \left(\frac{10K\pi }{3}\right)\) is:
[JEE Main 2026, 2 Apr (Shift 1)]
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Let \(f:\mathrm{ℝ}\to \mathrm{ℝ}\) be a function defined by
\(f(x)=\left\{\begin{matrix}{x}^{2}\sin \left(\frac{\pi }{{x}^{2}}\right), & ifx\neq 0, \\ 0, & ifx=0.\end{matrix}\right.\)
Then which of the following statements is TRUE?
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\(\frac{\sqrt{3}\csc {20}^{^\circ }−\sec {20}^{^\circ }}{\cos {20}^{^\circ }\cos {40}^{^\circ }\cos {60}^{^\circ }\cos {80}^{^\circ }}\) is equal to
[JEE Main 2026, 24 Jan (Shift 1)]
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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
[JEE Main 2024, 29 Jan (Shift 1)]
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If \(\tan A =\frac{1}{\sqrt{x\left(x^2+x+1\right)}}, \tan B =\frac{\sqrt{x}}{\sqrt{x^2+x+1}}\) and \(\tan C =\left(x^{-3}+x^{-2}+x^{-1}\right)^{1 / 2}, 0< A , B , C <\frac{\pi}{2}\), then \(A + B\) is equal to :
[JEE Main 2024, 1 Feb (Shift 1)]
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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 \(n \in N\), then \(\sum_{k=1}^n \frac{k}{2^k}\) is equal to :
[JEE Main 2024, 27 Jan (Shift 2)]
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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 \(n \in N\), then \(\sum_{k=1}^n \frac{k}{2^k}\) is equal to :
[JEE Main 2024, 27 Jan (Shift 2)]
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Let \(\frac{\pi }{2}
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The sum of all the integral values of \(p\) such that the equation \(3 \sin ^2 x+12 \cos x-3=p, x \in \mathbb{R}\), has at least one solution, is:
[JEE Main 2026, 5 Apr (Shift 1)]
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Let \(f(x)=3\sqrt{x-2}+\sqrt{4-x}\) be a real valued function. If \(\alpha\) and \(\beta\) are respectively the minimum and the maximum values of \(f\), then \({\alpha }^{2}+2{\beta }^{2}\) is equal to
[JEE Main 2024, 4 Apr (Shift 2)]
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If \(\theta \in[0,2 \pi]\) satisfying the system of equations \(2 \sin ^2 \theta=\cos 2 \theta\) and \(2 \cos ^2 \theta=3 \sin \theta\). Then the sum of all real values of \(\theta\) is
[JEE Main 2025]
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Let \(\tan A, \tan B\), where \(A, B \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), be the roots of the quadratic equation \(x^2-2 x-5=0\). Then \(20 \sin ^2\left(\frac{A+B}{2}\right)\) is equal to:
[JEE Main 2026, 5 Apr (Shift 1)]
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If \(\frac{\tan \left(A−B\right)}{\tan A}+\frac{{\sin }^{2}C}{{\sin }^{2}A}=1,A,B,C\in \left(0,\frac{\pi }{2}\right)\), then
[JEE Main 2026, 28 Jan (Shift 1)]
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The sum of the solutions \(x \in R\) of the equation \(\frac{3 \cos 2 x+\cos ^3 2 x}{\cos ^6 x-\sin ^6 x}=x^3-x^2+6\) is
[JEE Main 2024, 29 Jan (Shift 2)]
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The sum of the solutions \(x \in R\) of the equation \(\frac{3 \cos 2 x+\cos ^3 2 x}{\cos ^6 x-\sin ^6 x}=x^3-x^2+6\) is
[JEE Main 2024, 29 Jan (Shift 2)]
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The number of solutions of the equation \(\cos 2\theta \cos \frac{\theta }{2}+\cos \frac{5\theta }{2}=2{\cos }^{3}\frac{5\theta }{2}\) in \(\left[-\frac{\pi }{2},\frac{\pi }{2}\right]\) is :
[JEE Main 2025, 7 Apr (Shift 2)]
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The least value of \(\left({\text{cos}}^{2}\theta −6\text{sin}\theta \text{cos}\theta +3{\text{sin}}^{2}\theta +2\right)\) is
[JEE Main 2026, 23 Jan (Shift 2)]
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If \(\frac{\tan \left(A−B\right)}{\tan A}+\frac{{\sin }^{2}C}{{\sin }^{2}A}=1,A,B,C\in \left(0,\frac{\pi }{2}\right)\), then
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The value of \(\left(\sin 70^\circ \right)\left(\cot 10^\circ \cot 70^\circ -1\right)\) is
[JEE Main 2025, 23 Jan (Shift 1)]
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If \(2 \sin ^3 x+\sin 2 x \cos x+4 \sin x-4=0\) has exactly 3 solutions in the interval \(\left[0, \frac{ n \pi}{2}\right], n \in N\), then the roots of the equation \(x^2+ n x+( n -3)=0\) belong to :
[JEE Main 2024, 30 Jan (Shift 1)]
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If \(2 \sin ^3 x+\sin 2 x \cos x+4 \sin x-4=0\) has exactly 3 solutions in the interval \(\left[0, \frac{ n \pi}{2}\right], n \in N\), then the roots of the equation \(x^2+ n x+( n -3)=0\) belong to :
[JEE Main 2024, 30 Jan (Shift 1)]
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The least value of \(\left({\text{cos}}^{2}\theta −6\text{sin}\theta \text{cos}\theta +3{\text{sin}}^{2}\theta +2\right)\) is
[JEE Main 2026, 23 Jan (Shift 2)]
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The number of solutions of equation \((4-\sqrt{3})\sin x\) \(-2\sqrt{3}{\cos }^{2}x=-\frac{4}{1+\sqrt{3}},x\in \left[-2\pi ,\frac{5\pi }{2}\right]\) is
JEE MAINS [2025]
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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
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Let \(f(x)=6+16 \cos \left(\frac{\pi}{3}-x\right) \cos \left(\frac{\pi}{3}+x\right) \cos x \sin 3 x \cos 6 x\) if range of \(f(x)\) is \([\alpha, \beta]\) then distance of \((\alpha, \beta)\) from \(3 x+4 y+12=0\) is
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If \(10{\sin }^{4}\theta +15{\cos }^{4}\theta =6\), then the value of \(\frac{27{\csc }^{6}\theta +8{\sec }^{6}\theta }{16{\sec }^{8}\theta }\) is:
[JEE Main 2025, 4 Apr (Shift 1)]
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Suppose \( \theta \in\left[0, \frac{\pi}{4}\right]\) is a solution of \(4 \cos \theta-3 \sin \theta=1\). Then \(\cos \theta\) is equal to :
[JEE Main 2024, 5 Apr (Shift 1)]
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Let \(|\cos \theta \cos (60-\theta) \cos (60+\theta)| \leq \frac{1}{8}, \theta \in[0,2 \pi]\). Then, the sum of all \(\theta \in[0,2 \pi]\), where \(\cos 3 \theta\) attains its maximum value, is
[JEE Main 2024, 9 Apr (Shift 1)]
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If \(\theta \in [-2\pi ,2\pi ]\), then the number of solutions of \(2\sqrt{2}{\cos }^{2}\theta +(2-\sqrt{6})\mathrm{cosθ}-\sqrt{3}=0\), is equal to:
[JEE Main 2025, 2 Apr (Shift 1)]
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If \(f(\theta )={\sin }^{4}\theta +{\cos }^{2}\theta\), then range of \(f(\theta )\) is
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Let \(P=\left\{\theta \in [0,4\pi ]:{\tan }^{2}\theta \neq 1\right\}\) and \(S=\left\{a\in Z:2\left({\cos }^{8}\theta -{\sin }^{8}\theta \right)\right.\left.\sec 2\theta ={a}^{2},\theta \in P\right\}\). Then \(n(S)\) is:
[JEE Main 2026, 2 Apr (Shift 2)]
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Let \(f(x)=6+16 \cos \left(\frac{\pi}{3}-x\right) \cos \left(\frac{\pi}{3}+x\right) \cos x \sin 3 x \cos 6 x\) if range of \(f(x)\) is \([\alpha, \beta]\) then distance of \((\alpha, \beta)\) from \(3 x+4 y+12=0\) is
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The sum of all values of \(\theta \in[0,2 \pi]\) satisfying \(2 \sin ^2 \theta=\cos 2 \theta\) and \(2 \cos ^2 \theta=3 \sin \theta\) is
[JEE Main 2025, 22 Jan (Shift 2)]
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The sum of all values of \(\theta \in[0,2 \pi]\) satisfying \(2 \sin ^2 \theta=\cos 2 \theta\) and \(2 \cos ^2 \theta=3 \sin \theta\) is
[JEE Main 2025, 22 Jan (Shift 2)]
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If \(\tan 15^{\circ}\) and \(\tan 30^{\circ}\) are the roots of the equation \(x^2+p x+q=0\), then \(p q=\) :
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If \(\tan 15^{\circ}\) and \(\tan 30^{\circ}\) are the roots of the equation \(x^2+p x+q=0\), then \(p q=\) :
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Evaluate: \(\sum _{r=1}^{13}\frac{1}{\sin \left[\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right]\sin \left[\frac{\pi }{4}+\frac{r\pi }{6}\right]}\)
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Let \(\frac{\pi }{2} [JEE Advanced 2024]
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Number of solutions of \(\sqrt{3}\cos 2\theta +8\cos \theta +3\sqrt{3}=0,\theta \in [-3\pi ,2\pi ]\) is:
[JEE Main 2026, 23 Jan (Shift 1)]
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Let \(\frac{\pi }{2}<\theta <\pi\) and \(\text{cot}\theta =−\frac{1}{2\sqrt{2}}\). Then the value of \(\text{sin}\left(\frac{15\theta }{2}\right)\left(\text{cos}8\theta +\text{sin}8\theta \right)\) \(+\text{cos}\left(\frac{15\theta }{2}\right)\left(\text{cos}8\theta −\text{sin}8\theta \right)\) is equal to
[JEE Main 2026, 23 Jan (Shift 2)]
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If \(\sin x=-\frac{3}{5}\), where \(\pi [JEE Main 2024, 8 Apr (Shift 1)]
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If \(\sin x=-\frac{3}{5}\), where \(\pi [JEE Main 2024, 8 Apr (Shift 1)]
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If \(\sum_{r=1}^{13}\left\{\frac{1}{\sin \left(\frac{\pi}{4}+(r-1) \frac{\pi}{6}\right) \sin \left(\frac{\pi}{4}+\frac{r \pi}{6}\right)}\right\}=a \sqrt{3}+b, a, b \in Z\), then \(a^2+b^2\) is equal to :
[JEE Main 2025, 28 Jan (Shift 2)]
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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 :
[JEE Main 2024, 8 Apr (Shift 2)]
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The sum of the solutions \(x \in R\) of the equation \(\frac{3 \cos 2 x+\cos ^3 2 x}{\cos ^6 x-\sin ^6 x}=x^3-x^2+6\) is
[JEE Main 2024, 29 Jan (Shift 2)]
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The value of \(\left(\sin 70^\circ \right)\left(\cot 10^\circ \cot 70^\circ -1\right)\) is
[JEE Main 2025, 23 Jan (Shift 1)]
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Let \(f(x)=6+16 \cos \left(\frac{\pi}{3}-x\right) \cos \left(\frac{\pi}{3}+x\right) \cos x \sin 3 x \cos 6 x\) if range of \(f(x)\) is \([\alpha, \beta]\) then distance of \((\alpha, \beta)\) from \(3 x+4 y+12=0\) is
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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 \(n \in N\), then \(\sum_{k=1}^n \frac{k}{2^k}\) is equal to :
[JEE Main 2024, 27 Jan (Shift 2)]
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If \(\sin x+{\sin }^{2}x=1,x\in \left(0,\frac{\pi }{2}\right)\), then
\(\left({\cos }^{12}x+{\tan }^{12}x\right)+3\left({\cos }^{10}x+{\tan }^{10}x+{\cos }^{8}x+{\tan }^{8}x\right)+\left({\cos }^{6}x+{\tan }^{6}x\right)\) is equal to:
[JEE Main 2025, 29 Jan (Shift 2)]
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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
[JEE Main 2024, 1 Feb (Shift 2)]
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For \(\alpha, \beta \in(0, \frac{\pi} {2})\), 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
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Let \(f(x)=6+16 \cos \left(\frac{\pi}{3}-x\right) \cos \left(\frac{\pi}{3}+x\right) \cos x \sin 3 x \cos 6 x\) if range of \(f(x)\) is \([\alpha, \beta]\) then distance of \((\alpha, \beta)\) from \(3 x+4 y+12=0\) is
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If \(\theta \in[0,2 \pi]\) satisfying the system of equations \(2 \sin ^2 \theta=\cos 2 \theta\) and \(2 \cos ^2 \theta=3 \sin \theta\). Then the sum of all real values of \(\theta\) is
[JEE Main 2025]
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The sum of all value of x in [0, 2π], for which \(\text{sin}\left(x\right)+\text{sin}\left(2x\right)+\text{sin}\left(3x\right)+\text{sin}\left(4x\right)=0\) is equal to:
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For \(\alpha, \beta \in(0, \frac{\pi} {2})\), 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
[JEE Main 2024, 30 Jan (Shift 2)]
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Let \(\frac{\pi }{2}<\theta <\pi\) and \(\text{cot}\theta =−\frac{1}{2\sqrt{2}}\). Then the value of \(\text{sin}\left(\frac{15\theta }{2}\right)\left(\text{cos}8\theta +\text{sin}8\theta \right)\) \(+\text{cos}\left(\frac{15\theta }{2}\right)\left(\text{cos}8\theta −\text{sin}8\theta \right)\) is equal to
[JEE Main 2026, 23 Jan (Shift 2)]
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\(96 \cos \frac{\pi}{33} \cos \frac{2 \pi}{33} \cos \frac{4 \pi}{33} \cos \frac{8 \pi}{33} \cos \frac{16 \pi}{33}\) is equal to
[JEE Main 2023, 10 Apr (Shift 1)]
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Let in a right angled triangle, the smallest angle be \(\theta\). If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then \(\sin \theta\) is equal to:
[JEE Main 2021, 20 Jul (Shift 2)]
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The range of the function
\[\begin{aligned}f(x)=\log _{\sqrt{5}}\left(3+\cos \left(\frac{3 \pi}{4}\right.\right. & +x)+\cos \left(\frac{\pi}{4}+x\right) \\& \left.+\cos \left(\frac{\pi}{4}-x\right)-\cos \left(\frac{3 \pi}{4}-x\right)\right)\end{aligned}\]
[JEE Main 2021, 1 Sep (Shift 2)]
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Let \( S=\left\{x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right): 9^{1-\tan ^{2} x}+9^{\tan ^{2} x}=10\right\} \) and \( \beta=\sum_{x \in S} \tan ^{2}\left(\frac{x}{3}\right) \), then \( \frac{1}{6}(\beta-14)^{2} \) is equal to
[JEE Main 2023, 10 Apr (Shift 2)]
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If \( e^{\left(\cos ^{2} x+\cos ^{4} x+\cos ^{6} x+\ldots \infty\right) \log _{e} 2} \) satisfies the equation \( t^{2}-9 t+8=0 \), then the value of \( \frac{2 \sin x}{\sin x+\sqrt{3} \cos x} \) \( \left(0 [JEE Main 2021, 24 Feb (Shift 1)]
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\(\operatorname{cosec} 18^{\circ}\) is a root of the equation:
[JEE Main 2021, 31 Aug (Shift 1)]
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Let \(x+y=3−\cos 4\theta\) and \(x−y=4\sin 2\theta\) then the greatest of \(xy\) is
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\(\frac{1+\sin A−\cos A}{1+\sin A+\cos A}=\)
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If \( x=\sum_{n=0}^{\infty}(-1)^{n} \tan ^{2 n} \theta \) and \( y=\sum_{n=0}^{\infty} \cos ^{2 n} \theta \) for \( 0<\theta<\frac{\pi}{4} \), then
[JEE Main 2020, 9 Jan (Shift 2)]
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If \( \mathrm{L}=\sin ^{2}\left(\frac{\pi}{16}\right)-\sin ^{2}\left(\frac{\pi}{8}\right) \) and \( \mathrm{M}=\cos ^{2} \)\( \left(\frac{\pi}{16}\right)-\sin ^{2}\left(\frac{\pi}{8}\right) \), then
[JEE Main 2020, 5 Sep (Shift 2)]
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If \(15{\sin }^{4}\alpha +10{\cos }^{4}\alpha =6\), for some \(\alpha \in R\), then the value of \(27{\sec }^{6}\alpha +8{\csc }^{6}\alpha\) is equal to:
[JEE Main 2021, 18 Mar (Shift 2)]
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The value of \(\sin \left[n \pi+(-1)^n \frac{\pi}{4}\right], \mathrm{n} \in \mathrm{I}\) is
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The number of elements in the set \(S=\left\{\theta \in[0,2 \pi]: 3 \cos ^4 \theta-5 \cos ^2 \theta-2 \sin ^6 \theta+2=0\right\}\) is
[JEE Main 2023, 11 Apr (Shift 1)]
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The value of \(36\left(4 \cos ^2 9^{\circ}-1\right)\left(4 \cos ^2 27^{\circ}-1\right)\left(4 \cos ^2 81^{\circ}-1\right)\) \(\left(4 \cos ^2 243^{\circ}-1\right)\) is
[JEE Main 2023, 8 Apr (Shift 2)]
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Consider an obtuse angled triangle \(ABC\) in which the difference between the largest and the smallest angle is \(\frac{\pi }{2}\) and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1 .
Then the inradius of the triangle \(ABC\) is ____
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The set of all values of \(\lambda\) for which the equation cos2 2x – 2sin4 x – 2cos2 x = \(\lambda\) has a real solution x, is
[JEE Main 2023, 29 Jan (Shift 2)]
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The angle of elevation of the summit of a mountain from a point on the ground is \( 45^{\circ} \). After climbing up one \( \mathrm{km} \) towards the summit at an inclination of \( 30^{\circ} \) from the ground, the angle of elevation of the summit is found to be \( 60^{\circ} \). Then the height (in km) of the summit from the ground is :
[JEE Main 2020, 6 Sep (Shift 2)]
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If \(0 [JEE Main 2021, 25 Feb (Shift 2)]
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Number of solutions of the equation \( 32^{\tan ^{2} x}+32^{\sec ^{2} x}=81,0 \leq x \leq \frac{\pi}{4} \) is
[JEE Main 2021, 31 Aug (Shift 2)]
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If the solution of the equation \(\log _{\cos x} \cot x+4 \log _{\sin x} \tan x=1, x \in\left(0, \frac{\pi}{2}\right)\), is \(\sin ^{-1}\left(\frac{\alpha+\sqrt{\beta}}{2}\right)\), where \(\alpha, \beta\) are integers, then \(\alpha+\beta\) is equal to:
[JEE Main 2023, 30 Jan (Shift 1)]
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The value of \({\cos }^{2}10^\circ -\cos 10^\circ \cos 50^\circ +{\cos }^{2}50^\circ\) is
[JEE Main 2019, 9 Apr (Shift 1)]
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If \(\ \mathbf{y}=\frac{2 \sin \alpha}{1+\cos \alpha+\sin \alpha} \), then value of \(\ \frac{1-\cos \alpha+\sin \alpha}{1+\sin \alpha} \) is
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If \({\text{sec}}^{2}\theta =\frac{4}{3}\), then the general value of θ is (n \(\in\) Z)
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The equation \(\sin ^4 x-(k+2) \sin ^2 x-(k+3)=0\) possesses a solution if
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Let \(x+y=3-\cos 4 \theta\) and \(x-y=4 \sin 2 \theta\) then the greatest of \(xy\) is
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If \( 15 \sin ^{4} \alpha+10 \cos ^{4} \alpha=6 \), for some \( \alpha \in R \), then the value of \( 27 \sec ^{6} \alpha+8 \operatorname{cosec}^{6} \alpha \) is equal to:
[JEE Main 2021, 18 Mar (Shift 2)]
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The values of \( \mathrm{x} \) in \( (0, \pi) \) satisfying the equation \( \left|\begin{array}{ccc}1+\sin ^{2} x & \sin ^{2} x & \sin ^{2} x \\ \cos ^{2} x & 1+\cos ^{2} x & \cos ^{2} x \\ 4 \sin 2 x & 4 \sin 2 x & 1+4 \sin 2 x\end{array}\right|=0 \), are
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If \(\ \mathbf{A}+\mathbf{B}+\mathbf{C}=\frac{\pi}{2} \) then
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The least positive non-integral solution of the equation \(\sin \pi\left(x^2+x\right)=\sin \pi x^2\) is
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The angle of elevation of the top \(P\) of a tower from the feet of one person standing due South of the tower is \(45^{\circ}\) and from the feet of another person standing due west of the tower is \(30^{\circ}\). If the height of the tower is 5 meters, then the distance (in meters) between the two persons is equal to
[JEE Main 2023, 11 Apr (Shift 2)]
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If \(n\) is the number of solutions of the equation \(2\cos x\text{ }\left(4\sin \left(\frac{\pi }{4}+x\right)\text{ }\sin \left(\frac{\pi }{4}−x\right)−1\right)\)= 1, \(x\in \left[0,\pi \right]\) and \(S\) is the sum of all these solutions, then the ordered pair \((n,S)\) is
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\(\text{ If }\sin \theta +\cos \theta =\frac{1}{2}\text{, then }16(\sin (2\theta )+\cos (4\theta )+\sin (6\theta ))\text{ is equal to : }\)
[JEE Main 2021, 27 Jul (Shift 1)]
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The number of solutions of the equation \( x+2 \tan x=\frac{\pi}{2} \) in the interval \( [0,2 \pi] \) is:
[JEE Main 2021, 17 Mar (Shift 2)]
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In a triangle \(A B C\), if \(\cos A+2 \cos B+\cos C=2\) and the lengths of the sides opposite to the angles \(A\) and \(C\) are 3 and 7 respectively, then \(\cos A-\cos C\) is equal to
[JEE Main 2023, 12 Apr (Shift 1)]
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The angle of elevation of the top \(P\) of a tower from the feet of one person standing due South of the tower is \(45^{\circ}\) and from the feet of another person standing due west of the tower is \(30^{\circ}\). If the height of the tower is 5 meters, then the distance (in meters) between the two persons is equal to
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Consider an obtuse angled triangle \(ABC\) in which the difference between the largest and the smallest angle is \(\frac{\pi }{2}\) and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1 .
Let \(a\) be the area of the triangle \(ABC\). Then the value of \((64 a)^2\) is ____
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A man is observing from the top of tower a boat speeding towards the lower from a certain point \( \mathrm{A} \), with uniform speed. At that point, angle of depression of boat with man's eye is \( 30^{\circ} \) (ignore mans height). After Sailing for 20 seconds, towards the base of tower (same level as of water), the boat has reached a point \( B \), where angle of depression is \( 45^{\circ} \). Then time taken by boat (in seconds) from \(B\) to reach base of tower is:
[JEE Main 2021, 25 Feb (Shift 1)]
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The set of all values of \(\lambda\) for which the equation \(\cos ^2 2 x-2 \sin ^4 x-2 \cos ^2 x=\lambda\)
[JEE Main 2023, 29 Jan (Shift 2)]
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If the equation \( \cos ^{4} \theta+\sin ^{4} \theta+\lambda=0 \) has real solutions for \( \theta \), then \( \lambda \) lies in the interval
[JEE Main 2020, 2 Sep (Shift 2)]
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If \(\cot \alpha=1\) and \(\sec \beta=-\frac{5}{3}\), where \(\pi<\alpha<\frac{3 \pi}{2}\) and, \(\frac{\pi}{2}\) \(<\beta<\pi\), then the value of \(\tan (\alpha+\beta)\) and the quadrant in which \(\alpha+\beta\) lies, respectively are
[JEE Main 2022, 28 June (Shift 2)]
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\(\text{ The value of }36\left(4{\cos }^{2}9^\circ -1\right)\left(4{\cos }^{2}27^\circ -1\right)\left(4{\cos }^{2}81^\circ -1\right)\left(4{\cos }^{2}243^\circ -1\right)\text{ is }\)
[JEE Main 2023, 08 Apr (Shift 2)]
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The value of
\(2\sin \left(\frac{\pi }{8}\right)\sin \left(\frac{2\pi }{8}\right)\sin \left(\frac{3\pi }{8}\right)\sin \left(\frac{5\pi }{8}\right)\sin \left(\frac{6\pi }{8}\right)\sin \left(\frac{7\pi }{8}\right)\text{ is : }\)
[JEE Main 2021, 26 Aug (Shift 2)]
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Let \(S=\left\{x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right): 9^{1-\tan ^2 x}+9^{\tan ^2 x}=10\right\}\) and \(\beta=\sum_{x \in s} \tan ^2\left(\frac{x}{3}\right)\), then \(\frac{1}{6}(\beta-14)^2\) is equal to
[JEE Main 2023, 10 Apr (Shift 2)]
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All possible values of \( \theta \in[0,2 \pi] \) for which \( \sin 2 \theta+\tan 2 \theta \) \( >0 \) lie in:
[JEE Main 2021, 25 Feb (Shift 1)]
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The value of \( \cot \left(\frac{\pi}{24}\right) \) is:
[JEE Main 2021, 25 Jul (Shift 2)]
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The value of \( 2 \sin \left(\frac{\pi}{8}\right) \sin \left(\frac{2 \pi}{8}\right) \sin \left(\frac{3 \pi}{8}\right) \sin \left(\frac{5 \pi}{8}\right) \) \( \sin \left(\frac{6 \pi}{8}\right) \sin \left(\frac{7 \pi}{8}\right) \) is
[JEE Main 2021, 26 Aug (Shift 2)]
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If \(\tan 15^{\circ}+\frac{1}{\tan 75^{\circ}}+\frac{1}{\tan 105^{\circ}}+\tan 195^{\circ}=2 a\), then the value of \(\left(a+\frac{1}{a}\right)\) is:
[JEE Main 2023, 30 Jan (Shift 1)]
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Let \(\frac{\sin A}{\sin B}=\frac{\sin (A-C)}{\sin (C-B)}\), where A, B, C are angles of a triangle A B C. If the lengths of the sides opposite these angles are a , b , c respectively, then :-
[JEE Main 2021, 27 Aug (Shift 1)]
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If \(\tan \left(\frac{\pi}{9}\right), x, \tan \left(\frac{7 \pi}{18}\right)\) are in A.P and \(\tan \left(\frac{\pi}{9}\right), y, \tan \left(\frac{5 \pi}{18}\right)\) are also in A.P
Then, \(|x-2 y|=\)
[JEE Main 2021, 27 Jul (Shift 2)]
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If for \( x \in\left(0, \frac{\pi}{2}\right), \log _{10} \sin x+\log _{10} \cos x=-1 \) and \( \log _{10}(\sin x+\cos x)=\frac{1}{2}\left(\log _{10} n-1\right), n>0 \) then the value of \( n \) is equal to:
[JEE Main 2021, 16 Mar (Shift 1)]
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The number of solutions of \(\sin ^7 x+\cos ^7 x=1, x \in[0,4 \pi]\) is equal to :
[JEE Main 2021, 22 Jul (Shift 2)]
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The value of \( \cos ^{3}\left(\frac{\pi}{8}\right) \cdot \cos \left(\frac{3 \pi}{8}\right)+\sin ^{3}\left(\frac{\pi}{8}\right) \cdot \sin \left(\frac{3 \pi}{8}\right) \) is :
[JEE Main 2020, 9 Jan (Shift 1)]
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The sum of the solution of the equation \( \frac{\cos x}{1+\sin x}=|\tan 2 x|, x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)-\left\{\frac{\pi}{4},-\frac{\pi}{4}\right\} \) is:
[JEE Main 2021, 26 Aug (Shift 1)]
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The equation \({\sin }^{4}x−\left(k+2\right)\text{ }{\sin }^{2}x−\left(k+3\right)=0\) possesses a solution if
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If \(\tan 15^\circ +\frac{1}{\tan 75^\circ }+\frac{1}{\tan 105^\circ }+\tan {195}^{^\circ }=2a,\) then the value of \(\left(a+\frac{1}{a}\right)\) is equal to
[JEE Main 2023, 30 Jan (Shift 1)]
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Let \(f: R \rightarrow R\) be defined as \(f(x+y)+f(x-y)=2 f(x) f(y), f\left(\frac{1}{2}\right)=-1\) and \(f(0) \neq 0 .\) Then, the value of \(\sum_{k=1}^{20} \frac{1}{\sin (k) \sin (k+f(k))}\) is equal to:
[JEE Main 2021, 27 Jul (Shift 2)]
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If \( \mathrm{n} \) is the number of solutions of the equation \( 2 \cos x\left(4 \sin \left(\frac{\pi}{4}+x\right) \sin \left(\frac{\pi}{4}-x\right)-1\right)=1 \), \( x \in[0, \pi] \) and \( \mathrm{S} \) is the sum of all these solutions, then the ordered pair \( (n, S) \) is
[JEE Main 2021, 1 Sep (Shift 2)]
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If \( \sin \theta+\cos \theta=\frac{1}{2} \), then \( 16(\sin (2 \theta)+\cos (4 \theta) \)\( +\sin (6 \theta))= \)
[JEE Main 2021, 27 Jul (Shift 1)]
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Sum of all values of \( x \) in \( [0,2 \pi] \) for which \( \sin x+\sin 2 x+\sin 3 x+\sin 4 x=0 \), is equal to:
[JEE Main 2021, 25 Jul (Shift 1)]
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The number of solutions of \( \sin ^{7} x+\cos ^{7} x=1 \), \( x \in[0,4 \pi] \) is equal to
[JEE Main 2021, 22 Jul (Shift 2)]
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The sum of solutions of the equation \(\frac{\cos x}{1+\sin x}=|\tan 2x|\),\(x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)-\left\{\frac{\pi }{4},-\frac{\pi }{4}\right\}\) is:
[JEE Main 2021, 26 Aug (Shift 1)]
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The least positive non-integral solution of the equation \(\sin \pi\left(x^2+x\right)=\sin \pi x^2\) is
JEE MAINS[2020]
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Let \(f(\theta)=3\left(\sin ^4\left(\frac{3 \pi}{2}-\theta\right)+\sin ^4(3 \pi+\theta)\right)-2\left(1-\sin ^2 2 \theta\right)\) and \(S=\left\{\theta \in[0, \pi]: f^{\prime}(\theta)=-\frac{\sqrt{3}}{2}\right\}\). If \(4 \beta=\sum_{\theta \in S} \theta\), then \(f(\beta)\) is equal to
[JEE Main 2023, 29 Jan (Shift 1)]
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Two vertical poles are \( 150 \mathrm{~m} \) apart and height of one is three times that of other. If from the middle point of the line joining their feet, an observer finds the angle of elevation of their tops to be complementary, then height of shorter pole (in \( \mathrm{m} \) ) is:
[JEE Main 2021, 24 Feb (Shift 1)]
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If \( x=\sum_{n=0}^{\infty}(-1)^{n} \tan ^{2 n} \theta \) and \( y=\sum_{n=0}^{\infty} \cos ^{2 n} \theta \) for \( 0<\theta<\frac{\pi}{4} \), then
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Let \(f: R \rightarrow R\) be defined as \(f(x+y)+f(x-y)=2 f(x) f(y), f\left(\frac{1}{2}\right)=-1\). Then, the value of \(\sum_{k=1}^{20} \frac{1}{\sin (k) \sin (k+f(k))}\) is equal to:
[JEE Main 2021, 27 Jul (Shift 2)]
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