Trigonometric Equation – JEE Main Previous Year Question with Solutions
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Q. Let A and B denote the statements $\mathbf{A}: \cos \alpha+\cos \beta+\cos \gamma=0$ $\mathbf{B}: \sin \alpha+\sin \beta+\sin \gamma=0$ If $\cos (\beta-\gamma)+\cos (\gamma-\alpha)+\cos (\alpha-\beta)=-\frac{3}{2},$ then $:-$ (1) Both A and B are true (2) Both A and B are false (3) A is true and B is false (4) A is false and B is true [AIEEE 2009]

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Sol. (1) $\cos (\beta-\gamma)+\cos (\gamma-\alpha)+\cos (\alpha-\beta)=-\frac{3}{2}$ $2 \cos (\beta-\gamma)+2 \cos (\gamma-\alpha)+2 \cos (\alpha-\beta)=-3$ $1+1+1+2(\cos \beta \cos \gamma+\sin \beta \sin \gamma)+2(\cos \gamma \cos \alpha+\sin \gamma \sin \alpha)+2(\cos \alpha \cos \beta+\sin \alpha \sin \beta)$ $=0$ $\left(\sin ^{2} \alpha+\cos ^{2} \alpha\right)+\left(\sin ^{2} \beta+\cos ^{2} \beta\right)+\left(\sin ^{2} \gamma+\cos ^{2} \gamma\right)+2 \cos \alpha \cos \beta+2 \cos \beta \cos \gamma+2 \cos$ $\gamma \cos \alpha$ $+2 \sin \alpha \sin \beta+2 \sin \beta \sin \gamma+2 \sin \gamma \sin \alpha=0$ $\left(\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma+2 \sin \alpha \sin \beta+2 \sin \beta \sin \gamma\right.$ $+2 \sin \gamma \sin \alpha)+\left(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma\right.$ $+2 \cos \alpha \cos \beta+\cos \beta \cos \gamma+\cos \gamma \cos \alpha)=0$ $(\sin \alpha+\sin \beta+\sin \gamma)^{2}+(\cos \alpha+\cos \beta+\cos \gamma)^{2}=0$ Only Possible when $\sin \alpha+\sin \beta+\sin \gamma=0$ $\cos \alpha+\cos \beta+\cos \gamma=0$

Q. The possible values of $\theta \in(0, \pi)$ such that $\sin (\theta)+\sin (4 \theta)+\sin (7 \theta)=0$ are: (1) $\frac{2 \pi}{9}, \frac{\pi}{4}, \frac{4 \pi}{9}, \frac{\pi}{2}, \frac{3 \pi}{4}, \frac{8 \pi}{9}$ (2) $\frac{\pi}{4}, \frac{5 \pi}{12}, \frac{\pi}{2}, \frac{2 \pi}{3}, \frac{3 \pi}{4}, \frac{8 \pi}{9}$ (3) $\frac{2 \pi}{9}, \frac{\pi}{4}, \frac{\pi}{2}, \frac{2 \pi}{3}, \frac{3 \pi}{4}, \frac{35 \pi}{36}$ (4) $\frac{2 \pi}{9}, \frac{\pi}{4}, \frac{\pi}{2}, \frac{2 \pi}{3}, \frac{3 \pi}{4}, \frac{8 \pi}{9}$ [AIEEE 2011]

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Sol. (1) $\sin \theta+\sin 4 \theta+\sin 7 \theta=0$ $2 \sin \left(\frac{\theta+7 \theta}{2}\right) \cos \left(\frac{7 \theta-\theta}{2}\right)+\sin 4 \theta=0$ $\Rightarrow \sin 4 \theta[2 \cos 3 \theta+1]=0$ $\Rightarrow \sin 4 \theta=0 \Rightarrow 4 \theta=0, \pi, 2 \pi, 3 \pi, 4 \pi$ $\Rightarrow \theta=0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3 \pi}{4}, \pi$ but 0 and $\pi$ are not included. and $2 \cos 3 \theta+1=0 \Rightarrow \cos 3 \theta=\frac{-1}{2}$ $\Rightarrow 3 \theta=\frac{2 \pi}{3}, \frac{4 \pi}{3}, \frac{8 \pi}{3}, \frac{10 \pi}{3} \quad \Rightarrow \quad \theta=\frac{2 \pi}{9}, \frac{4 \pi}{9}, \frac{8 \pi}{9}, \frac{10 \pi}{9}$ but $\frac{10 \pi}{9} \notin(0, \pi)$ So, $\theta=\frac{\pi}{4}, \frac{\pi}{2}, \frac{3 \pi}{4}, \frac{2 \pi}{9}, \frac{4 \pi}{9}, \frac{8 \pi}{9}$

Q. If $0 \leq x<2 \pi,$ then the number of real values of $x,$ which satisfy the equation $\cos x+\cos 2 x+\cos 3 x+\cos 4 x=0,$ is : – (1) 9        (2) 3           (3) 5          (4) 7 [JEE Mains 2016]

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Sol. (4) $2 \cos 2 x \cos x+2 \cos 3 x \cos x=0$ $\Rightarrow 2 \cos x(\cos 2 x+\cos 3 x)=0$ $2 \cos x 2 \cos 5 x / 2 \cos x / 2=0$ $x=\frac{\pi}{2}, \frac{3 \pi}{2}, \pi, \frac{\pi}{5}, \frac{3 \pi}{5}, \frac{7 \pi}{5}, \frac{9 \pi}{5}$ 7 Solutions

Q. If sum of all the solutions of the equation $8 \cos x \cdot\left(\cos \left(\frac{\pi}{6}+x\right) \cdot \cos \left(\frac{\pi}{6}-x\right)-\frac{1}{2}\right)=1$ in $[0, \pi]$ is $k \pi,$ then $k$ is equal to : (1) $\frac{13}{9}$ (2) $\frac{8}{9}$ (3) $\frac{20}{9}$ ( 4)$\frac{2}{3}$ [JEE Mains 2016]

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Sol. (1) $8 \cos x\left(\cos ^{2} \frac{\pi}{6}-\sin ^{2} x-\frac{1}{2}\right)=1$ $\Rightarrow 8 \cos x\left(\frac{1}{4}-\left(1-\cos ^{2} x\right)\right)=1$ $\Rightarrow 8 \cos x\left(\cos ^{2} x-\frac{3}{4}\right)=1$ $\Rightarrow 2 \cos 3 x=1 \Rightarrow \cos 3 x=\frac{1}{2}$ $\therefore 3 x+2 n \pi \pm \frac{\pi}{3}, n \in I$ $\Rightarrow \mathrm{x}=\frac{2 \mathrm{n} \pi}{3} \pm \frac{\pi}{9}$ $\ln \mathrm{x} \in[0, \pi]: \mathrm{x}=\frac{\pi}{9}, \frac{2 \pi}{3}+\frac{\pi}{9}, \frac{2 \pi}{3}-\frac{\pi}{9}$ only $\operatorname{sum}=\frac{13 \pi}{9}$

• April 15, 2021 at 2:03 pm

pure 15 saal me 4 questions hi aae hai kya jee main bc?

107
• February 19, 2021 at 8:03 pm

Aur questions nahi they ya fer jaga nahi tha daal ne ke liye?

34
• September 7, 2021 at 6:54 pm

🤣🤣🤣🤣

0
• January 2, 2021 at 11:03 am

provide more questions

10
• October 14, 2020 at 2:39 pm

Very less Qns

3
• October 21, 2020 at 3:46 pm

Good questions but add more question from jee main 2017-2020 papers

9
• October 8, 2020 at 11:52 am

You could have added more questions……

1
• September 23, 2020 at 5:59 pm

Very less questions

2
• September 3, 2020 at 12:04 pm

Good

1
• August 27, 2020 at 8:41 pm

Kishan Ka Sala Hun

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• August 27, 2020 at 11:43 am

Thank you.

2
• August 15, 2020 at 5:44 pm

This app is so worst that it is not useful for anyone because
1.solutions are not clear
2.size of letters are very small

5
• May 29, 2020 at 8:38 am

Thanks sir

1
• May 13, 2020 at 1:32 pm

Good app ! Super I like this app the most

1
• March 3, 2020 at 5:41 pm

2019 2020 ke jameen ke previous year paper with solution Dal do please

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