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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 $\operatorname{sum}=\frac{13 \pi}{9}$
Aur questions nahi they ya fer jaga nahi tha daal ne ke liye?
provide more questions
Very less Qns
Plss add more
Good questions but add more question from jee main 2017-2020 papers
You could have added more questions……
Very less questions
Good
Kishan Ka Sala Hun
Thank you.
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
Thanks sir
Good app ! Super I like this app the most
2019 2020 ke jameen ke previous year paper with solution Dal do please