Q. If $\frac{\sin ^{4} x}{2}+\frac{\cos ^{4} x}{3}=\frac{1}{5},$ then
(A) $\tan ^{2} x=\frac{2}{3}$
(B) $\frac{\sin ^{8} x}{8}+\frac{\cos ^{8} x}{27}=\frac{1}{125}$
(C) $\tan ^{2} \mathrm{x}=\frac{1}{3}$
(D) $\frac{\sin ^{8} x}{8}+\frac{\cos ^{8} x}{27}=\frac{2}{125}$
[JEE 2009, 4 + 4]
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Sol. (A,B) (a) $\frac{\tan ^{4} x}{2}+\frac{1}{3}=\frac{\sec ^{4} x}{5}$ put $\tan ^{2} \mathrm{x}=\mathrm{t}$ on solving we get $t=2 / 3$ $\Rightarrow \sin ^{2} x=\frac{2}{5} \quad \Rightarrow \quad \cos ^{2} x=\frac{3}{5}$
Q. For $0<\theta<\frac{\pi}{2},$ the solution(s) of $\sum_{m=1}^{6} \csc \left(\theta+\frac{(m-1) \pi}{4}\right) \csc \left(\theta+\frac{m \pi}{4}\right)=4 \sqrt{2}$ is (are)
(A) $\frac{\pi}{4}$
(B) $\frac{\pi}{6}$
(C) $\frac{\pi}{12}$
(D) $\frac{5 \pi}{12}$
[JEE 2009, 4 + 4]
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Sol. (C,D)
Q. The maximum value of the expression $\frac{1}{\sin ^{2} \theta+3 \sin \theta \cos \theta+5 \cos ^{2} \theta}$ is
[JEE 2010,3+3]
Q. Two parallel chords of a circle of radius 2 are at a distance $\sqrt{3}+1$ apart. If the chords subtend at the center, angles of $\frac{\pi}{k}$ and $\frac{2 \pi}{k}$ where $\mathrm{k}>0$, Glue of $[\mathrm{k}]$ is –
[JEE 2010,3+3]
Q. Let $\mathrm{P}=\{\theta: \sin \theta-\cos \theta=\sqrt{2} \cos \theta\}$ and $\mathrm{Q}=\{\theta: \sin \theta+\cos \theta=\sqrt{2} \sin \theta\}$ be two sets. Then
[JEE 2011,3]
[JEE 2011,3]
Q. The value of $\sum_{\mathrm{k}=1}^{13} \frac{1}{\sin \left(\frac{\pi}{4}+\frac{(\mathrm{k}-1) \pi}{6}\right) \sin \left(\frac{\pi}{4}+\frac{\mathrm{k} \pi}{6}\right)}$ is equal to
(A) $3-\sqrt{3}$
(B) $2(3-\sqrt{3})$
(C) $2(\sqrt{3}-1)$
(D) $2(2+\sqrt{3})$
[JEE Advance 2016]
Q. Let $\alpha$ and $\beta$ be nonzero real numbers such that $2(\cos \beta-\cos \alpha)+\cos \alpha \cos \beta=1$. Then which of the following is/are true ?
(A) $\tan \left(\frac{\alpha}{2}\right)-\sqrt{3} \tan \left(\frac{\beta}{2}\right)=0$
(B) $\sqrt{3} \tan \left(\frac{\alpha}{2}\right)-\tan \left(\frac{\beta}{2}\right)=0$
(C) $\tan \left(\frac{\alpha}{2}\right)+\sqrt{3} \tan \left(\frac{\beta}{2}\right)=0$
(D) $\sqrt{3} \tan \left(\frac{\alpha}{2}\right)+\tan \left(\frac{\beta}{2}\right)=0$
[JEE Advance 2017]
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Sol. (A,C)
