PrepBharat

1. If \[\tan x+\tan\left(x+\pi/3\right)+\tan\left(x+2\pi/3\right)=3\]       , then
a) \[\tan x=1\]
b) \[\tan 2x=1\]
c) \[\tan 3x=1\]
d) none of these.

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2. The equation cos \[2x+a\sin x=2a-7\]     possesses a solution if
a) a < 2
b) \[2 \leq a\leq6\]
c) a> 6
d) a is any integer

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3. \[\sin47^{\circ}+\sin61^{\circ}-\sin11^{\circ}-\sin25^{\circ}\]       is equal to
a) \[\sin36^{\circ}\]
b) \[\cos36^{\circ}\]
c) \[\sin7^{\circ}\]
d) \[\cos7^{\circ}\]

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4.If tan \[\alpha=1/7\]   and sin \[\beta=1/\sqrt{10}\]   where \[0<\alpha,\beta<\pi/2\]    , then \[2\beta\] is equal to
a) \[\pi/4-\alpha\]
b) \[3\pi/4-\alpha\]
c) \[\pi/8-\alpha/2\]
d) \[3\pi/8-\alpha/2\]

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5. If \[3\pi/4<\alpha<\pi\]   , then \[\sqrt{2 \cot \alpha+\frac{1}{\sin^{2}\alpha}}\]
is equal to
a) \[1+\cot \alpha\]
b) \[-1-\cot \alpha\]
c) \[1-\cot \alpha\]
d) \[-1+\cot \alpha\]

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6. If \[\cos\left(\theta-\alpha\right)=a\]    and \[\sin\left(\theta-\beta\right)=b\left(0<\theta-\alpha,\theta-\beta<\pi/2\right)\]       , then \[\cos^{2}\left(\alpha-\beta\right)+2b\sin\left(\alpha-\beta\right)\]
is equal to
a) \[4a^{2}b^{2}\]
b) \[a^{2}-b^{2}\]
c) \[a^{2}+b^{2}\]
d) \[-a^{2}b^{2}\]

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7. The expression \[\frac{\cos 6x+6\cos 4x+15\cos 2x+10}{\cos 5x+5\cos 3x+10\cos x}\]
is equal to
a) \[\cos 2x\]
b) \[2\cos x\]
c) \[\cos^{2}x\]
d) \[1+\cos x\]

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8. If \[\tan\left(\pi\cos\theta\right)=\cot\left(\pi\sin\theta\right)\]      then \[\cos\left(\theta-\pi/4\right)\]    is equal to
a) \[\pm\frac{1}{2\sqrt{2}}\]
b) \[\pm\frac{1}{\sqrt{2}}\]
c) \[\pm\sqrt{2}\]
d) \[\pm2\sqrt{2}\]

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9. If \[\tan\theta_{1},\tan\theta_{2},\tan\theta_{3}\tan\theta_{4}\]      are the roots of the equation \[x^{4}-x^{3}\sin2\beta+x^{2}\cos2\beta-x\cos\beta-\sin\beta=0\]
then \[\tan\left(\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}\right)\]     is equal to
a) \[\sin\beta\]
b) \[\cos\beta\]
c) \[\tan\beta\]
d) \[\cot\beta\]

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10. The expression \[\cos^{2}\phi+\cos^{2}\left( a+\phi\right)-2cos a\cos\phi\cos\left( a+\phi\right)\]
is independent of
a) \[\phi\]
b) a
c) both a and \[\phi\]
d) none of a and \[\phi\]

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