1. If the value of
\[\sin\frac{\pi}{14}\sin\frac{3\pi}{14}\sin\frac{5\pi}{14}\sin\frac{7\pi}{14}\sin\frac{9\pi}{14}\sin\frac{11\pi}{14}\sin\frac{13\pi}{14}\]
is equal to \[k^{2}\], then k is equal to
a) -1/8
b) 1/8
c) 1/64
d) 1
Explanation:
2. If \[k=\sin\pi/18 \sin5\pi/18\sin7\pi/18\] , then
the numerical value of k is equal to
a) 1/2
b) 1/4
c) 1/8
d) 1/18
Explanation:
3. If A and B are acute positive angles
satisfying the equations \[3\sin^{2} A+2\sin^{2} B=1\] and
\[3\sin 2A-2\sin 2B=0\]
then A + 2B is equal to
a) \[\pi/4\]
b) \[\pi/2\]
c) \[3\pi/4\]
d) \[2\pi/3\]
Explanation: From the given relations, we have
4. If \[\alpha,\beta,\gamma\] are acute angles and \[\cos\theta=\sin\beta/\sin\alpha,\cos\phi=\sin\gamma/\sin\alpha\]
and \[\cos\left(\theta-\phi\right)=\sin\beta\sin\gamma\]
then \[\tan^{2}\alpha-\tan^{2}\beta-\tan^{2}\gamma\] is equal to
a) -1
b) 0
c) 1
d) none of these
Explanation: From the third relation we get
5.If \[A=\begin{bmatrix}\cos^{2}\alpha & \cos\alpha\sin\alpha \\\cos\alpha\sin\alpha &\sin^{2}\alpha \end{bmatrix}\]
and \[A=\begin{bmatrix}\cos^{2}\beta & \cos\beta\sin\beta \\\cos\beta\sin\beta &\sin^{2}\beta \end{bmatrix}\]
are two matrices such that AB is the null matrix, then
a) \[\alpha=\beta\]
b) \[\cos\left(\alpha-\beta\right)=0\]
c) \[\sin\left(\alpha-\beta\right)=0\]
d) none of these
Explanation: AB = 0
6. If \[\tan\beta=\frac{n \sin\alpha \cos\alpha}{1-n\sin^{2}\alpha}\]
, then \[\tan\left(\alpha-\beta\right)\]
is equal to
a) \[n \tan\alpha\]
b) \[\left(1-n\right) \tan\alpha\]
c) \[\left(1+n\right) \tan\alpha\]
d) none of these
Explanation:
7. If \[\frac{\cos\theta}{a}=\frac{\sin\theta}{b}\] , then
\[\frac{a}{\sec2\theta}+\frac{b}{cosec 2\theta}\]
is equal to
a) a
b) b
c) a/b
d) a+b
Explanation:
8. If \[\sin\alpha+\sin\beta+\sin\gamma=0\] and \[\cos\alpha+\cos\beta+\cos\gamma=0\]
, then value of \[\cos\left(\alpha-\beta\right)+\cos\left(\beta-\gamma\right)+\cos\left(\gamma-\alpha\right)\]
a) -3/2
b) -1
c) -1/2
d) 0
Explanation:
9. The value of
\[\sum_{k=1}^{13}\frac{1}{\sin\left(\frac{\pi}{4}+\frac{\left(k-1\right)\pi}{6}\right)\sin\left(\frac{\pi}{4}+\frac{k\pi}{6}\right)}\]
is equal to
a) \[3-\sqrt{3}\]
b) \[2\left(3-\sqrt{3}\right)\]
c) \[2\left(\sqrt{3}-1\right)\]
d) \[2\left(2+\sqrt{3}\right)\]
Explanation:
10. If \[k_{1}=\tan27\theta-\tan\theta\]
and \[k_{2}=\frac{\sin\theta}{\cos3\theta}+\frac{\sin3\theta}{\cos9\theta}+\frac{\sin9\theta}{\cos27\theta}\]
then,
a) \[k_{1}=k_{2}\]
b) \[k_{1}=2k_{2}\]
c) \[k_{1}+k_{2}=2\]
d) \[k_{2}=2k_{1}\]
Explanation: We can write
