In the triangle $A B C$, the angle $\angle B A C$ is a root of the equation $$\sqrt { 3 } \cos x + \sin x = 1 / 2$$ Then the triangle $A B C$ is
(A) obtuse angled
(B) right angled
(C) acute angled but not equilateral
(D) equilateral

The set of all solutions of the equation $\cos 2 \theta = \sin \theta + \cos \theta$ is given by
(A) $\theta = 0$
(B) $\theta = n \pi + \frac { \pi } { 2 }$, where $n$ is any integer
(C) $\theta = 2 n \pi$ or $\theta = 2 n \pi - \frac { \pi } { 2 }$ or $\theta = n \pi - \frac { \pi } { 4 }$, where $n$ is any integer
(D) $\theta = 2 n \pi$ or $\theta = n \pi + \frac { \pi } { 4 }$, where $n$ is any integer

The number of solutions of the equation $\sin ( 7 x ) + \sin ( 3 x ) = 0$ with $0 \leq x \leq 2 \pi$ is
(A) 9
(B) 12
(C) 15
(D) 18.

In the range $0 \leq x \leq 2 \pi$, the equation $\cos ( \sin ( x ) ) = \frac { 1 } { 2 }$ has
(A) 0 solutions.
(B) 2 solutions.
(C) 4 solutions.
(D) infinitely many solutions.

Let $S$ be the set of those real numbers $x$ for which the identity $$\sum _ { n = 2 } ^ { \infty } \cos ^ { n } x = ( 1 + \cos x ) \cot ^ { 2 } x$$ is valid, and the quantities on both sides are finite. Then
(A) $S$ is the empty set.
(B) $S = \{ x \in \mathbb { R } : x \neq n \pi$ for all $n \in \mathbb { Z } \}$.
(C) $S = \{ x \in \mathbb { R } : x \neq 2 n \pi$ for all $n \in \mathbb { Z } \}$.
(D) $S = \{ x \in \mathbb { R } : x \neq ( 2 n + 1 ) \pi$ for all $n \in \mathbb { Z } \}$.

The set of all solutions of the equation $\cos 2 \theta = \sin \theta + \cos \theta$ is given by
(a) $\theta = 0$.
(b) $\theta = n \pi + \frac { \pi } { 2 }$, where $n$ is any integer.
(c) $\theta = 2 n \pi$ or $\theta = 2 n \pi - \frac { \pi } { 2 }$ or $\theta = n \pi - \frac { \pi } { 4 }$, where $n$ is any integer.
(d) $\theta = 2 n \pi$ or $\theta = n \pi + \frac { \pi } { 4 }$, where $n$ is any integer.

11. The number of integral values of $k$ for which the equation $7 \cos x + 5 \sin x = 2 k +$ 1 has a solution is
(A) 4
(B) 8
(C) 10
(D) 12

The number of solutions of the pair of equations $$2\sin^2\theta - \cos 2\theta = 0$$ $$2\cos^2\theta - 3\sin\theta = 0$$ in the interval $[0, 2\pi]$ is
(A) 0
(B) 1
(C) 2
(D) 4

For $0<\theta<\frac{\pi}{2}$, the solution(s) of $$\sum_{m=1}^{6}\operatorname{cosec}\left(\theta+\frac{(m-1)\pi}{4}\right)\operatorname{cosec}\left(\theta+\frac{m\pi}{4}\right)=4\sqrt{2}$$ is(are)
(A) $\frac{\pi}{4}$
(B) $\frac{\pi}{6}$
(C) $\frac{\pi}{12}$
(D) $\frac{5\pi}{12}$

The number of values of $\theta$ in the interval $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ such that $\theta \neq \frac { \mathrm { n } \pi } { 5 }$ for $\mathrm { n } = 0 , \pm 1 , \pm 2$ and $\tan \theta = \cot 5 \theta$ as well as $\sin 2 \theta = \cos 4 \theta$ is

The number of all possible values of $\theta$, where $0 < \theta < \pi$, for which the system of equations $$\begin{gathered} ( y + z ) \cos 3 \theta = ( x y z ) \sin 3 \theta \\ x \sin 3 \theta = \frac { 2 \cos 3 \theta } { y } + \frac { 2 \sin 3 \theta } { z } \\ ( x y z ) \sin 3 \theta = ( y + 2 z ) \cos 3 \theta + y \sin 3 \theta \end{gathered}$$ have a solution $\left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right)$ with $y _ { 0 } z _ { 0 } \neq 0$, is

For $x \in (0, \pi)$, the equation $\sin x + 2\sin 2x - \sin 3x = 3$ has
(A) infinitely many solutions
(B) three solutions
(C) one solution
(D) no solution

The number of distinct solutions of the equation $$\frac { 5 } { 4 } \cos ^ { 2 } 2 x + \cos ^ { 4 } x + \sin ^ { 4 } x + \cos ^ { 6 } x + \sin ^ { 6 } x = 2$$ in the interval $[ 0,2 \pi ]$ is

Let $f : [ 0,2 ] \rightarrow \mathbb { R }$ be the function defined by
$$f ( x ) = ( 3 - \sin ( 2 \pi x ) ) \sin \left( \pi x - \frac { \pi } { 4 } \right) - \sin \left( 3 \pi x + \frac { \pi } { 4 } \right)$$
If $\alpha , \beta \in [ 0,2 ]$ are such that $\{ x \in [ 0,2 ] : f ( x ) \geq 0 \} = [ \alpha , \beta ]$, then the value of $\beta - \alpha$ is $\_\_\_\_$

$$\lim_{x\rightarrow 2}\left(\frac{\sqrt{1-\cos\{2(x-2)\}}}{x-2}\right)$$
(1) equals $\sqrt{2}$
(2) equals $-\sqrt{2}$
(3) equals $\frac{1}{\sqrt{2}}$
(4) does not exist

The number of solutions of the equation $\sin 2 x - 2 \cos x + 4 \sin x = 4$ in the interval $[ 0,5 \pi ]$ is :
(1) 3
(2) 5
(3) 4
(4) 6

The number of values of $\alpha$ in $[ 0,2 \pi ]$ for which $2 \sin ^ { 3 } \alpha - 7 \sin ^ { 2 } \alpha + 7 \sin \alpha = 2$, is:
(1) 3
(2) 1
(3) 6
(4) 4

If $12 \cot^2\theta - 31 \csc\theta + 32 = 0$, then the value of $\sin\theta$ is:
(1) $\frac{3}{5}$ or $1$
(2) $\frac{2}{3}$ or $-\frac{2}{3}$
(3) $\frac{4}{5}$ or $\frac{3}{4}$
(4) $\pm\frac{1}{2}$

The number of distinct real roots of the equation $\tan^{2}x - \sec^{10}x + 1 = 0$ in the interval $\left(0, \frac{\pi}{3}\right)$ is: (1) 0 (2) 1 (3) 2 (4) 3

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

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) $\frac { 20 } { 9 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 13 } { 9 }$
(4) $\frac { 8 } { 9 }$

If $0 \leq x < \frac{\pi}{2}$, then the number of values of $x$ for which $\sin x - \sin 2x + \sin 3x = 0$, is:
(1) 4
(2) 3
(3) 2
(4) 1

Let $S = \left\{ \theta \in [ - 2 \pi , 2 \pi ] : 2 \cos ^ { 2 } \theta + 3 \sin \theta = 0 \right\}$. Then the sum of the elements of $S$ is:
(1) $\pi$
(2) $\frac { 13 \pi } { 6 }$
(3) $\frac { 5 \pi } { 3 }$
(4) $2 \pi$

The number of solutions of $\sin ^ { 7 } x + \cos ^ { 7 } x = 1 , x \in [ 0,4 \pi ]$ is equal to
(1) 11
(2) 7
(3) 5
(4) 9

The sum of solutions of the equation $\frac { \cos x } { 1 + \sin x } = | \tan 2 x | , x \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right) - \left\{ - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right\}$