Given $\cos x + \cos y + \cos z = \frac { 3 \sqrt { } 3 } { 2 }$ and $\sin x + \sin y + \sin z = \frac { 3 } { 2 }$ then show that $x = \frac { \pi } { 6 } + 2 k \pi , y = \frac { \pi } { 6 } + 2 \ell \pi , z = \frac { \pi } { 6 } + 2 m \pi$ for some $k , \ell , m \in \mathbf { Z }$.

For two constants $a$ ($1 \leq a \leq 2$) and $b$, the function $f(x) = \sin(ax + b + \sin x)$ satisfies the following conditions. (가) $f(0) = 0$ and $f(2\pi) = 2\pi a + b$ (나) The minimum positive value of $t$ such that $f'(0) = f'(t)$ is $4\pi$. Let $A$ be the set of all values of $\alpha$ in the open interval $(0, 4\pi)$ where the function $f(x)$ has a local maximum. If $n$ is the number of elements in set $A$ and $\alpha_{1}$ is the smallest element in set $A$, then $n\alpha_{1} - ab = \frac{q}{p}\pi$. What is the value of $p + q$? [4 points]

The number of roots of the equation $x ^ { 2 } + \sin ^ { 2 } x = 1$ in the closed interval $\left[ 0 , \frac { \pi } { 2 } \right]$ is
(a) 0
(b) 1
(c) 2
(d) 3

The number of roots of the equation $x ^ { 2 } + \sin ^ { 2 } x = 1$ in the closed interval $\left[ 0 , \frac { \pi } { 2 } \right]$ is
(a) 0
(b) 1
(c) 2
(d) 3

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

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

Find the minimum value of $$|\sin x + \cos x + \tan x + \cot x + \sec x + \operatorname{cosec} x|$$ for real numbers $x$ not multiple of $\pi/2$.

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 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

The total number of real solutions of the equation
$$\theta = \tan ^ { - 1 } ( 2 \tan \theta ) - \frac { 1 } { 2 } \sin ^ { - 1 } \left( \frac { 6 \tan \theta } { 9 + \tan ^ { 2 } \theta } \right)$$
is (Here, the inverse trigonometric functions $\sin ^ { - 1 } x$ and $\tan ^ { - 1 } x$ assume values in $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ and ( $- \frac { \pi } { 2 } , \frac { \pi } { 2 }$ ), respectively.)

(A)

1

(B)

2

(C)

3

(D)

5

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

If the solution of the equation $\log_{\cos x} \cot x + 4\log_{\sin x} \tan x = 1, \quad x \in \left(0, \frac{\pi}{2}\right)$ is $\sin^{-1}\frac{\alpha + \sqrt{\beta}}{2}$, where $\alpha, \beta$ are integers, then $\alpha + \beta$ is equal to:
(1) 3
(2) 5
(3) 6
(4) 4

Let $S = \left\{ x \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right) : 9 ^ { 1 - \tan ^ { 2 } x } + 9 ^ { \tan ^ { 2 } x } = 10 \right\}$ and $\beta = \sum _ { x \in S } \tan ^ { 2 } \frac { x } { 3 }$, then $\frac { 1 } { 6 } ( \beta - 14 ) ^ { 2 }$ is equal to
(1) 16
(2) 8
(3) 64
(4) 32

Let $S = \{ \theta \in [ 0,2 \pi ) : \tan ( \pi \cos \theta ) + \tan ( \pi \sin \theta ) = 0 \}$, then $\sum _ { \theta \in S } \sin ^ { 2 } \left( \theta + \frac { \pi } { 4 } \right)$ is equal to

The number of solutions, of the equation $e^{\sin x} - 2e^{-\sin x} = 2$ is
(1) 2
(2) more than 2
(3) 1
(4) 0

If $2 \sin ^ { 3 } x + \sin 2 x \cos x + 4 \sin x - 4 = 0$ has exactly 3 solutions in the interval $\left[ 0 , \frac { \mathrm { n } \pi } { 2 } \right] , \mathrm { n } \in \mathrm { N }$, then the roots of the equation $x ^ { 2 } + n x + ( n - 3 ) = 0$ belong to :
(1) $( 0 , \infty )$
(2) $( - \infty , 0 )$
(3) $\left( - \frac { \sqrt { 17 } } { 2 } , \frac { \sqrt { 17 } } { 2 } \right)$
(4) $Z$

Given that $x \in [ 0, 2 \pi )$, $$\cos ( 5 x ) = \cos ( 3 x ) \cdot \cos ( 2 x )$$ How many different solutions does the equation have?\ A) 3\ B) 6\ C) 8\ D) 11\ E) 12