For $x$ satisfying the equation $3 \cos 2 x + 17 \cos x = 0$, find the value of $\tan ^ { 2 } x$. [3 points]

When $\sin \theta + 3 \cos \theta = 0$ and $\cos ( \pi - \theta ) > 0$, what is the value of $\sin \theta$? [3 points]
(1) $\frac { 3 \sqrt { 10 } } { 10 }$
(2) $\frac { \sqrt { 10 } } { 5 }$
(3) 0
(4) $- \frac { \sqrt { 10 } } { 5 }$
(5) $- \frac { 3 \sqrt { 10 } } { 10 }$

4. Use a 2B pencil to answer multiple-choice questions, and use a black pen, marker, or ballpoint pen to answer non-multiple-choice questions.
I. Fill-in-the-Blank Questions (Total Score: 48 points, 4 points each)
1. If $\operatorname { tg } \alpha = \frac { 1 } { 2 }$, then $\operatorname { tg } \left( \alpha + \frac { \pi } { 4 } \right) =$ $\_\_\_\_$.
2. A parabola has vertex at $(2,0)$ and directrix $x = -1$. Its focus is at $\_\_\_\_$.
3. Let $A = \left\{ 5 , \log _ { 2 } ( a + 3 ) \right\}$ and $B = \{ a , b \}$. If $A \cap B = \{ 2 \}$, then $A \cup B =$ $\_\_\_\_$.
4. For a geometric sequence $\left\{ a _ { n } \right\} ( n \in \mathbb{N} )$ with common ratio $q = - \frac { 1 } { 2 }$, if $\lim _ { n \rightarrow \infty } \left( a _ { 1 } + a _ { 3 } + a _ { 5 } + \cdots + a _ { 2 n - 1 } \right) = \frac { 8 } { 3 }$, then $a _ { 1 } =$ $\_\_\_\_$.

Let $-\frac{\pi}{6} < \theta < -\frac{\pi}{12}$. Suppose $\alpha_1$ and $\beta_1$ are the roots of the equation $x^2 - 2x\sec\theta + 1 = 0$ and $\alpha_2$ and $\beta_2$ are the roots of the equation $x^2 + 2x\tan\theta - 1 = 0$. If $\alpha_1 > \beta_1$ and $\alpha_2 > \beta_2$, then $\alpha_1 + \beta_2$ equals
(A) $2(\sec\theta - \tan\theta)$
(B) $2\sec\theta$
(C) $-2\tan\theta$
(D) $0$

If $5\tan^2 x - \cos^2 x = 2\cos 2x + 9$, then the value of $\cos 4x$ is
(1) $-\dfrac{3}{5}$
(2) $\dfrac{1}{3}$
(3) $\dfrac{2}{9}$
(4) $-\dfrac{7}{9}$

If an angle $A$ of a $\triangle A B C$ satisfies $5 \cos A + 3 = 0$, then the roots of the quadratic equation $9 x ^ { 2 } + 27 x + 20 = 0$ are
(1) $\sec A , \cot A$
(2) $\sec A , \tan A$
(3) $\tan A , \cos A$
(4) $\sin A , \sec A$

If $\tan A$ and $\tan B$ are the roots of the quadratic equation $3 x ^ { 2 } - 10 x - 25 = 0$, then the value of $3 \sin ^ { 2 } ( A + B ) - 10 \sin ( A + B ) \cos ( A + B ) - 25 \cos ^ { 2 } ( A + B )$ is :
(1) - 25
(2) 10
(3) - 10
(4) 25

Let in a right angled triangle, the smallest angle be $\theta$. If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then $\sin \theta$ is equal to:
(1) $\frac { \sqrt { 5 } + 1 } { 4 }$
(2) $\frac { \sqrt { 5 } - 1 } { 2 }$
(3) $\frac { \sqrt { 2 } - 1 } { 2 }$
(4) $\frac { \sqrt { 5 } - 1 } { 4 }$

Two poles $A B$ of length $a$ metres and $C D$ of length $a + b ( b \neq a )$ metres are erected at the same horizontal level with bases at $B$ and $D$. If $B D = x$ and $\tan \angle A C B = \frac { 1 } { 2 }$, then: (1) $x ^ { 2 } + 2 ( a + 2 b ) x - b ( a + b ) = 0$ (2) $x ^ { 2 } + 2 ( a + 2 b ) x + a ( a + b ) = 0$ (3) $x ^ { 2 } - 2 a x + b ( a + b ) = 0$ (4) $x ^ { 2 } - 2 a x + a ( a + b ) = 0$

Let $\mathrm { S } = \left\{ \theta \in [ - \pi , \pi ] - \left\{ \pm \frac { \pi } { 2 } \right\} : \sin \theta \tan \theta + \tan \theta = \sin 2 \theta \right\}$. If $T = \sum _ { \theta \in S } \cos 2 \theta$, then $T + n ( S )$ is equal to
(1) $7 + \sqrt { 3 }$
(2) 5
(3) $8 + \sqrt { 3 }$
(4) 9

Let a vertical tower $A B$ of height $2 h$ stands on a horizontal ground. Let from a point $P$ on the ground a man can see upto height $h$ of the tower with an angle of elevation $2 \alpha$. When from $P$, he moves a distance $d$ in the direction of $\overrightarrow { A P }$, he can see the top $B$ of the tower with an angle of elevation $\alpha$. If $d = \sqrt { 7 } h$, then $\tan \alpha$ is equal to
(1) $\sqrt { 5 } - 2$
(2) $\sqrt { 3 } - 1$
(3) $\sqrt { 7 } - 2$
(4) $\sqrt { 7 } - \sqrt { 3 }$

On the coordinate plane, $O$ is the origin, and points $A(1,0)$ and $B(-2,0)$ are given. There are also two points $P$ and $Q$ in the upper half-plane satisfying $\overline{AP} = \overline{OA}$, $\overline{BQ} = \overline{OB}$, $\angle POQ$ is a right angle. Let $\angle AOP = \theta$.
The length of line segment $\overline{OP}$ is which of the following options? (Single choice question, 3 points)
(1) $\sin\theta$
(2) $\cos\theta$
(3) $2\sin\theta$
(4) $2\cos\theta$
(5) $\cos 2\theta$

The following question appeared in an examination:
Given that $\tan x = \sqrt { 3 }$, find the possible values of $\sin 2 x$.
A student gave the following answer:
$$\begin{aligned} & \tan x = \sqrt { 3 } \text { so } x = 60 ^ { \circ } \text { and } 2 x = 120 ^ { \circ } \\ & \text { therefore } \sin 2 x = \frac { \sqrt { 3 } } { 2 } \end{aligned}$$
Which one of the following statements is correct?
A $\frac { \sqrt { 3 } } { 2 }$ is the only possible value, and this is fully supported by the reasoning given in the student's answer.
B $\frac { \sqrt { 3 } } { 2 }$ is the only possible value, but the reasoning given should consider other possible values of $x$ for which $\tan x = \sqrt { 3 }$.
C $\frac { \sqrt { 3 } } { 2 }$ is the only possible value, but the reasoning given should consider other possible values of $x$ for which $\sin 2 x = \frac { \sqrt { 3 } } { 2 }$.
D $\frac { \sqrt { 3 } } { 2 }$ is not the only possible value because the reasoning given should have considered other possible values of $x$ for which $\tan x = \sqrt { 3 }$.
E $\frac { \sqrt { 3 } } { 2 }$ is not the only possible value because the reasoning given should have considered other possible values of $x$ for which $\sin 2 x = \frac { \sqrt { 3 } } { 2 }$.

Find the value of
$$\sum_{k=0}^{90} \sin(10 + 90k)^{\circ}$$

ABCD is a square $| \mathrm { AB } | = 3$ units $| \mathrm { BE } | = | \mathrm { CF } | = 1$ unit $m ( \widehat { F A E } ) = x$
According to the given information above, what is the value of $\cot \mathrm { x }$?
A) $\frac { 6 } { 5 }$
B) $\frac { 8 } { 5 }$
C) $\frac { 7 } { 6 }$
D) $\frac { 9 } { 7 }$
E) $\frac { 11 } { 8 }$

Let $a \in \left( \frac { \pi } { 6 } , \frac { \pi } { 4 } \right)$. Given that
$$\begin{aligned} & x = \tan a \\ & y = \tan ( 2 a ) \\ & z = \tan ( 3 a ) \end{aligned}$$
Which of the following is the correct ordering of these numbers?
A) $x < y < z$
B) $x < z < y$
C) $y < x < z$
D) $z < x < y$
E) $z < y < x$

Let $0 < a < \dfrac{\pi}{2}$ and
$$\cos^{2} a - \cos(2a) = \sin(2a)$$
For the value of $a$ satisfying this equality, which of the following is correct?
A) $\tan a = \dfrac{1}{5}$ B) $\cot a = \dfrac{2}{\sqrt{5}}$ C) $\cos a = \dfrac{1}{\sqrt{5}}$ D) $\operatorname{cosec} a = \sqrt{5}$ E) $\sin(2a) = \dfrac{3}{5}$

Let $x, y$ and $z$ be distinct elements of the set $\left\{\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}\right\}$ such that
$$\sin x < \tan y < \sec z$$
Which of the following is the correct ordering of $x$, $y$ and $z$?
A) $x < y < z$ B) $y < x < z$ C) $y < z < x$ D) $z < x < y$ E) $z < y < x$