Consider the system of equations
$$\left\{ \begin{array} { l } \cos 2 \alpha + \cos 2 \beta = \frac { 4 } { 15 } \\ \cos \alpha \cos \beta = - \frac { 2 \sqrt { 15 } } { 15 } \end{array} \right.$$
where $0 \leqq \alpha \leqq \pi , \quad 0 \leqq \beta \leqq \pi$, and $\alpha < \beta$ with
$$| \cos \alpha | \geqq | \cos \beta |$$
We want to find the values of $\cos \alpha$ and $\cos \beta$.
Using the double angle formula, from (1) we obtain
$$\cos ^ { 2 } \alpha + \cos ^ { 2 } \beta = \frac { \text { アイ } } { \text { ウエ } }$$
Also, from (2), $\cos ^ { 2 } \alpha \cos ^ { 2 } \beta = \frac { \square } { 15 }$.
Therefore, using condition (3),
$$\cos ^ { 2 } \alpha = \frac { \boldsymbol { \text { カ } }} { \boldsymbol { y } } , \quad \cos ^ { 2 } \beta = \frac { \boldsymbol { \text { ク } }} { \boldsymbol { \text { ケ } } }$$
From (2) and the conditions $0 \leqq \alpha \leqq \pi , 0 \leqq \beta \leqq \pi , \alpha < \beta$,
[2] On the coordinate plane, take point $\mathrm { A } \left( 0 , \frac { 3 } { 2 } \right)$, and on the graph of the function $y = \log _ { 2 } x$, take two points $\mathrm { B } \left( p , \log _ { 2 } p \right), \mathrm { C } \left( q , \log _ { 2 } q \right)$. When point C divides segment AB internally in the ratio $1 : 2$, we want to find the values of $p$ and $q$.
By the condition on the domain of the logarithm, $p >$ タ, $q >$ タ. Here, for a logarithm $\log _ { a } b$, $a$ is called the base and $b$ is called the argument.
The coordinates of the point that divides segment AB internally in the ratio $1 : 2$ are expressed in terms of $p$ as
Since this coincides with the coordinates of C,
holds.
Equation (5) can be rewritten as
$$p = \frac { \square } { \square } q \text { 衣 }$$
Solving the system of equations formed by (4) and (6), and noting that $p >$ □タ, $q >$ □タ, we have
$$p = \square \sqrt { \square } , \quad q = \square \sqrt { \square }$$
The $y$-coordinate of C is $\log _ { 2 }$ (□ヒ $\sqrt { }$ □フ). When this value is rounded to one decimal place (rounding the second decimal place), we get □ヘ. Choose the correct answer for □ヘ from the following options (0) through (b). Here, $\log _ { 10 } 2 = 0.3010$ , $\log _ { 10 } 3 = 0.4771 , \log _ { 10 } 7 = 0.8451$.
(0) 0.3 (1) 0.6 (2) 0.9 (3) 1.3 (4) 1.6 (5) 1.9 (6) 2.3 (7) 2.6 (8) 2.9 (9) 3.3 (a) 3.6 (b) 3.9

Let $C$ be the parabola $y = x ^ { 2 } + 1$ with origin O, and let P be the point $(a, 2a)$.
(1) Find the equation of the line passing through point P and tangent to parabola $C$.
The equation of the tangent line at point $(t, t ^ { 2 } + 1)$ on $C$ is
$$y = \square t x - t ^ { 2 } + 1$$
If this line passes through P, then $t$ satisfies the equation
$$t ^ { 2 } - \square a t + \text { エ } a - \text { オ } = 0$$
so $t = \square a -$ カ y. Therefore
When $a \neq$ ケ, there are 2 tangent lines to $C$ passing through P, and their equations are
$$y = ( \square a - \square ) x - \text { シ } a ^ { 2 } + \text { ス } a$$
a ⋯⋯⋯(1)
and
$$y = \text { せ } x$$
(2) Let $\ell$ be the line represented by equation (1) in (1). If the intersection of $\ell$ and the $y$-axis is $\mathrm { R } ( 0 , r )$, then $r = -$ シ $a ^ { 2 } +$ ス $a$. For $r > 0$,
ソ $< a <$ タ
タ and in this case, the area $S$ of triangle OPR is
$$S = \text { チ } \left( a ^ { \text {ツ } } - a \text { 园 } \right)$$
When ソ $< a <$ タ , examining the increase and decrease of $S$ , we find that $S$ attains its maximum value at $a = \frac { \text { ト } } { \text { ナ } }$
(3) When □ソ $< a <$ □タ, the area $T$ of the region enclosed by the parabola $C$, the line $\ell$ in (2), and the two lines $x = 0, x = a$ is
$$T = \frac { 1 } { \square } \text { ハ } a ^ { 3 } - \square a ^ { 2 } + \square$$
ト
In the range $\frac { 1 } { \text { ナ } } \leqq a <$ タ, $T$ is □ヘ.
In the range, $T$ is □ヘ. Choose the correct answer for □ヘ from the following options (0) through (5).
(0) decreasing (1) attains a local minimum but not a local maximum (2) increasing (3) attains a local maximum but not a local minimum (4) constant (5) attains both a local minimum and a local maximum

In the following, all terms of the sequences under consideration are real numbers.
(1) For a geometric sequence $\left\{ s _ { n } \right\}$ with first term 1 and common ratio 2,
$$s _ { 1 } s _ { 2 } s _ { 3 } = \square , \quad s _ { 1 } + s _ { 2 } + s _ { 3 } = \square$$
(2) Let $\left\{ s _ { n } \right\}$ be a geometric sequence with first term $x$ and common ratio $r$. Let $a, b$ be real numbers (with $a \neq 0$), and suppose the first three terms of $\left\{ s _ { n } \right\}$ satisfy
$$\begin{aligned} & s _ { 1 } s _ { 2 } s _ { 3 } = a ^ { 3 } \\ & s _ { 1 } + s _ { 2 } + s _ { 3 } = b \end{aligned}$$
Then
$$x r = \square$$
Furthermore, using (2) and (3), we find the relation satisfied by $r, a, b$:
$$\text { エ } r ^ { 2 } + ( \text { オ } - \text { カ } ) r + \text { 倍 } = 0$$
Since there exists a real number $r$ satisfying (4),
$$\text { ク } a ^ { 2 } + \text { ケ } a b - b ^ { 2 } \leqq 0$$
Conversely, when $a, b$ satisfy (5), we can find the values of $r, x$ using (3) and (4).
(3) When $a = 64 , b = 336$, consider the geometric sequence $\left\{ s _ { n } \right\}$ satisfying conditions (1) and (2) in (2) with common ratio greater than 1. Using (3) and (4), we find the common ratio $r$ and first term $x$ of $\left\{ s _ { n } \right\}$: $r = \square , x =$ サシ.
Using $\left\{ s _ { n } \right\}$, define the sequence $\left\{ t _ { n } \right\}$ by
$$t _ { n } = s _ { n } \log _ { \square } s _ { n } \quad ( n = 1,2,3 , \cdots )$$
Then the general term of $\left\{ t _ { n } \right\}$ is $t _ { n } = ( n +$ ス $) \cdot$ コ $^ { n + }$ セ. The sum $U _ { n }$ of the first $n$ terms of $\left\{ t _ { n } \right\}$ is found by computing $U _ { n } - \square U _ { n }$:

On the coordinate plane, take point $\mathrm { A } ( 2,0 )$, and on the circle centered at origin O with radius 2, take points $\mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F }$ such that points $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F }$ are the vertices of a regular hexagon in order. Here, B is in the first quadrant.
(1) The coordinates of point B are (ア ア (ア $)$, and the coordinates of point D are $( -$ ウ, $0 )$.
(2) Let M be the midpoint of segment BD, and let N be the intersection of line AM and line CD. We want to find $\overrightarrow { \mathrm { ON } }$.
$\overrightarrow { \mathrm { ON } }$ can be expressed in two ways using real numbers $r, s$: $\overrightarrow { \mathrm { ON } } = \overrightarrow { \mathrm { OA } } + r \overrightarrow { \mathrm { AM } } , \overrightarrow { \mathrm { ON } } = \overright