For the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ with eccentricity $e$, write out a value of $e$ that satisfies the condition ``the line $y = kx$ intersects the hyperbola at four distinct points'' and give one such value $\_\_\_\_$ .

Throughout this part $A$ and $B$ denote real symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We denote by $\lambda _ { 1 } \leqslant \lambda _ { 2 }$ the eigenvalues of $A$.
We consider the set $\Gamma \subset \mathbb { R } ^ { 2 }$ defined by the equation $\langle A X , X \rangle = 1$.
II.B.1) Characterize the conditions on the $\lambda _ { i }$ for which this set is: a) empty; b) the union of two lines; c) an ellipse; d) a hyperbola.
II.B.2) Represent on the same figure the sets $\Gamma$ obtained for $A$ diagonal with $\lambda _ { 1 } \in \{ - 4 , - 1,0,1 / 4,1 \}$ and $\lambda _ { 2 } = 1$.

138- The rotation matrix $A$, with the relation $\begin{bmatrix} x \\ y \end{bmatrix} = A \cdot \begin{bmatrix} x' \\ y' \end{bmatrix}$, transforms the conic equation $5x^2 + 24xy - 2y^2 = 12$ into standard form with respect to $x'$ and $y'$. What is the tangent of the rotation angle?
(1) $\dfrac{2}{3}$ (2) $\dfrac{3}{4}$ (3) $\dfrac{4}{3}$ (4) $\dfrac{3}{2}$

140- The area of the graph of the curve $0 = 3x^2 + \sqrt{3}xy + 2y^2 - 10 = 0$ is:
\[ (1)\quad 6\pi \qquad (2)\quad 7\pi \qquad (3)\quad \frac{10\pi}{3} \qquad (4)\quad \frac{20\pi}{\sqrt{21}} \]

Match the conics in Column I with the statements/expressions in Column II.
Column I
(A) Circle
(B) Parabola
(C) Ellipse
(D) Hyperbola
Column II
(p) The locus of the point $( h , k )$ for which the line $h x + k y = 1$ touches the circle $x ^ { 2 } + y ^ { 2 } = 4$
(q) Points $z$ in the complex plane satisfying $| z + 2 | - | z - 2 | = \pm 3$
(r) Points of the conic have parametric representation $x = \sqrt { 3 } \left( \frac { 1 - t ^ { 2 } } { 1 + t ^ { 2 } } \right) , y = \frac { 2 t } { 1 + t ^ { 2 } }$
(s) The eccentricity of the conic lies in the interval $1 \leq x < \infty$
(t) Points $z$ in the complex plane satisfying $\operatorname { Re } ( z + 1 ) ^ { 2 } = | z | ^ { 2 } + 1$

For the hyperbola $\frac { x ^ { 2 } } { \cos ^ { 2 } \alpha } - \frac { y ^ { 2 } } { \sin ^ { 2 } \alpha } = 1$, which of the following remains constant when $\alpha$ varies?
(1) eccentricity
(2) directrix
(3) abscissae of vertices
(4) abscissae of foci

Consider the following two equations
$$\begin{gathered} \left( \log _ { 4 } 2 \sqrt { x } \right) ^ { 2 } + \left( \log _ { 4 } 2 \sqrt { y } \right) ^ { 2 } = \log _ { 2 } ( \sqrt [ 4 ] { 2 } \cdot x \sqrt { y } ) \\ \sqrt [ 3 ] { x } \cdot \sqrt [ 4 ] { y } = 2 ^ { k } \end{gathered}$$
We are to find the range of values which the constant $k$ can take so that there exist positive real numbers $x , y$ which satisfy (1) and (2) simultaneously.
Set $X = \log _ { 2 } x$ and $Y = \log _ { 2 } y$. Let us express (1) and (2) in terms of $X$ and $Y$. First we consider (1). Since
$$\log _ { 4 } 2 \sqrt { x } = \frac { \log _ { 2 } x + \mathbf { A } } { \mathbf { B } }$$
and
$$\log _ { 2 } ( \sqrt [ 4 ] { 2 } \cdot x \sqrt { y } ) = \frac { \mathbf { C } } { \mathbf { D } } + \log _ { 2 } x + \frac { \log _ { 2 } y } { \mathbf { E } } ,$$
(1) reduces to
$$( X - \mathbf { F } ) ^ { 2 } + ( Y - \mathbf { G } ) ^ { 2 } = \mathbf { H I } .$$
In the same way, (2) reduces to
$$4 X + \mathbf { J } Y = \mathbf { K } \mathbf { L } k .$$
Since the distance $d$ from the center of the circle (3) to the straight line (4) on the $XY$-plane is given by
$$d = \frac { | \mathbf { M N } - \mathbf { O P } k | } { \mathbf { Q } } ,$$
the range of values which $k$ can take is
$$\mathbf { R } \leq k \leqq \mathbf { S } .$$

7. Let the ellipses $\Gamma_{1}: \frac{x^{2}}{5^{2}} + \frac{y^{2}}{3^{2}} = 1$, $\Gamma_{2}: \frac{x^{2}}{5^{2}} + \frac{y^{2}}{3^{2}} = 2$, $\Gamma_{3}: \frac{x^{2}}{5^{2}} + \frac{y^{2}}{3^{2}} = \frac{2x}{5}$ have major axis lengths $l_{1}$, $l_{2}$, $l_{3}$ respectively. Which of the following options is correct?
(1) $l_{1} = l_{2} = l_{3}$
(2) $l_{1} = l_{2} < l_{3}$
(3) $l_{1} < l_{2} < l_{3}$
(4) $l_{1} = l_{3} < l_{2}$
(5) $l_{1} < l_{3} < l_{2}$

II. Multiple-Choice Questions (25 points)

Instructions: For questions 8 through 12, each of the five options is independent, and at least one option is correct. Select all correct options and mark them on the ``Answer Sheet''. No points are deducted for incorrect answers. Full marks (5 points) are awarded for all five options correct; 2.5 points are awarded if only one option is incorrect; no points are awarded if two or more options are incorrect.

A flashlight's light beam forms a right circular cone with a light divergence angle of $60^{\circ}$, as shown in the figure. The wall is perpendicular to the floor, and their intersection is a straight line $L$. The flashlight is directed perpendicular to $L$, meaning the axis of the right circular cone is perpendicular to $L$. If the edge of the light beam on the wall is part of a parabola, then the edge of the light beam on the floor is part of which of the following shapes?
(1) Two intersecting lines (2) Circle (3) Parabola (4) Ellipse with unequal major and minor axes (5) Hyperbola

From the following conic sections on the coordinate plane, select the options that intersect all vertical lines.
(1) $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$
(2) $\frac{x^{2}}{9} - \frac{y^{2}}{4} = 1$
(3) $-\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$
(4) $y = \frac{4}{9}x^{2}$
(5) $x = \frac{4}{9}y^{2}$

In space, two intersecting lines $L , M$ form an angle of $24 ^ { \circ }$ . Rotating $M$ around $L$ one complete revolution generates a right circular cone surface. A plane $E$ is parallel to line $L$. What is the cross-section formed by plane $E$ and this cone surface?
(1) Hyperbola
(2) Parabola
(3) Ellipse (with unequal major and minor axes)
(4) Circle
(5) Two intersecting lines