There is a wooden block where $ACFD$ and $ABED$ are two congruent isosceles trapezoids, and $BCFE$ is a rectangle. Let the projection of point $A$ on line $BC$ be $M$ and its projection on plane $BCFE$ be $P$. Given that $\overline{AD} = 30$, $\overline{CF} = 40$, $\overline{AP} = 15$, and $\overline{BC} = 10$. Place plane $BCFE$ on a horizontal table, and call any plane parallel to $BCFE$ a horizontal plane. Using the fact that the projection of $\overline{AD}$ on plane $BCFE$ has length 30, find $\tan \angle AMP$. (Fill-in-the-blank question, 2 points)

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

An electronic billboard continuously alternates between playing advertisements A and B ($A$, $B$, $A$, $B \ldots$), with each advertisement playing for $T$ minutes (where $T$ is an integer). A person passes by just as advertisement A starts playing. 30 minutes later, the person returns to the location and sees advertisement B just starting to play. Select the options that could be the value of $T$.
(1) $15$ (2) $10$ (3) $8$ (4) $6$ (5) $5$

On a globe with center $O$, there are five points $A$, $B$, $C$, $D$, $E$. Points $A$, $B$, $C$ are all on the equator with longitudes of East $0^{\circ}$, $60^{\circ}$, and $90^{\circ}$ respectively. Points $D$ and $E$ are both on the $30^{\circ}$ North latitude line with longitudes of East $0^{\circ}$ and $180^{\circ}$ respectively. Select the correct options.
(1) The length of the equator equals the sum of the lengths of the meridians at East $0^{\circ}$ and $180^{\circ}$ (2) The length of the $45^{\circ}$ North latitude line equals $\frac{1}{2}$ of the equator's length (3) The shortest path length from $A$ to $B$ along the equator equals the path length from $D$ to the North Pole along the East $0^{\circ}$ meridian (4) The path length from $D$ to $E$ along the $30^{\circ}$ North latitude line equals the sum of the path lengths from $D$ to the North Pole along the East $0^{\circ}$ meridian and from the North Pole to $E$ along the East $180^{\circ}$ meridian (5) The line passing through the North Pole and point $A$ is perpendicular to the line passing through the North Pole and point $C$

On a square piece of paper, there is a point $P$ that is 6 cm from the left edge and 8 cm from the bottom edge. Now fold the bottom-left corner $O$ of the paper inward to point $P$, as shown in the figure. The area of the folded triangle is $\square$ square centimeters.

Consider all sequences composed of only the three digits 0, 1, 2. The length $n$ of a sequence refers to the sequence consisting of $n$ digits (which may repeat). Let $a(n)$ be the total count of consecutive pairs of zeros (i.e., 00) appearing in all sequences of length $n$. For example, among sequences of length 3 containing consecutive zeros, there are 000, 001, 002, 100, 200. Among these, 000 contributes 2 occurrences of 00, and each of the others contributes 1 occurrence of 00, so $a(3) = 6$. The value of $a(5)$ is $\square$.

In a shooting game, a player must avoid obstacles to shoot a target. A rectangular coordinate system is set up on the game screen with the lower left corner $O$ of the screen as the origin, the lower edge of the screen as the $x$-axis, and the left edge of the screen as the $y$-axis. The target is placed at point $P ( 12,10 )$. There are two walls in the screen (wall thickness is negligible), one wall extends horizontally from point $A ( 10,5 )$ to point $B ( 15,5 )$, and another wall extends horizontally from point $C ( 0,6 )$ to point $D ( 9,6 )$, as shown in the schematic diagram on the right. If a player at point $Q$ can shoot the target at point $P$ in a straight line without being blocked by the two walls, which of the following options could be the coordinates of point $Q$?
(1) $( 6,3 )$
(2) $( 7,3 )$
(3) $( 8,5 )$
(4) $( 9,1 )$
(5) $( 9,2 )$

A resident of a building displays a Christmas tree-shaped light decoration on the building's exterior wall, as shown in the figure. From a certain point $P$ on the fifth floor exterior wall, small light bulbs are pulled to the two ends $A , B$ of the fourth floor to form an isosceles triangle $P A B$, where $\overline { P A } = \overline { P B }$; light bulbs are pulled to the two ends $C , D$ of the third floor to form an isosceles triangle $P C D$; light bulbs are pulled to the two ends $E , F$ of the second floor to form an isosceles triangle $PEF$. Assume each floor has equal height and each floor has equal length. If the length of the line segment cut out by the fifth floor inside triangle $P A B$ is $\frac { 1 } { 3 }$ of the floor length, what fraction of the floor length is the length of the line segment cut out by the fifth floor inside triangle $PEF$? (Light decoration thickness is negligible)
(1) $\frac { 1 } { 7 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 5 }$
(4) $\frac { 2 } { 9 }$
(5) $\frac { 1 } { 4 }$

As shown in the figure, consider a rectangular stone block with a vertex $A$ and a face containing point $A$. Let the midpoints of the edges of this face be $B , E , F , D$ respectively. Another face of the rectangular block containing point $B$ has its edge midpoints as $B , C , H , G$ respectively. Given that $\overline { B C } = 8$ and $\overline { B D } = \overline { D C } = 9$. The stone block is now cut to remove eight corners, such that the cutting plane for each corner passes through the midpoints of the three adjacent edges of that corner.
How many faces does the stone block have after cutting the corners? (Single choice question, 3 points)
(1) Octahedron
(2) Decahedron
(3) Dodecahedron
(4) Tetradecahedron
(5) Hexadecahedron

As shown in the figure, consider a rectangular stone block with a vertex $A$ and a face containing point $A$. Let the midpoints of the edges of this face be $B , E , F , D$ respectively. Another face of the rectangular block containing point $B$ has its edge midpoints as $B , C , H , G$ respectively. Given that $\overline { B C } = 8$ and $\overline { B D } = \overline { D C } = 9$. The stone block is now cut to remove eight corners, such that the cutting plane for each corner passes through the midpoints of the three adjacent edges of that corner.
Find the area of $\triangle B C D$. (Non-multiple choice question, 4 points)

As shown in the figure, consider a rectangular stone block with a vertex $A$ and a face containing point $A$. Let the midpoints of the edges of this face be $B , E , F , D$ respectively. Another face of the rectangular block containing point $B$ has its edge midpoints as $B , C , H , G$ respectively. Given that $\overline { B C } = 8$ and $\overline { B D } = \overline { D C } = 9$. The stone block is now cut to remove eight corners, such that the cutting plane for each corner passes through the midpoints of the three adjacent edges of that corner.
Find the length of $\overline { A D }$ and the volume of tetrahedron $A B C D$, and find the height from vertex $A$ to the base plane when $\triangle B C D$ is the base of the tetrahedron. (Volume of pyramid $= \frac { \text{Base area} \times \text{Height} } { 3 }$) (Non-multiple choice question, 8 points)

In space, there is a regular cube $ABCD-EFGH$, where vertices $A, B, C, D$ lie on the same plane, and $\overline{AE}$ is one of its edges, as shown in the figure. Among the following options, select the plane that is perpendicular to both plane $BGH$ and plane $CFE$.
(1) Plane $ADH$
(2) Plane $BCD$
(3) Plane $CDG$
(4) Plane $DFG$
(5) Plane $DFH$

The Earth is a sphere. Five points $A , B , C , D , E$ on the Earth's surface have the following latitude and longitude coordinates, for example, point $A$ is located at longitude 0 degrees, north latitude 60 degrees.

Location	Longitude 0 degrees	Longitude 180 degrees
North latitude 60 degrees	$A$	$B$
North latitude 30 degrees	$C$	$D$
Latitude 0 degrees	$E$

A great circle is the circle formed by the intersection of a plane passing through the center of the sphere with the sphere's surface. The shorter arc formed by two distinct points on the sphere on a great circle is the shortest path. Based on the above, select the correct options.
(1) ``The shortest path length from the North Pole to $A$'' equals ``the shortest path length from the North Pole to $B$''
(2) ``The shortest path length from $A$ to $B$'' equals ``the shortest path length from $C$ to $D$''
(3) The shortest path from $A$ to $E$ must pass through $C$
(4) The shortest path from $C$ to $D$ must pass through the North Pole
(5) The ratio of ``the shortest path length from $E$ to the North Pole'' to ``the shortest path length from $C$ to $D$'' is $2 : 3$

A liquid crystal display consists of red, green, and blue LED bulbs. The lighting cycle rules for each color bulb are as follows:
Red: ``On for 3 seconds, then off for 1 second, then on for 2 seconds'' Green: ``On for 6 seconds, then off for 2 seconds'' Blue: ``On for $k$ seconds, then off for ($15 - k$) seconds'', where $k$ is a positive integer. If at a certain moment all three colors of bulbs simultaneously begin their respective cycles, and the display always has lights on, with the switching time between on and off being negligibly short, then the minimum value of $k$ is

Let $A$ and $b$ be defined as
$$A = \left( \begin{array} { r r r } - 3 & 0 & 0 \\ - 2 & - 3 & 1 \\ 2 & - 3 & - 3 \end{array} \right) , b = \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) .$$
The partial derivative of a scalar-valued function $f ( x )$ with respect to $x = \left( \begin{array} { l l l } x _ { 1 } & x _ { 2 } & x _ { 3 } \end{array} \right) ^ { T }$ is defined as
$$\frac { \partial } { \partial x } f ( x ) = \left( \frac { \partial } { \partial x _ { 1 } } f ( x ) \quad \frac { \partial } { \partial x _ { 2 } } f ( x ) \quad \frac { \partial } { \partial x _ { 3 } } f ( x ) \right)$$
and a stationary point of $f ( x )$ is defined as $x$ satisfying $\frac { \partial } { \partial x } f ( x ) = \left( \begin{array} { l l l } 0 & 0 & 0 \end{array} \right) . x ^ { T }$ denotes the transpose of $x$. Answer the following questions.
(1) Find the characteristic polynomial of $A$.
(2) $C$ is given as $C = A ^ { 5 } + 9 A ^ { 4 } + 30 A ^ { 3 } + 36 A ^ { 2 } + 30 A + 9 I$ by using $A$ and an identity matrix $I$. Calculate $C$.
(3) Calculate the partial derivative of $x ^ { T } A x$ with respect to $x$.
(4) Find a symmetric matrix $\tilde { A }$ that satisfies equation $x ^ { T } A x = x ^ { T } \tilde { A } x$ for any vector $x$. Find eigenvalues $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } \left( \lambda _ { 1 } \geq \lambda _ { 2 } \geq \lambda _ { 3 } \right)$, and eigenvectors $v _ { 1 } , v _ { 2 } , v _ { 3 }$. Choose the eigenvectors such that $V = \left( v _ { 1 } v _ { 2 } v _ { 3 } \right)$ becomes an orthogonal matrix.
(5) Prove that $x ^ { T } A x \leq 0$ holds for any real vector $x$.
(6) Find a stationary point of function $g ( x ) = x ^ { T } A x + 2 b ^ { T } x$.

Problem 5
I. A function $f ( x )$ is continuous and defined on the interval $0 \leq x \leq \pi$. If $f ( x )$ is extended to the interval $- \pi \leq x \leq \pi$ as an odd function, it can be expanded in the following Fourier sine series:
$$\begin{aligned} & f ( x ) \doteq \sum _ { n = 1 } ^ { \infty } \left( b _ { n } \sin n x \right) \\ & b _ { n } = \frac { 2 } { \pi } \int _ { 0 } ^ { \pi } f ( x ) \sin n x \, d x \quad ( n = 1,2,3 , \cdots ) \end{aligned}$$
Here, $f ( 0 ) = f ( \pi ) = 0$.

1. Find the Fourier sine series for the following function $f ( x )$: $$f ( x ) = x ( \pi - x ) \quad ( 0 \leq x \leq \pi )$$
2. Derive the following equation using the result obtained in Question I.1, $$\frac { 1 } { 1 ^ { 3 } } - \frac { 1 } { 3 ^ { 3 } } + \frac { 1 } { 5 ^ { 3 } } - \frac { 1 } { 7 ^ { 3 } } + \cdots = \frac { \pi ^ { 3 } } { 32 }$$

II. A two-variable function $f ( x , y )$ is continuous and defined in the region $0 \leq x \leq \pi$ and $0 \leq y \leq \pi$. Using a similar method to Question I, $f ( x , y )$ can be expanded in the following double Fourier sine series:
$$\begin{aligned} & f ( x , y ) = \sum _ { m = 1 } ^ { \infty } \sum _ { n = 1 } ^ { \infty } \left( B _ { m n } \sin m x \sin n y \right) \\ & B _ { m n } = \frac { 4 } { \pi ^ { 2 } } \int _ { 0 } ^ { \pi } \int _ { 0 } ^ { \pi } f ( x , y ) \sin m x \sin n y \, d x \, d y \quad ( m , n = 1,2,3 , \cdots ) \end{aligned}$$
Here, $f ( 0 , y ) = f ( \pi , y ) = f ( x , 0 ) = f ( x , \pi ) = 0$.

1. Find the double Fourier sine series for the following function $f ( x , y )$: $$f ( x , y ) = x ( \pi - x ) \sin y \quad ( 0 \leq x \leq \pi , 0 \leq y \leq \pi )$$
2. Function $u ( x , y , t )$ is defined in the region $0 \leq x \leq \pi , 0 \leq y \leq \pi$ and $t \geq 0$. Obtain the solution for the following partial differential equation of $u ( x , y , t )$ by the method of separation of variables: $$\frac { \partial u } { \partial t } = c ^ { 2 } \left( \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } \right)$$ where $c$ is a positive constant and the following boundary and initial conditions apply: $$\begin{aligned} & u ( 0 , y , t ) = u ( \pi , y , t ) = u ( x , 0 , t ) = u ( x , \pi , t ) = 0 \\ & u ( x , y , 0 ) = x ( \pi - x ) \sin y \end{aligned}$$

A real-valued function $u ( x , t )$ is defined in $0 \leq x \leq 1$ and $t \geq 0$. Here, $x$ and $t$ are independent. Suppose solving the following partial differential equation:
$$\frac { \partial u } { \partial t } = \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } ,$$
under the following conditions:
$$\begin{aligned} \text { Boundary condition : } & u ( 0 , t ) = u ( 1 , t ) = 0 , \\ \text { Initial condition : } & u ( x , 0 ) = x - x ^ { 2 } . \end{aligned}$$
Since the constant function $u ( x , t ) = 0$ is obviously a solution of the partial differential equation, consider the other solutions. Answer the following questions.
(1) Calculate the following expression, where $n$ and $m$ are positive integers.
$$\int _ { 0 } ^ { 1 } \sin ( n \pi x ) \sin ( m \pi x ) \mathrm { d } x$$
(2) Suppose $u ( x , t ) = \xi ( x ) \tau ( t )$, where $\xi ( x )$ is a function only of $x$ and $\tau ( t )$ is a function only of $t$. Express the ordinary differential equations for $\xi$ and $\tau$ using an arbitrary constant $C$. You may use that $f ( x )$ and $g ( t )$ are constant functions when $f ( x )$ and $g ( t )$ satisfy $f ( x ) = g ( t )$ for arbitrary $x$ and $t$.
(3) Solve the ordinary differential equations in question (2). Next, show that a solution of partial differential equation $(*)$ which satisfies the boundary condition is given by the following $u _ { n } ( x , t )$, and express $\alpha$ and $\beta$ using a positive integer $n$.
$$u _ { n } ( x , t ) = e ^ { \alpha t } \sin ( \beta x )$$
(4) The solution of partial differential equation $(*)$ which satisfies the boundary and initial conditions is represented by the linear combination of $u _ { n } ( x , t )$ as shown below. Obtain $c _ { n }$. You may use the result of question (1).
$$u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } u _ { n } ( x , t )$$

Problem 3

Answer the following questions. Here, $i , e$, and $\log$ denote the imaginary unit, the base of the natural logarithm, and the natural logarithm, respectively. I. Consider the definite integral $I$ expressed as
$$I = \int _ { 0 } ^ { 2 \pi } \frac { \cos \theta d \theta } { ( 2 + \cos \theta ) ^ { 2 } }$$

1. Find a complex function $G ( z )$ of a complex variable $z$ when we rewrite $I$ as an integral of a complex function as

$$\oint _ { | z | = 1 } G ( z ) d z$$
where the integration path is a unit circle in the counter clockwise direction.
2. Find all poles and the respective orders and residues.
3. Evaluate the integral $I$. II. Let a function of a real variable $\theta$ with real parameters $\alpha$ and $\beta$ be
$$f ( \theta ; \alpha , \beta ) = 1 + e ^ { 2 i \beta } + \alpha e ^ { i ( \theta + \beta ) }$$
Consider the definite integral
$$F ( \alpha , \beta ) = \int _ { 0 } ^ { 2 \pi } d \theta \frac { d } { d \theta } [ \log f ( \theta ; \alpha , \beta ) ]$$

1. Find a complex function $G ( z )$ of a complex variable $z$ when we rewrite $F ( \alpha , \beta )$ as an integral of a complex function as

$$\oint _ { | z | = 1 } G ( z ) d z$$
where the integration path is a unit circle in the counter clockwise direction.
2. Find all poles and the respective orders and residues.
3. Evaluate $F ( \alpha , \beta )$ by classifying cases with respect to $\alpha$ and $\beta$. Ignore the case in which the integration path passes through any poles.

Problem 5

The Laplace transform $F ( s ) = L [ f ( t ) ]$ of a function $f ( t )$, where $t \geq 0$, is defined as
$$F ( s ) = \int _ { 0 } ^ { \infty } f ( t ) e ^ { - s t } d t$$
Here, $s$ is a complex number, and $e$ is the base of the natural logarithm. Answer the following questions. Show the derivation process with your answer. I. Prove the following relations:

1. $L \left[ t ^ { n } \right] = \frac { n ! } { s ^ { n + 1 } }$, where $n$ is a natural number.
2. $L \left[ \frac { d f ( t ) } { d t } \right] = s F ( s ) - f ( 0 )$, where $f ( t )$ is a differentiable function.
3. $L \left[ e ^ { a t } f ( t ) \right] = F ( s - a )$, where $a$ is a real number.

II. Solve the following differential equation using a Laplace transformation for $t \geq 0$:
$$t \frac { d ^ { 2 } f ( t ) } { d t ^ { 2 } } + ( 1 + 3 t ) \frac { d f ( t ) } { d t } + 3 f ( t ) = 0 , \quad f ( 0 ) = 1 , \left. \quad \frac { d f } { d t } \right| _ { t = 0 } = - 3$$
You can use the relation $L [ t f ( t ) ] = - \frac { d } { d s } F ( s )$, if necessary. III. The point $\mathrm { P } ( x ( t ) , y ( t ) )$, which satisfies the following simultaneous differential equations, passes through the point $( a , b )$ when $t = 0$. $a$ and $b$ are real numbers.
$$\left\{ \begin{array} { l } \frac { d x ( t ) } { d t } = - x ( t ) \\ \frac { d y ( t ) } { d t } = x ( t ) - 2 y ( t ) \end{array} \right.$$

1. Solve the simultaneous differential equations using a Laplace transformation for $t \geq 0$.
2. Express the relation between $x$ and $y$ by eliminating $t$ from the solution of III. 1.
3. For both $( a , b ) = ( 1,1 )$ and $( - 1,1 )$, draw the trajectories of point P when $t$ varies continuously from 0 to infinity.

Problem 6

A product factory manufactures 2 types of products: product-I and product-II. Part-A is necessary for product-I, and both part-$A$ and part-$B$ are necessary for product-II. There are parts that have standard quality and parts that do not have standard quality among part-$A$ and part-$B$. All parts are delivered from the part factory to the product factory, but there is no quality check of any part. The qualities of part-$A$ and part-$B$ are independent, and they will not affect each other. The probabilities that part-$A$ and part-$B$ have standard quality are $a$ and $b$, respectively.
A final quality inspection is made in the product factory for product-I and for product-II before shipment. The inspection judges whether the quality of each product meets the standard or not. The inspections will not affect each other. The product inspection is not perfect: namely, products that have standard quality pass the product inspection as acceptable with the probability $x$. The products that do not have standard quality pass the product inspection as acceptable with the probability $y$.
Answer the following questions: I. A product-I is randomly sampled and inspected once. Here, the probability that product-I can be manufactured with standard quality is defined as follows:

• The probability that product-I has standard quality is $c$ if part-$A$ has standard quality.
• Product-I will never have standard quality if part-A does not have standard quality.

1. Show the probability that the selected product-I passes the product inspection as acceptable.
2. Show the probability that the selected product-I actually has standard quality after it has passed the product inspection as acceptable.

II. A product-II is randomly sampled and inspected $n$ times. Here, the probability that product-II can be manufactured with standard quality is defined as follows:

• The probability that product-II has standard quality is $c$ if both part-$A$ and part-$B$ have standard quality.
• The probability that product-II has standard quality is $d$ if only either part-$A$ or part-$B$ has standard quality.
• Product-II will never have standard quality if both part-$A$ and part-$B$ do not have standard quality.

1. Show the probability that the selected product-II has standard quality.
2. Show the probability that the selected product-II actually has standard quality after it has passed all product inspections (i.e., $n$ times) as acceptable.

Consider to solve the following simultaneous linear equation:
$$A x = b$$
where $\boldsymbol { A } \in \mathcal { R } ^ { m \times n } , \boldsymbol { b } \in \mathcal { R } ^ { m }$ are a constant matrix and a vector, and $\boldsymbol { x } \in \mathcal { R } ^ { n }$ is an unknown vector. Answer the following questions.
(1) An $m \times ( n + 1 )$ matrix $\overline { \boldsymbol { A } } = ( \boldsymbol { A } \mid \boldsymbol { b } )$ is made by adding a column vector after the last column of matrix $\boldsymbol { A }$. In the case of $\boldsymbol { A } = \left( \begin{array} { c c c } 1 & 0 & - 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{array} \right)$ and $\boldsymbol { b } = \left( \begin{array} { c } 2 \\ 4 \\ 2 \end{array} \right)$, $\overline { \boldsymbol { A } } = \left( \begin{array} { c c c c } 1 & 0 & - 1 & 2 \\ 1 & 1 & 0 & 4 \\ 0 & 1 & 1 & 2 \end{array} \right)$ is obtained. Let the $i$-th column vector of the matrix $\overline { \boldsymbol { A } }$ be $\boldsymbol { a } _ { i } ( i = 1,2,3,4 )$.
(i) Find the maximum number of linearly independent vectors among $\boldsymbol { a } _ { 1 } , \boldsymbol { a } _ { 2 }$ and $\boldsymbol { a } _ { 3 }$.
(ii) Show that $a _ { 4 }$ can be represented as a linear sum of $a _ { 1 } , a _ { 2 }$ and $a _ { 3 }$, by obtaining scalars $x _ { 1 }$ and $x _ { 2 }$ that satisfy $\boldsymbol { a } _ { 4 } = x _ { 1 } \boldsymbol { a } _ { 1 } + x _ { 2 } \boldsymbol { a } _ { 2 } + \boldsymbol { a } _ { 3 }$.
(iii) Find the maximum number of linearly independent vectors among $\boldsymbol { a } _ { 1 } , \boldsymbol { a } _ { 2 } , \boldsymbol { a } _ { 3 }$ and $a _ { 4 }$.
(2) Show that the solution of the simultaneous linear equation exists when $\operatorname { rank } \overline { \boldsymbol { A } } = \operatorname { rank } \boldsymbol { A }$, for arbitrary $m , n , \boldsymbol { A }$ and $\boldsymbol { b }$.
(3) There is no solution when $\operatorname { rank } \overline { \boldsymbol { A } } > \operatorname { rank } \boldsymbol { A }$. When $m > n , \operatorname { rank } \boldsymbol { A } = n$ and $\operatorname { rank } \overline { \boldsymbol { A } } > \operatorname { rank } \boldsymbol { A }$, obtain $\boldsymbol { x }$ that minimizes the squared norm of the difference between the left hand side and the right hand side of the simultaneous linear equation, namely $\| \boldsymbol { b } - \boldsymbol { A } \boldsymbol { x } \| ^ { 2 }$.
(4) When $m < n$ and $\operatorname { rank } \boldsymbol { A } = m$, there exist multiple solutions for the simultaneous linear equation for arbitrary $\boldsymbol { b }$. Obtain $\boldsymbol { x }$ that minimizes $\| \boldsymbol { x } \| ^ { 2 }$ among them, by adopting the method of Lagrange multipliers and using the simultaneous linear equation as the constraint condition.
(5) Show that there exists a unique $\boldsymbol { P } \in \mathcal { R } ^ { n \times m }$ that satisfies the following four equations for arbitrary $m , n$ and $\boldsymbol { A }$.
$$\begin{array} { r } \boldsymbol { A P A } = \boldsymbol { A } \\ \boldsymbol { P A P } = \boldsymbol { P } \\ ( \boldsymbol { A P } ) ^ { T } = \boldsymbol { A P } \\ ( \boldsymbol { P A } ) ^ { T } = \boldsymbol { P A } \end{array}$$
(6) Show that both $\boldsymbol { x }$ obtained in (3) and $\boldsymbol { x }$ obtained in (4) are represented in the form of $x = P b$.

I. Find the general solutions of the following differential equations.

1. $\frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } - 3 y = e ^ { x } \cos x$
2. $\frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { 1 } { x } \frac { d y } { d x } + \frac { 4 } { x ^ { 2 } } y = \left( \frac { 2 \log x } { x } \right) ^ { 2 }$

II. Answer the following questions for the partial differential equation represented in Equation (3) and the boundary conditions represented in Equations (4)-(7):
$$\begin{aligned} & \frac { \partial ^ { 2 } u ( x , y ) } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u ( x , y ) } { \partial y ^ { 2 } } = 0 \quad ( 0 \leq x , 0 \leq y \leq 1 ) \\ & \left\{ \begin{array} { l } \lim _ { x \rightarrow + \infty } u ( x , y ) = 0 \\ \left. \frac { \partial u ( x , y ) } { \partial y } \right| _ { y = 0 } = 0 \\ u ( x , 1 ) = 0 \\ \left. \frac { \partial u ( x , y ) } { \partial x } \right| _ { x = 0 } = 1 + \cos \pi y \end{array} \right. \end{aligned}$$

1. Find the solution which satisfies Equations (3) and (4) in the form of $u ( x , y ) = X ( x ) \cdot Y ( y )$.
2. Find the solution satisfying Equations (5) and (6) for the solution of Question II.1.
3. Find the solution of the partial differential equation (3) satisfying all the boundary conditions given in Equations (4)-(7), using the solution of Question II.2.

Let $f _ { 1 }$ be a positive constant function on $[ 0,1 ]$ with $f _ { 1 } ( x ) = c$, and let $p$ and $q$ be positive real numbers with $1 / p + 1 / q = 1$. Moreover, let $\left\{ f _ { n } \right\}$ be the sequence of functions on $[ 0,1 ]$ defined by
$$f _ { n + 1 } ( x ) = p \int _ { 0 } ^ { x } \left( f _ { n } ( t ) \right) ^ { 1 / q } \mathrm {~d} t \quad ( n = 1,2 , \ldots )$$
Answer the following questions.
(1) Let $\left\{ a _ { n } \right\}$ and $\left\{ c _ { n } \right\}$ be the sequences of real numbers defined by $a _ { 1 } = 0 , c _ { 1 } = c$ and
$$\begin{aligned} & a _ { n + 1 } = q ^ { - 1 } a _ { n } + 1 \quad ( n = 1,2 , \ldots ) \\ & c _ { n + 1 } = \frac { p \left( c _ { n } \right) ^ { 1 / q } } { a _ { n + 1 } } \quad ( n = 1,2 , \ldots ) \end{aligned}$$
Show that $f _ { n } ( x ) = c _ { n } x ^ { a _ { n } }$.
(2) Let $g _ { n }$ be the function on $[ 0,1 ]$ defined by $g _ { n } ( x ) = x ^ { a _ { n } } - x ^ { p }$ for $n \geq 2$. Noting that $a _ { n } \geq 1$ holds true for $n \geq 2$, show that $g _ { n }$ attains its maximum at a point $x = x _ { n }$, and find the value of $x _ { n }$.
(3) Show that $\lim _ { n \rightarrow \infty } g _ { n } ( x ) = 0$ for any $x \in [ 0,1 ]$.
(4) Let $d _ { n }$ be defined by $d _ { n } = \left( c _ { n } \right) ^ { q ^ { n } }$. Show that $d _ { n + 1 } / d _ { n }$ converges to a finite positive value as $n \rightarrow \infty$. You may use the fact that $\lim _ { t \rightarrow \infty } ( 1 - 1 / t ) ^ { t } = 1 / \mathrm { e }$.
(5) Find the value of $\lim _ { n \rightarrow \infty } c _ { n }$.
(6) Show that $\lim _ { n \rightarrow \infty } f _ { n } ( x ) = x ^ { p }$ for any $x \in [ 0,1 ]$.

Answer the following questions concerning complex functions defined over the $z$-plane ( $z = x + i y$ ), where $i$ denotes the imaginary unit.
I. For the function $f ( z ) = \frac { z } { \left( z ^ { 2 } + 1 \right) ( z - 1 - i a ) }$, where $a$ is a positive real number:

1. Find all the poles and respective residues of $f ( z )$.
2. Using the residue theorem, calculate the definite integral $$\int _ { - \infty } ^ { \infty } \frac { x } { \left( x ^ { 2 } + 1 \right) ( x - 1 - i a ) } d x$$

II. Consider the function $g ( z ) = \frac { z } { \left( z ^ { 2 } + 1 \right) ( z - 1 ) }$ and the closed counter-clockwise path of integration $C$, which consists of the upper half circle $C _ { 1 }$ with radius $R \left( z = R e ^ { i \theta } , 0 \leq \theta \leq \pi \right)$, the line segment $C _ { 2 }$ on the real axis $( z = x , - R \leq x \leq 1 - r )$, the lower half circle $C _ { 3 }$ with its center at $z = 1 \left( z = 1 - r e ^ { i \theta } , 0 \leq \theta \leq \pi \right)$, and the line segment $C _ { 4 }$ on the real axis $( z = x , 1 + r \leq x \leq R )$. Here, $e$ denotes the base of the natural logarithm, and let $r > 0 , r \neq \sqrt { 2 }$ and $R > 1 + r$.
Answer the following questions.

1. Calculate the integral $\int _ { C } g ( z ) d z$.
2. Using the result from Question II.1, calculate the following value $$\lim _ { \varepsilon \rightarrow + 0 } \left[ \int _ { - \infty } ^ { 1 - \varepsilon } g ( x ) d x + \int _ { 1 + \varepsilon } ^ { \infty } g ( x ) d x \right]$$

I. Consider surfaces presented by the following sets of equations, with parameters $u$ and $v$ in a three-dimensional orthogonal coordinate system $x y z$. Show the equations for the surfaces without the parameters and sketch them. Here, $a , b$, and $c$ are non-zero real constants.

1. $x = a u \cosh v , y = b u \sinh v , z = u ^ { 2 }$.
2. $x = a \frac { u - v } { u + v } , y = b \frac { u v + 1 } { u + v } , z = c \frac { u v - 1 } { u + v }$.

II. In a three-dimensional orthogonal coordinate system $x y z$, consider the surface $S$ represented by the following equation, where $a$ and $b$ are real constants.
$$z = x ^ { 2 } - 2 y ^ { 2 } + a x + b y$$

1. Determine the normal vector at a point $( x , y , z )$ on the surface $S$.
2. Determine the equation for the surface $T$ which is obtained by rotating the surface $S$ around the $z$-axis by $\pi / 4$. Here, the positive direction of rotation is counter-clockwise when looking at the origin from the positive side of the $z$-axis.
3. Consider the surface $S ^ { \prime }$, which is the portion of the surface $S$ in $- 1 \leq x \leq 1$ and $- 1 \leq y \leq 1$. Determine the area of the projection of the surface $S ^ { \prime }$ onto the $y z$ plane.
4. Calculate the length of the perimeter for the surface $S ^ { \prime }$ when $a = b = 0$.
5. Calculate the Gaussian curvature of the surface $S$ at the point $\left( 0 , \frac { 1 } { 4 } , - \frac { 1 } { 8 } \right)$ when $a = b = 0$.