Problem 1

I. Find the value of the following definite integral:
$$I = \int _ { 2 } ^ { 4 } \frac { d x } { \sqrt { ( x - 2 ) ( 4 - x ) } }$$
II. Find the general solution and the singular solution of the following differential equation:
$$y = x \frac { d y } { d x } + \frac { d y } { d x } + \left( \frac { d y } { d x } \right) ^ { 2 }$$
III. Find the general solution of the following differential equation:
$$x ^ { 2 } \frac { d ^ { 2 } y } { d x ^ { 2 } } - x \frac { d y } { d x } - 8 y = x ^ { 2 }$$

Problem 2

Answer the following questions about the square matrix $A$ of order 3:
$$A = \left( \begin{array} { c c c } 3 & 0 & 1 \\ - 1 & 2 & - 1 \\ - 2 & - 2 & 1 \end{array} \right)$$
I. Find all eigenvalues of $A$. II. Find the matrix $A ^ { n }$, where $n$ is a natural number. III. The square matrix $\boldsymbol { B }$ of order 3 is diagonalizable and meets $\boldsymbol { A } \boldsymbol { B } = \boldsymbol { B } \boldsymbol { A }$. Prove that any eigenvector $\boldsymbol { p }$ of $\boldsymbol { A }$ is also an eigenvector of $\boldsymbol { B }$. IV. Find the square matrix $\boldsymbol { B }$ of order 3 that meets $\boldsymbol { B } ^ { 2 } = \boldsymbol { A }$, where $\boldsymbol { B }$ is diagonalizable and all eigenvalues of $\boldsymbol { B }$ are positive. V. The square matrix $\boldsymbol { X }$ of order 3 is diagonalizable and meets $\boldsymbol { A } \boldsymbol { X } = \boldsymbol { X } \boldsymbol { A }$. When $\operatorname { tr } ( \boldsymbol { A } \boldsymbol { X } ) = d$, find the maximum of $\operatorname { det } ( \boldsymbol { A } \boldsymbol { X } )$ as a function of $d$.
Here, $d$ is positive real and all eigenvalues of $X$ are positive. In addition, $\operatorname { tr } ( M )$ is the trace (the sum of the main diagonal elements) of the square matrix $\boldsymbol { M }$, and $\operatorname { det } ( \boldsymbol { M } )$ is the determinant of the matrix $\boldsymbol { M }$.

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 4

For the real numbers $\theta$ and $\alpha$ within the regions $0 \leq \theta < 2 \pi$ and $0 \leq \alpha \leq \pi$, consider the line L that passes through two points: point $\mathrm { P } ( \cos \theta$, $\sin \theta , 1 )$ and point $\mathrm { Q } ( \cos ( \theta + \alpha ) , \sin ( \theta + \alpha ) , - 1 )$ in a three-dimensional Cartesian coordinate system $x y z$. I. Represent the line L as a linear function of a parameter $t$. Here, the point on the line L at $t = 0$ should represent the point Q and the point at $t = 1$ should represent the point P. II. Find the surface S swept by the line L as an equation of $x , y$ and $z$ when $\theta$ varies in the region $0 \leq \theta < 2 \pi$. Let C be the intersection lines of the surface S with the plane $y = 0$. Find the equation of C in terms of $x$ and $z$, and sketch the shape of C.
Next, examine the Gaussian curvature of the surface S. Generally, when the position vector $r$ of a point R on a curved surface is represented using parameters $u$ and $v$ by
$$\boldsymbol { r } ( u , v ) = ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) ,$$
the Gaussian curvature $K$ is represented as the following equation:
$$K = \frac { \left( \boldsymbol { r } _ { u u } \cdot \boldsymbol{e} \right) \left( \boldsymbol { r } _ { v v } \cdot \boldsymbol { e } \right) - \left( \boldsymbol { r } _ { u v } \cdot \boldsymbol { e } \right) ^ { 2 } } { \left( \boldsymbol { r } _ { u } \cdot \boldsymbol { r } _ { u } \right) \left( \boldsymbol { r } _ { v } \cdot \boldsymbol { r } _ { v } \right) - \left( \boldsymbol { r } _ { u } \cdot \boldsymbol { r } _ { v } \right) ^ { 2 } } ,$$
where $\boldsymbol { r } _ { u }$ and $\boldsymbol { r } _ { v }$ are first-order partial differentials of $\boldsymbol { r } ( u , v )$ with respect to the parameters $u$ and $v$, and $\boldsymbol { r } _ { u u } , \boldsymbol { r } _ { u v }$ and $\boldsymbol { r } _ { v v }$ are second-order partial differentials of $\boldsymbol { r } ( u , v )$ with respect to the parameters $u$ and $v$. $( \boldsymbol { a } \cdot \boldsymbol { b } )$ represents the inner product of two three-dimensional vectors $a$ and $b$, and $e$ is the unit vector of the normal direction at the point R. III. Let the point W be the intersection of the surface S and the $x$ axis in the region $x > 0$. Calculate the Gaussian curvature of S at the point W for $\alpha$ within the region $0 \leq \alpha < \pi$. IV. For $\alpha$ within the region $0 \leq \alpha < \pi$, prove that the Gaussian curvature is less than or equal to 0 at arbitrary points on the surface $S$.

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.