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 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 $S$.