todai-math

Papers (28)

2025

2024

problem1 1 problem2 1 problem3 1 todai-engineering-math 6

2023

problem1 1 problem2 1 problem3 1 todai-engineering-math 6

2022

problem1 1 problem2 1 problem3 1 todai-engineering-math__paper1 2 todai-engineering-math__paper2 2 todai-engineering-math__paper3 7

2021

todai-engineering-math__paper1 6 todai-engineering-math__paper2 3 todai-engineering-math__paper3 3

2020

todai-engineering-math 6

2019

todai-engineering-math 6

2018

ist 3 todai-engineering-math 6

2017

ist 3 todai-engineering-math 6

2016

todai-engineering-math 6 translated 3

2015

ist 3

2018 todai-engineering-math

6 maths questions

Q1 Second order differential equations Solving non-homogeneous second-order linear ODE View

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

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

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

Find the solution which satisfies Equations (3) and (4) in the form of $u ( x , y ) = X ( x ) \cdot Y ( y )$.
Find the solution satisfying Equations (5) and (6) for the solution of Question II.1.
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.

Q2 Invariant lines and eigenvalues and vectors Properties of eigenvalues under matrix operations View

I. Suppose that $\lambda$ is an eigenvalue of a regular matrix $\boldsymbol { P }$, prove that:

$\lambda$ is not zero.
$\lambda ^ { - 1 }$ is an eigenvalue of $\boldsymbol { P } ^ { - 1 }$ and $\lambda ^ { n }$ is an eigenvalue of $\boldsymbol { P } ^ { n }$, where $n$ is a positive integer.

II. Suppose $\boldsymbol { P }$ is an orthogonal matrix. When the following symmetric matrix $\boldsymbol { A }$ can be diagonalized by $\boldsymbol { P }$, find the matrix $\boldsymbol { P }$ and obtain the diagonalized matrix.
$$A = \left( \begin{array} { c c c } 2 & - 1 & 1 \\ - 1 & 2 & - 1 \\ 1 & - 1 & 2 \end{array} \right)$$
III. When a matrix $\boldsymbol { P }$, and vectors $\boldsymbol { r }$ and $\boldsymbol { x }$ are given as
$$\boldsymbol { P } = \left( \begin{array} { c c c } 1 & 1 & 1 \\ p & p ^ { 2 } & p ^ { 3 } \\ q & q ^ { 2 } & q ^ { 3 } \end{array} \right) , \quad \boldsymbol { r } = \left( \begin{array} { c } r \\ r ^ { 2 } \\ r ^ { 3 } \end{array} \right) , \quad \boldsymbol { x } = \left( \begin{array} { c } x \\ y \\ z \end{array} \right) ,$$
where $p , q$, and $r$ are non-zero real numbers that differ from each other.

Find the condition that $p$ and $q$ must satisfy in order for $\boldsymbol { P }$ to be a regular matrix.
When $\boldsymbol { P } ^ { \mathrm { T } } \boldsymbol { x } = \boldsymbol { r }$ has a single solution, obtain $\boldsymbol { x }$. Here, $\boldsymbol { P } ^ { \mathrm { T } }$ is the transposed matrix of $\boldsymbol { P }$.

IV. The matrix $\boldsymbol { P } _ { n }$ is an $n$-th order square matrix ( $n \geq 2$ ), as shown below, where $p$ and $q$ are real numbers that differ from each other.
$$\boldsymbol { P } _ { n } = \left( \begin{array} { c c c c c c } p + q & q & 0 & \cdots & 0 & 0 \\ p & p + q & \ddots & \ddots & \vdots & \vdots \\ 0 & p & \ddots & \ddots & 0 & \vdots \\ \vdots & 0 & \ddots & \ddots & q & 0 \\ \vdots & \vdots & \ddots & \ddots & p + q & q \\ 0 & 0 & \cdots & 0 & p & p + q \end{array} \right)$$

Obtain the recurrence formula satisfied by the determinant of $\boldsymbol { P } _ { n }$, $\left| \boldsymbol { P } _ { n } \right|$.
Express the determinant $\left| \boldsymbol { P } _ { n } \right|$ in terms of $p , q$, and $n$, using the recurrence formula in Question IV.1.

Q3 Complex numbers 2 Contour Integration and Residue Calculus View

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:

Find all the poles and respective residues of $f ( z )$.
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.

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

Q4 Vector Product and Surfaces View

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.

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

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.
$$z = x ^ { 2 } - 2 y ^ { 2 } + a x + b y$$

Determine the normal vector at a point $( x , y , z )$ on the surface $S$.
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.
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.
Calculate the length of the perimeter for the surface $S ^ { \prime }$ when $a = b = 0$.
Calculate the Gaussian curvature of the surface $S$ at the point $\left( 0 , \frac { 1 } { 4 } , - \frac { 1 } { 8 } \right)$ when $a = b = 0$.

Q5 Complex numbers 2 Properties of Analytic/Entire Functions View

Let $f ( t )$ be a periodic function of period $T , f ( t + T ) = f ( t ) ( T > 0 )$, and be expanded in the complex Fourier series as follows:
$$f ( t ) = \sum _ { n = - \infty } ^ { \infty } F _ { n } \exp \left( - i \omega _ { n } t \right)$$
Here, $i$ is the imaginary unit, and $t$ is a real number. Answer the following questions.
I. Express $\omega _ { n }$ using $T$ and $n$.
II. Let $M$ be a positive-integer constant and $\delta ( t )$ be the delta function, and define
$$\hat { f } ( t ) = \sum _ { m = 0 } ^ { M - 1 } f ( t ) \delta ( t - m \Delta t ) , \quad \Delta t = \frac { T } { M }$$
Express
$$\lim _ { \varepsilon \rightarrow + 0 } \int _ { - \varepsilon } ^ { T - \varepsilon } \hat { f } ( t ) \exp \left( i \omega _ { k } t \right) d t$$
using $F _ { n } ( n = - \infty , \cdots , - 1,0,1 , \cdots , \infty )$. Here $k$ is an arbitrary integer.
III. $\Delta t$ is given in Question II. Express $F _ { j } ( j = 0,1,2 , \cdots , M - 1 )$ using
$$f ( 0 ) , \quad f ( \Delta t ) , \quad f ( 2 \Delta t ) , \cdots , \quad f ( ( M - 1 ) \Delta t )$$
when $F _ { n } = 0 ( n < 0$ or $n \geq M )$.
IV. Calculate $F _ { j }$ in Question III when $f ( l \Delta t ) = ( - 1 ) ^ { l } \quad ( l = 0,1,2 , \cdots , M - 1 )$.

Q6 Permutations & Arrangements Distribution of Objects into Bins/Groups View

There are $n$ children queuing in a line. You have $m$ candies and will begin handing out 1 or 2 candies to each child, starting from the first child in the line. You hand out the candies until reaching the end of the line or until there are no candies left. Answer the following questions. Note that $n$ and $m$ are positive integers.
I. Show the number of distribution patterns of candies if $n = m = 4$.
II. Show the number of distribution patterns of candies if $m \geq 2 n$.
III. Define $X _ { m }$ as the number of distribution patterns of candies if $n \geq m$. Show the recurrence formula satisfied by $X _ { m }$.
IV. Obtain $X _ { m }$ using the recurrence formula in Question III.
V. Consider the situation where the number of children is larger than the number of candies. Define $P ( m )$ as the ratio of the number of distribution patterns (where the distribution finishes by giving 2 candies) to the total number of distribution patterns. $P ( m )$ converges as $m$ increases. Compute the convergence value.
VI. Consider the situation where $m \geq 2 n$. The following rules are added to the way of handing out the candies: For the first child in the line, the probability of receiving 1 candy is $1 / 2$ and the probability of receiving 2 candies is $1 / 2$. If a child receives 1 candy, the probability of the next child receiving 1 candy is $1 / 2$ and the probability of receiving 2 candies is $1 / 2$. If a child receives 2 candies, the probability of the next child receiving 1 candy is $3 / 4$ and the probability of receiving 2 candies is $1 / 4$. Compute the probability that the $n$-th child in the line receives 2 candies.