todai-math

Papers (28)
2025
problems1-3 3 todai-engineering-math 6
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
2024 todai-engineering-math

6 maths questions

Q1 First order differential equations (integrating factor) View
Problem 1
I. Find the general solution $y ( x )$ of the following differential equation:
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ( 1 - y ) ,$$
where $0 < y < 1$.
II. Find the value of the following definite integral, $I$ :
$$I = \int _ { - 1 } ^ { 1 } \frac { \arccos \left( \frac { x } { 2 } \right) } { \cos ^ { 2 } \left( \frac { \pi } { 3 } x \right) } \mathrm { d } x$$
where $0 \leq \arccos \left( \frac { x } { 2 } \right) \leq \pi$.
III. For any positive variable $x$, we define $f ( x )$ and $g ( x )$ respectively as
$$f ( x ) = \sum _ { m = 0 } ^ { \infty } \frac { 1 } { ( 2 m ) ! } x ^ { 2 m }$$
and
$$g ( x ) = \frac { \mathrm { d } } { \mathrm {~d} x } f ( x )$$
For any non-negative integer $n , I _ { n } ( x )$ is defined as
$$I _ { n } ( x ) = \int _ { 0 } ^ { x } \left\{ \frac { g ( X ) } { f ( X ) } \right\} ^ { n } \mathrm {~d} X$$
Here, you may use
$$\exp ( x ) = \sum _ { m = 0 } ^ { \infty } \frac { 1 } { m ! } x ^ { m }$$
  1. Calculate $f ( x ) ^ { 2 } - g ( x ) ^ { 2 }$.
  2. Express $I _ { n + 2 } ( x )$ using $I _ { n } ( x )$.
Q2 Invariant lines and eigenvalues and vectors Diagonalize a matrix explicitly View
Problem 2
Answer the following questions about a real symmetric matrix, $\boldsymbol { A }$ :
$$A = \left( \begin{array} { l l l } 0 & 1 & 2 \\ 1 & 0 & 2 \\ 2 & 2 & 3 \end{array} \right)$$
I. Find all the different eigenvalues of matrix $\boldsymbol { A } , \lambda _ { 1 } , \cdots , \lambda _ { r } \left( \lambda _ { 1 } < \cdots < \lambda _ { r } \right)$.
II. Find all the eigenspaces $W \left( \lambda _ { 1 } \right) , \cdots , W \left( \lambda _ { r } \right)$ corresponding to $\lambda _ { 1 } , \cdots , \lambda _ { r }$, respectively.
III. Find an orthonormal basis, $\boldsymbol { b } _ { 1 } , \boldsymbol { b } _ { 2 } , \boldsymbol { b } _ { 3 }$, which belongs to either of $W \left( \lambda _ { 1 } \right) , \cdots , W \left( \lambda _ { r } \right)$, obtained in Question II.
IV. Find the spectral decomposition of $A$ :
$$A = \sum _ { i = 1 } ^ { r } \lambda _ { i } P _ { i }$$
where $\boldsymbol { P } _ { i }$ is the projection matrix onto $W \left( \lambda _ { i } \right)$.
V. Find $A ^ { n }$, where $n$ is any positive integer.
Q3 Complex numbers 2 Contour Integration and Residue Calculus View
Problem 3
Answer the following questions. Here, for any complex value $z , \bar { z }$ is the complex conjugate of $z$, arg $z$ is the argument of $z , | z |$ is the absolute value of $z$, and $i$ is the imaginary unit.
I. Sketch the region of $z$ on the complex plane that satisfies the following:
$$z \bar { z } + \sqrt { 2 } ( z + \bar { z } ) + 3 i ( z - \bar { z } ) + 2 \leq 0$$
II. Answer the following questions on the complex valued function $f ( z )$ below.
$$f ( z ) = \frac { z ^ { 2 } - 2 } { \left( z ^ { 2 } + 2 i \right) z ^ { 2 } }$$
  1. Find all the poles of $f ( z )$ as well as the orders and residues at the poles.
  2. By applying the residue theorem, find the value of the following integral $I _ { 1 }$. Here, the integration path $C$ is the circle on the complex plane in the counterclockwise direction which satisfies $| z + 1 | = 2$. $$I _ { 1 } = \oint _ { C } f ( z ) \mathrm { d } z$$

III. Answer the following questions.
  1. Let $g ( z )$ be a complex valued function, which satisfies $$\lim _ { | z | \rightarrow \infty } g ( z ) = 0$$ for $0 \leq \arg z \leq \pi$. Let $C _ { R }$ be the semicircle, with radius $R$, in the upper half of the complex plane with the center at the origin. Show $$\lim _ { R \rightarrow \infty } \int _ { C _ { R } } e ^ { i a z } g ( z ) \mathrm { d } z = 0$$ where $a$ is a positive real number.
  2. Find the value of the following integral, $I _ { 2 }$ : $$I _ { 2 } = \int _ { 0 } ^ { \infty } \frac { \sin x } { x } \mathrm {~d} x$$
Q4 Parametric differentiation View
Problem 4
I. In the two-dimensional orthogonal $x y$ coordinate system, consider the curve $L$ represented by the following equations with the parameter $t ( 0 \leq t \leq 2 \pi )$. Here, $a$ is a positive real constant.
$$\begin{aligned} & x ( t ) = a ( t - \sin t ) \\ & y ( t ) = a ( 1 - \cos t ) \end{aligned}$$
  1. Obtain the length of the curve $L$ when $t$ varies in the range of $0 \leq t \leq 2 \pi$.
  2. Obtain the curvature at an arbitrary point of the curve $L$.

Here, $t = 0$ and $t = 2 \pi$ are excluded.
II. In the three-dimensional orthogonal $x y z$ coordinate system, consider the curved surface represented by the following equations with the parameters $u$ and $v$ ( $u$ and $v$ are real numbers).
$$\begin{aligned} & x ( u , v ) = \sinh u \cos v \\ & y ( u , v ) = 2 \sinh u \sin v \\ & z ( u , v ) = 3 \cosh u \end{aligned}$$
  1. Express the curved surface by an equation without the parameters.
  2. Sketch the $x y$-plane view at $z = 5$ and the $x z$-plane view at $y = 0$, respectively, of the curved surface. In the sketches, indicate the values at the intersection with each of the axes.
  3. Express the unit normal vector $\boldsymbol { n }$ of the curved surface by $u$ and $v$. Here, the $z$-component of $\boldsymbol { n }$ should be positive.
  4. Let $\kappa$ be the Gaussian curvature at the point $u = v = 0$. Calculate the absolute value of $\kappa$.
Q5 Taylor series Derive series via differentiation or integration of a known series View
Problem 5
I. We consider a continuous and absolutely integrable function $f ( t )$ of a real variable $t$ and denote the Fourier transform of the function $f ( t )$ as $\mathcal { F } \{ f ( t ) \}$. We define a function $F ( \omega )$ by the following formula:
$$F ( \omega ) = \mathcal { F } \{ f ( t ) \} = \int _ { - \infty } ^ { \infty } f ( t ) \exp ( - i \omega t ) \mathrm { d } t$$
where $\omega$ is a real variable and $i$ is the imaginary unit.
  1. We define $g ( t ) = f ( a t )$ for a constant $a$ satisfying $a > 0$ and $$G ( \omega ) = \mathcal { F } \{ g ( t ) \} = \mathcal { F } \{ f ( a t ) \}$$ Express $G ( \omega )$ using the function $F$.
  2. When $f ( t ) = \exp \left( - t ^ { 2 } \right)$ and $a = 2$, sketch the graph of the Fourier transformed functions $F ( \omega )$ and $G ( \omega )$ defined by Equation (2) as a function of $\omega$ to show the difference between them.
  3. We define $h ( t ) = f ( t ) \exp ( - i b t )$ for a constant $b$ satisfying $b > 0$ and $$H ( \omega ) = \mathcal { F } \{ h ( t ) \} = \mathcal { F } \{ f ( t ) \exp ( - i b t ) \}$$ Express $H ( \omega )$ using the function $F$.
  4. When $f ( t ) = \exp \left( - t ^ { 2 } \right)$ and $b = 2$, sketch the graph of the Fourier transformed functions $F ( \omega )$ and $H ( \omega )$ defined by Equation (3) as a function of $\omega$ to show the difference between them.

II. Let $N$ be a positive integer. We define a discrete Fourier transform $D _ { 1 } , \cdots , D _ { N }$ by the following formula:
$$D _ { m } = \frac { 1 } { \sqrt { N } } \sum _ { n = 1 } ^ { N } c _ { n } \exp \left( - i \frac { 2 \pi } { N } n m \right)$$
for a complex sequence $c _ { 1 } , \cdots , c _ { N }$. Here, $m$ is an integer satisfying $1 \leq m \leq N$.
  1. Calculate $S \left( n , n ^ { \prime } \right)$ : $$S \left( n , n ^ { \prime } \right) = \frac { 1 } { N } \sum _ { m = 1 } ^ { N } \exp \left\{ i \frac { 2 \pi } { N } \left( n - n ^ { \prime } \right) m \right\}$$ Here, $n$ is an integer satisfying $1 \leq n \leq N$, and $n ^ { \prime }$ is an integer satisfying $1 \leq n ^ { \prime } \leq N$.
  2. Let $U _ { m n }$ be a complex number satisfying $$D _ { m } = \sum _ { n = 1 } ^ { N } U _ { m n } c _ { n }$$ Show that the matrix $\mathbf { U } = \left[ U _ { m n } \right] _ { 1 \leq m \leq N , 1 \leq n \leq N }$ is a unitary matrix.
  3. Derive an equation for the inverse discrete Fourier transform $c _ { n }$ from $D _ { 1 } , \cdots , D _ { N }$. Here, $n$ is an integer satisfying $1 \leq n \leq N$.
  4. For any complex value $z , \bar { z }$ is the complex conjugate of $z$. We define $Q$ by $$Q = \sum _ { n = 1 } ^ { N } \left( \overline { c _ { n } } c _ { n + 1 } + \overline { c _ { n + 1 } } c _ { n } \right)$$ Express $Q$ in terms of $D _ { m }$ and $\overline { D _ { m } }$. Here, we impose the condition $c _ { N + 1 } = c _ { 1 }$.
Q6 Discrete Probability Distributions Markov Chain and Transition Matrix Analysis View
Problem 6
Consider an electric vehicle charging station with a single charger installed and let us observe the number of vehicles at the station at regular time intervals.
Arriving vehicles at the station are lined up in the queue in the order of arrival, and only the first vehicle in the queue can be charged. In the interval between one observation and the next observation, assume that one new vehicle arrives with probability $p ( 0 < p < 1 )$, and that the vehicle charging at the head of the queue completes charging with probability $q ( 0 < q < 1 )$. Here, assume that $p$ and $q$ are constants and $p + q < 1$.
The queue can accommodate $N ( N \geq 2 )$ vehicles, including the vehicle being charged at the head of the queue, and the $( N + 1 )$-th vehicle shall give up and leave the station without queuing up. The vehicle which completes charging leaves the station immediately.
In the interval between one observation and the next observation, either only one or no vehicles arrive at the station and either only one or no vehicles complete charging. Moreover, assume that both arrival of new vehicle and completion of charging for the first vehicle do not occur together in any one interval.
I. When there are $i ( 0 < i < N )$ vehicles in the queue, find the probability for the following condition: no new vehicle arrives and the first vehicle does not leave in the interval between one observation and the next observation.
Let $\pi _ { i } ( 0 \leq i \leq N )$ be the probability that $i$ vehicles are in the queue in the steady state.
II. Express the relationship between $\pi _ { i }$ and $\pi _ { i + 1 }$. Here, $i \leq N - 1$.
III. Express $\pi _ { i }$ using $p , q$ and $N$.
IV. Find the expected value of the number of vehicles at the station in the steady state using $p , q$ and $N$. Here, $p < q$.