todai-math

Papers (33)

2025

2024

liberal-arts_official 4 problem1 1 science 6 science_official 6 todai-engineering-math 4

2023

problem1 1 problem2 1 problem3 1 science 5 todai-engineering-math 5

2022

problem1 1 problem2 1 problem3 1 science 6 todai-engineering-math__paper1 2 todai-engineering-math__paper2 2 todai-engineering-math__paper3 7

2021

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

2020

science 6 todai-engineering-math 5

2019

todai-engineering-math 5

2018

ist 1 todai-engineering-math 2

2017

ist 2 todai-engineering-math 3

2016

todai-engineering-math 5 translated 3

2015

ist 2

2020 todai-engineering-math

5 maths questions

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

I. Answer the following questions about the differential equation:
$$\cos x \frac { d ^ { 2 } y } { d x ^ { 2 } } - \sin x \frac { d y } { d x } - \frac { y } { \cos x } = 0 \quad \left( - \frac { \pi } { 2 } < x < \frac { \pi } { 2 } \right) .$$

A particular solution of Eq. (1) is of the form of $y = ( \cos x ) ^ { m }$ (m is a constant). Find the constant $m$.
Find the general solution of Eq. (1), using the solution of Question I.1.

II. Find the value of the following integral:
$$I = \int _ { 1 } ^ { \infty } x ^ { 5 } e ^ { - x ^ { 4 } + 2 x ^ { 2 } - 1 } d x$$
Note that, for a positive constant $\alpha$, the relation $\int _ { 0 } ^ { \infty } e ^ { - \alpha x ^ { 2 } } d x = \frac { 1 } { 2 } \sqrt { \frac { \pi } { \alpha } }$ holds.
III. Express the general solution of the following differential equation in the form of $f ( x , y ) = C$ ( $C$ is a constant) using an appropriate function $f ( x , y )$ :
$$\left( x ^ { 3 } y ^ { n } + x \right) \frac { d y } { d x } + 2 y = 0 \quad ( x > 0 , y > 0 )$$
where $n$ is an arbitrary real constant.

Q2 Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties View

Consider the following matrix $\boldsymbol { A }$ :
$$A = \left( \begin{array} { c c c } 1 & - 2 & - 1 \\ - 2 & 1 & 1 \\ - 1 & 1 & \alpha \end{array} \right)$$
where $\alpha$ is a real number. In the following, the transpose of a vector $\boldsymbol { v }$ is denoted by $\boldsymbol { v } ^ { \mathrm { T } }$.
I. Obtain $\alpha$ when the sum of the three eigenvalues of the matrix $A$ is 7.
II. Obtain $\alpha$ when the product of the three eigenvalues of the matrix $\boldsymbol { A }$ is $- 16$.
III. Let $\| \boldsymbol { A } \|$ be the maximum of $\boldsymbol { x } ^ { \mathrm { T } } \boldsymbol { A x }$ for the set of real vectors $\boldsymbol { x } = \left( \begin{array} { l } x _ { 1 } \\ x _ { 2 } \\ x _ { 3 } \end{array} \right)$ that satisfy $\boldsymbol { x } ^ { \mathrm { T } } \boldsymbol { x } = 1$. Obtain $\alpha$ when $\| \boldsymbol { A } \| = 4$.
IV. In the following questions, $\alpha = 4$.

Obtain all eigenvalues of the matrix $\boldsymbol { A }$ and their corresponding normalized eigenvectors.
Find the range of $\boldsymbol { y } ^ { \mathrm { T } } \boldsymbol { A } \boldsymbol { y }$ for the real vectors $\boldsymbol { y } = \left( \begin{array} { l } y _ { 1 } \\ y _ { 2 } \\ y _ { 3 } \end{array} \right)$ that satisfy $\boldsymbol { y } ^ { \mathrm { T } } \boldsymbol { y } = 1$ and $y _ { 1 } - y _ { 2 } - 2 y _ { 3 } = 0$.
Find the range of $\boldsymbol { z } ^ { \mathrm { T } } \boldsymbol { A } \boldsymbol { z }$ for the real vectors $\boldsymbol { z } = \left( \begin{array} { c } z _ { 1 } \\ z _ { 2 } \\ z _ { 3 } \end{array} \right)$ that satisfy $\boldsymbol { z } ^ { \mathrm { T } } \boldsymbol { z } = 1$ and $z _ { 1 } + z _ { 2 } + z _ { 3 } = 0$.

Q3 Complex numbers 2 Contour Integration and Residue Calculus View

In the following, $z$ denotes a complex number, and $x$ and $\varepsilon$ denote real numbers. The imaginary unit is denoted by $i$.
I. Answer the following questions about the function $f _ { n } ( z ) = 1 / \left( z ^ { n } - 1 \right)$. Here, $n$ is an integer greater than or equal to 2.

For the case that $n = 3$, find all singularities of $f _ { n } ( z )$.
Calculate the residue value at a singularity $p _ { 0 }$ of $f _ { n } ( z )$ and give a simple expression of the residue in terms of $n$ and $p _ { 0 }$.
For a contour $C$ given by the closed curve $| z | = 2$ and oriented in the counter-clockwise direction, evaluate the contour integral $\oint _ { C } f _ { n } ( z ) d z$.

II. Obtain the following limit value:
$$\lim _ { \varepsilon \rightarrow + 0 } \left[ \int _ { - \infty } ^ { 1 - \varepsilon } \frac { 1 } { x ^ { 3 } - 1 } d x + \int _ { 1 + \varepsilon } ^ { \infty } \frac { 1 } { x ^ { 3 } - 1 } d x \right]$$
III. Obtain the following limit value:
$$\lim _ { \varepsilon \rightarrow + 0 } \left[ \int _ { 0 } ^ { 1 - \varepsilon } \frac { \cos x } { x ^ { 4 } - 1 } d x + \int _ { 1 + \varepsilon } ^ { \infty } \frac { \cos x } { x ^ { 4 } - 1 } d x \right]$$
IV. Obtain the following limit value:
$$\lim _ { \varepsilon \rightarrow + 0 } \left[ \int _ { 0 } ^ { 1 - \varepsilon } \frac { \sin \left( x ^ { 2 } - \frac { \pi } { 4 } \right) } { x ^ { 4 } - 1 } d x + \int _ { 1 + \varepsilon } ^ { \infty } \frac { \sin \left( x ^ { 2 } - \frac { \pi } { 4 } \right) } { x ^ { 4 } - 1 } d x \right]$$

Q4 Vectors 3D & Lines Multi-Part 3D Geometry Problem View

In the three-dimensional orthogonal coordinate system $x y z$, the unit vectors along the $x , y$, and $z$ directions are $\mathbf { i } , \mathbf { j }$, and $\mathbf { k }$, respectively. Using the parameter $\theta ( 0 \leq \theta \leq \pi )$, we define two curves by their vector functions $\mathbf { P } ( \theta )$ and $\mathbf { Q } ( \theta )$ :
$$\begin{aligned} & \mathbf { P } ( \theta ) = x ( \theta ) \mathbf { i } + y ( \theta ) \mathbf { j } \\ & \mathbf { Q } ( \theta ) = \mathbf { P } ( \theta ) + z ( \theta ) \mathbf { k } \end{aligned}$$
where
$$\begin{aligned} & x ( \theta ) = \frac { 3 } { 2 } \cos ( \theta ) - \frac { 1 } { 2 } \cos ( 3 \theta ) \\ & y ( \theta ) = \frac { 3 } { 2 } \sin ( \theta ) - \frac { 1 } { 2 } \sin ( 3 \theta ) \end{aligned}$$
Here, $z ( \theta )$ is a continuous function satisfying $z ( 0 ) > 0$ and $z ( \pi ) < 0$, and the curve parametrized by $\mathbf { Q } ( \theta )$ lies on the sphere of radius 2, centered at the origin $( 0,0,0 )$ of the coordinate system. The positive direction of a curve corresponds to increasing values of the parameter $\theta$. Note that the curvature is the reciprocal of the radius of curvature. Answer the following questions.
I. As $\theta$ is varied from 0 to $\pi$, calculate the arc length of the curve parametrized by $\mathbf { P } ( \theta )$.
II. Obtain $z ( \theta )$.
III. Let $\alpha$ be the angle between the tangent of the curve parametrized by $\mathbf { Q } ( \theta )$ and the unit vector $\mathbf { k }$. Calculate $\cos ( \alpha )$.
IV. Find the curvature $\kappa _ { P } ( \theta )$ of the curve parametrized by $\mathbf { P } ( \theta )$. Here, $\theta = 0$ and $\theta = \pi$ are excluded.
V. Let $\kappa _ { Q } ( \theta )$ be the curvature of the curve parametrized by $\mathbf { Q } ( \theta )$. Express $\kappa _ { Q } ( \theta )$ in terms of $\kappa _ { P } ( \theta )$ and $\alpha$. Here, $\theta = 0$ and $\theta = \pi$ are excluded.

Q6 Discrete Probability Distributions Probability Computation for Compound or Multi-Stage Random Experiments View

Consider a game where points are awarded in $n$ independent trials. In each trial, either $+1$ or $-1$ is awarded and both outcomes have the same probability of $1/2$. Let $X _ { k }$ be the point awarded in the $k ^ { \text {th } }$ trial $( 1 \leq k \leq n )$, and $S _ { k } = \sum _ { i = 1 } ^ { k } X _ { i }$. In the following questions, $n$ is an even integer such that $n \geq 4$, and $t$ is an even integer such that $2 \leq t \leq n$.
I. Obtain the probability for $S _ { 4 } = 0$.
II. Let $P _ { n } ( t )$ be the probability for $S _ { n } = t$. Find $P _ { n } ( t )$.
III. Let $P _ { n } ^ { + } ( t )$ be the probability for $S _ { 1 } = 1$ and $S _ { n } = t$. Find $P _ { n } ^ { + } ( t )$.
IV. Let $P _ { n } ^ { - } ( t )$ be the probability for $S _ { 1 } = - 1$ and $S _ { n } = t$. Find $P _ { n } ^ { - } ( t )$.
V. Let $Q _ { n } ( t )$ be the probability that all of the variables $\left\{ S _ { j } \right\}$ $( j = 1,2 , \cdots , n - 1 )$ are greater than zero and $S _ { n } = t$. Express $Q _ { n } ( t )$ with $P _ { n } ^ { + } ( t )$ and $P _ { n } ^ { - } ( t )$. Then, express $Q _ { n } ( t )$ with $P _ { n } ( t )$.
VI. Obtain the probability that all of the variables $\left\{ S _ { j } \right\} ( j = 1,2 , \cdots , n )$ are greater than zero.