todai-math

Papers (33)
2025
problems1-3 3 todai-engineering-math 5
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
2024 todai-engineering-math

4 maths questions

Q1 Differential equations Solving Separable DEs with Initial Conditions 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 Argand & Loci 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 $\kappa$.