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
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2017
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2016
todai-engineering-math 6 translated 3
2015
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2023 todai-engineering-math

6 maths questions

Q1 Second order differential equations Solving non-homogeneous second-order linear ODE View
Answer all the following questions.
I. Find the following limit value:
$$\lim _ { x \rightarrow 0 } \frac { b ^ { x } - c ^ { x } } { a x } \quad ( a , b , c > 0 )$$
II. Find the general solutions of the following differential equations.
$$\begin{aligned} & \text { 1. } \frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { y } { x } = \log x \quad ( x > 0 ) \\ & \text { 2. } \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} x } - 2 y = 2 x ^ { 2 } + 2 x \end{aligned}$$
III. Let $a _ { n }$ be defined by
$$a _ { n } = \frac { n ! } { n ^ { n + \frac { 1 } { 2 } } e ^ { - n } }$$
where $n$ is a positive integer and $e$ is the base of natural logarithm. Find $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { a _ { n + 1 } }$. Note that the function $y = x ^ { - 1 } ( x > 0 )$ is convex downward.
Q2 Invariant lines and eigenvalues and vectors Diagonalize a matrix explicitly View
Consider expressing the following matrix $\boldsymbol { A }$ in a form of $\boldsymbol { A } = \boldsymbol { P } \boldsymbol { D } \boldsymbol { P } ^ { - 1 }$, using a diagonal matrix $\boldsymbol { D }$ and a regular matrix $\boldsymbol { P }$. Here, $a$ is a real number.
$$A = \left( \begin{array} { l l l } 2 & 1 & 0 \\ 1 & 3 & a \\ 0 & a & 2 \end{array} \right)$$
I. When $a = 1$, find a diagonal matrix $D$.
II. When $a = 1$, prove $\boldsymbol { x } ^ { \mathrm { T } } \boldsymbol { A } \boldsymbol { x } > 0$ for any three-dimensional non-zero real vector $\boldsymbol { x }$. $\boldsymbol { x } ^ { \mathrm { T } }$ represents the transpose of $\boldsymbol { x }$.
III. Find the condition of $a$ which satisfies $\boldsymbol { x } ^ { \mathrm { T } } \boldsymbol { A } \boldsymbol { x } > 0$ for any three-dimensional non-zero real vector $\boldsymbol { x }$.
IV. Assume that $a$ satisfies the condition obtained in Question III.
For a real vector $\boldsymbol { b } = \left( \begin{array} { c } a \\ 0 \\ - 1 \end{array} \right)$, express the minimum value of the function $f ( \boldsymbol { x } ) = \boldsymbol { x } ^ { \mathrm { T } } \boldsymbol { A } \boldsymbol { x } - \boldsymbol { b } ^ { \mathrm { T } } \boldsymbol { x }$ by using $a$.
Q3 Complex numbers 2 Contour Integration and Residue Calculus View
In the following, $z = x + i y$ and $w = u + i v$ represent complex numbers, where $i$ is the imaginary unit, and $x , y , u$ and $v$ are real numbers.
I. In order to evaluate the integral
$$I = \int _ { - \infty } ^ { \infty } \frac { 1 } { x ^ { 6 } + 1 } \mathrm {~d} x$$
consider the complex function $f ( z ) = \frac { 1 } { z ^ { 6 } + 1 }$.
1. Find all singularities of $f ( z )$. 2. By applying the residue theorem, determine the value of $I$.
II. Two domains, which are banded and semi-infinite on the complex $z$-plane, are defined as:
$$D _ { 1 } = \left\{ x + i y \left\lvert \, 0 \leq x \leq \frac { \pi } { 2 } \right. , y \geq 0 \right\} \text { and } D _ { 2 } = \left\{ x + i y \mid x \geq 0 , - \frac { \pi } { 2 } \leq y \leq 0 \right\}$$
Consider the mapping $w = g ( z )$ from the complex $z$-plane to the complex $w$-plane with an analytic function $g ( z )$. Let $D _ { 1 } ^ { * }$ and $D _ { 2 } ^ { * }$ be the images of $D _ { 1 }$ and $D _ { 2 }$, respectively, through this mapping.
1. When $g ( z ) = \cos z$, sketch the domain $D _ { 1 } ^ { * }$. 2. When $g ( z ) = ( \cosh z ) ^ { 3 }$, sketch the domain $D _ { 2 } ^ { * }$.
Q4 Volumes of Revolution Volume of a Region Defined by Inequalities in 3D View
In the three-dimensional orthogonal $x y z$ coordinate system, consider the region $V$ that satisfies Equations (1) and (2).
$$\begin{aligned} & x ^ { 2 } + y ^ { 2 } - z ^ { 2 } \geq 0 \\ & x ^ { 2 } + y ^ { 2 } + 2 x \leq 0 \end{aligned}$$
I. Sketch the cross-sectional shape of the region $V$ at $z = 1$.
II. Obtain the surface area of the region $V$.
Q5 Taylor series Derive series via differentiation or integration of a known series View
Answer all the following questions.
I. Let a periodic function $f ( x )$ satisfy the condition $f ( x + \pi ) = f ( x - \pi )$. Find the Fourier series expansion of $f ( x )$ for each case, where $f ( x )$ is expressed as follows for the interval $- \pi \leq x \leq \pi$.
1. $f ( x ) = x \quad ( - \pi < x < \pi ) , \quad f ( - \pi ) = f ( \pi ) = 0$ 2. $f ( x ) = x ^ { 2 }$
For the Fourier series expansion, the following equations should be used.
$$\begin{aligned} & f ( x ) = \frac { a _ { 0 } } { 2 } + \sum _ { n = 1 } ^ { \infty } \left( a _ { n } \cos n x + b _ { n } \sin n x \right) \\ & a _ { 0 } = \frac { 1 } { \pi } \int _ { - \pi } ^ { \pi } f ( x ) \mathrm { d } x \\ & a _ { n } = \frac { 1 } { \pi } \int _ { - \pi } ^ { \pi } f ( x ) \cos n x \mathrm {~d} x \\ & b _ { n } = \frac { 1 } { \pi } \int _ { - \pi } ^ { \pi } f ( x ) \sin n x \mathrm {~d} x \end{aligned}$$
II. Consider the function shown in Figure 5.1 as
$$V ( t ) = A \left| \sin \left( \frac { \omega t } { 2 } \right) \right| \quad \left( \omega = \frac { 2 \pi } { T } \right) .$$
Note that $A$ and $T$ are positive real numbers. The complex Fourier series expansion of $V ( t )$ is given as
$$V ( t ) = - \frac { 2 A } { \pi } \sum _ { n = - \infty } ^ { \infty } \frac { 1 } { 4 n ^ { 2 } - 1 } e ^ { i n \omega t }$$
Let $I ( t )$ be the periodic solution that satisfies the ordinary differential equation
$$L \frac { \mathrm {~d} I ( t ) } { \mathrm { d } t } + R I ( t ) = V ( t )$$
Note that $L$ and $R$ are positive real numbers. Find the coefficient $C _ { n }$, when the complex Fourier series expansion of $I ( t )$ is expressed as
$$I ( t ) = \sum _ { n = - \infty } ^ { \infty } C _ { n } e ^ { i n \omega t }$$
Q6 Exponential Distribution View
Consider a light whose state alternately and repeatedly switches between the OFF (no light) state and the ON (light) state. For each OFF and ON state, the duration, represented by $T _ { 0 }$ and $T _ { 1 }$ respectively, changes at each transition and is independent.
By using $t$, which represents elapsed time from the initiation of each state, $T _ { 0 }$ and $T _ { 1 }$ follow the exponential distribution whose probability density functions are described respectively as
$$f _ { 0 } ( t ) = \lambda _ { 0 } e ^ { - \lambda _ { 0 } t } \quad \left( \lambda _ { 0 } > 0 \right)$$
and
$$f _ { 1 } ( t ) = \lambda _ { 1 } e ^ { - \lambda _ { 1 } t } \quad \left( \lambda _ { 1 } > 0 \right)$$
Here, for example, $P _ { 0 } ( a , b )$, which is the probability that the condition $a \leq T _ { 0 } \leq b ( 0 \leq a \leq b )$ is satisfied, can be calculated as
$$P _ { 0 } ( a , b ) = \int _ { a } ^ { b } f _ { 0 } ( t ) \mathrm { d } t$$
Assume that the light switches from the ON state to the OFF state at time $\tau = 0$. Answer the following questions.
I. Calculate the expected value and the standard deviation of $T _ { 0 }$.
II. Calculate the expected value and the standard deviation of $T _ { 0 } + T _ { 1 }$.
III. Consider a situation where time tends towards infinity and the condition $\left( \lambda _ { 0 } + \lambda _ { 1 } \right) \tau \rightarrow \infty$ approximately holds.
1. Calculate the probability that the light is in the OFF state. 2. Calculate the expected value of the remaining time from the current state to the next state transition of the light.
IV. At the time $\tau = \tau _ { x } \left( \tau _ { x } > 0 \right)$, calculate the probability that the light is in the ON state for the first occasion after $\tau = 0$.