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
2025 problems1-3

3 maths questions

Q1 Matrices Linear Transformation and Endomorphism Properties View
A line on a two-dimensional plane can be expressed as $\alpha x + \beta y + \gamma = 0$, where $( x , y )$ is a point on the line in the Cartesian coordinate system. We call the column vector $( \alpha , \beta , \gamma ) ^ { \mathrm { T } }$ a coefficient vector of the line. Answer the following questions. Note that the coefficient vector in your answer must satisfy $\alpha ^ { 2 } + \beta ^ { 2 } = 1$.
(1) Find a coefficient vector of a line that passes through a point $\vec { a }$ and is perpendicular to a unit vector $\vec { v }$ on a two-dimensional plane.
(2) Let a line B pass through a point $\vec { b }$ and be perpendicular to a unit vector $\vec { n }$. Given a line $A$, let the line $A ^ { \prime }$ be the mirror transformation of the line $A$ over the line $B$. Using $\vec { b }$ and $\vec { n }$, write a three-dimensional square matrix that transforms a coefficient vector of the line A to a coefficient vector of the line $\mathrm { A } ^ { \prime }$.
(3) Find the determinant of the matrix derived in Question (2).
(4) Consider the movement of the line $\mathrm { D } _ { t }$ whose coefficient vector changes with the real variable $t$ as $\left( 4 t , 4 t ^ { 2 } - 1 , t \right) ^ { \mathrm { T } }$. This line passes through a point regardless of $t$. Find the coordinate of that point.
(5) Suppose that, with the mirror transformation over a line $M _ { t }$, which also changes with $t$, the line $\mathrm { D } _ { t }$ in Question (4) is transformed to the line with a coefficient vector $( 0,1 , - t ) ^ { \mathrm { T } }$. Find the coefficient vector $\left( \alpha _ { t } , \beta _ { t } , \gamma _ { t } \right) ^ { \mathrm { T } }$ of the line $\mathrm { M } _ { t }$, where $\alpha _ { t } > 0$ and $\beta _ { t } > 0$ for $t > 0$.
(6) When $t$ changes from 0 to $+ \infty$, consider the region where the line $\mathrm { M } _ { t }$ in Question (5) can exist. Describe the region using a simple mathematical expression and draw a diagram of the region.
Q2 Systems of differential equations View
Let $t$ be a real independent variable, and let $a ( t ) , x ( t )$ and $y ( t )$ be real-valued functions. Answer the following questions.
(1) Let $a ( t )$ be a continuous and periodic function of $t$ whose period is $T$. Find the initial-value-problem solution $x ( t )$ of the ordinary differential equation
$$\frac { \mathrm { d } } { \mathrm {~d} t } x ( t ) = a ( t ) x ( t ) ,$$
where $x ( 0 ) = x _ { 0 } \neq 0$.
(2) Find the necessary and sufficient condition on $a ( t )$ such that the solution $x ( t )$ in Question (1) is a periodic solution with a period $T$.
(3) Find the initial-value-problem solutions $x ( t )$ and $y ( t )$ of the simultaneous ordinary differential equations
$$\begin{aligned} \frac { \mathrm { d } } { \mathrm {~d} t } x ( t ) & = - k x ( t ) + \sin ( t ) \cos ( t ) y ( t ) \\ \frac { \mathrm { d } } { \mathrm {~d} t } y ( t ) & = ( - k + \sin ( t ) ) y ( t ) \end{aligned}$$
where $k$ is a real constant, $x ( 0 ) = x _ { 0 } \neq 0$ and $y ( 0 ) = y _ { 0 } \neq 0$.
(4) Explain briefly how $x ( t )$ and $y ( t )$ converge as $t \rightarrow \infty$ when $k > 0$ in Question (3).
(5) Draw a schematic graph of the solution trajectory in Question (3) on the $y x$-plane where $k = 0 , x _ { 0 } = 2$ and $y _ { 0 } = 1$.
Q3 Sequences and series, recurrence and convergence Closed-form expression derivation View
A coin with the probability of coming up heads equal to $p$ is tossed $n$ times. Let $a _ { n }$ be the probability that the coin never comes up heads twice in a row, and let $b _ { n }$ be the probability that the coin never comes up heads three times in a row. Here, let $a _ { 1 } = 1$ and $b _ { 1 } = b _ { 2 } = 1$. Answer the following questions.
(1) Obtain $a _ { 2 }$ as a function of $p$.
(2) When $n \geq 3$, describe $a _ { n }$ by using $p , a _ { n - 1 }$ and $a _ { n - 2 }$.
(3) When $p = \frac { 2 } { 3 }$, obtain all the pairs $( \alpha , \beta )$ of real numbers that satisfy the following recursion:
$$a _ { n } + \alpha a _ { n - 1 } = \beta \left( a _ { n - 1 } + \alpha a _ { n - 2 } \right)$$
(4) When $p = \frac { 2 } { 3 }$, obtain $a _ { n }$ as a function of $n$.
(5) Obtain $b _ { 3 }$ as a function of $p$.
(6) When $n \geq 4$, describe $b _ { n }$ by using $p , b _ { n - 1 } , b _ { n - 2 }$ and $b _ { n - 3 }$.
(7) When $p = \frac { 3 } { 4 }$, show that the following equation holds for any positive integer $n$:
$$\begin{aligned} b _ { n } = & \frac { 9 } { 8 } \left( \frac { 3 } { 4 } \right) ^ { n - 1 } - \frac { ( - 1 ) ^ { n - 1 } } { 8 } \left( \frac { \sqrt { 3 } } { 4 } \right) ^ { n - 1 } \cos ( ( n - 1 ) \theta ) \\ & - \frac { \sqrt { 2 } \mathrm { i } } { 8 } \left\{ \left( \frac { - 1 + \sqrt { 2 } \mathrm { i } } { 4 } \right) ^ { n - 1 } - \left( \frac { - 1 - \sqrt { 2 } \mathrm { i } } { 4 } \right) ^ { n - 1 } \right\} \end{aligned}$$
where i is the imaginary unit and $\theta$ is the angle that satisfies $\cos \theta = \frac { 1 } { \sqrt { 3 } }$ and $\sin \theta = \frac { \sqrt { 2 } } { \sqrt { 3 } }$.