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
2023 science

5 maths questions

Q1 Integration by Substitution Substitution to Evaluate Limit of an Integral Expression View
1
(1) For a positive integer $k$, let $A_k = \displaystyle\int_{\sqrt{k\pi}}^{\sqrt{(k+1)\pi}} |\sin(x^2)|\,dx$. Prove that the following inequality holds: $$\frac{1}{\sqrt{(k+1)\pi}} \leqq A_k \leqq \frac{1}{\sqrt{k\pi}}$$
(2) For a positive integer $n$, let $B_n = \dfrac{1}{\sqrt{n}}\displaystyle\int_{\sqrt{n\pi}}^{\sqrt{2n\pi}} |\sin(x^2)|\,dx$. Find the limit $\displaystyle\lim_{n\to\infty} B_n$.
%% Page 2
Q2 Probability Definitions Conditional Probability and Bayes' Theorem View
2

Balls are drawn one at a time from a bag containing 3 black balls, 4 red balls, and 5 white balls, and all 12 drawn balls are arranged in a horizontal row in the order they were drawn. Each ball in the bag is equally likely to be drawn.
  1. [(1)] Find the probability $p$ that no two red balls are adjacent to each other.
  2. [(2)] Given that no two red balls are adjacent to each other, find the conditional probability $q$ that no two black balls are adjacent to each other.

%% Page 3
Q3 Circles Chord Length and Chord Properties View
3

Let $a$ be a real number, and let $C$ be the circumference of the circle with center $(0,\, a)$ and radius $1$ in the coordinate plane.
  1. [(1)] Find the range of $a$ such that $C$ is entirely contained in the region represented by the inequality $y > x^2$.
  2. [(2)] Suppose $a$ is in the range found in (1). Let $S$ be the part of $C$ satisfying $x \geq 0$ and $y < a$. For a point $\mathrm{P}$ on $S$, let $L_{\mathrm{P}}$ be the length of the chord cut off from the tangent line to $C$ at $\mathrm{P}$ by the parabola $y = x^2$. Find the range of $a$ such that there exist two distinct points $\mathrm{Q}$, $\mathrm{R}$ on $S$ satisfying $L_{\mathrm{Q}} = L_{\mathrm{R}}$.

%% Page 4
Q4 Vectors 3D & Lines Multi-Part 3D Geometry Problem View
4 (See the solution/explanation page)
Consider the four points $\mathrm{O}(0,\ 0,\ 0)$, $\mathrm{A}(2,\ 0,\ 0)$, $\mathrm{B}(1,\ 1,\ 1)$, $\mathrm{C}(1,\ 2,\ 3)$ in coordinate space.
(1) Find the coordinates of the point P satisfying $\overrightarrow{\mathrm{OP}} \perp \overrightarrow{\mathrm{OA}}$, $\overrightarrow{\mathrm{OP}} \perp \overrightarrow{\mathrm{OB}}$, $\overrightarrow{\mathrm{OP}} \cdot \overrightarrow{\mathrm{OC}} = 1$.
(2) Drop a perpendicular from point P to line AB, and let H be the intersection of that perpendicular with line AB. Express $\overrightarrow{\mathrm{OH}}$ in terms of $\overrightarrow{\mathrm{OA}}$ and $\overrightarrow{\mathrm{OB}}$.
(3) Define point Q by $\overrightarrow{\mathrm{OQ}} = \dfrac{3}{4}\overrightarrow{\mathrm{OA}} + \overrightarrow{\mathrm{OP}}$, and consider the sphere $S$ centered at Q with radius $r$.
Find the range of $r$ such that $S$ has a common point with triangle OHB. Here, triangle OHB lies in the plane containing the three points O, H, B, and consists of the boundary and its interior.
%% Page 5
Q5 Factor & Remainder Theorem Proof of Polynomial Divisibility or Identity View
5 Go to the solutions page
Consider the polynomial $f(x) = (x-1)^2(x-2)$.
  1. [(1)] Let $g(x)$ be a polynomial with real coefficients, and let $r(x)$ be the remainder when $g(x)$ is divided by $f(x)$. Show that the remainder when $g(x)^7$ is divided by $f(x)$ equals the remainder when $r(x)^7$ is divided by $f(x)$.
  2. [(2)] Let $a$, $b$ be real numbers, and let $h(x) = x^2 + ax + b$. Let $h_1(x)$ be the remainder when $h(x)^7$ is divided by $f(x)$, and let $h_2(x)$ be the remainder when $h_1(x)^7$ is divided by $f(x)$. Find all pairs $a$, $b$ such that $h_2(x)$ equals $h(x)$.

%% Page 6