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 science_official

6 maths questions

Q1 Vectors Introduction & 2D Angle or Cosine Between Vectors View
Let $A(0, -1, 1)$ be a point in coordinate space. Suppose a point $P$ in the $xy$-plane satisfies all of the following conditions (i), (ii), (iii).
  • [(i)] $P$ is different from the origin $O$.
  • [(ii)] $\angle AOP \geq \dfrac{2}{3}\pi$
  • [(iii)] $\angle OAP \leq \dfrac{\pi}{6}$

Sketch the region that $P$ can occupy in the $xy$-plane.
Q2 Differentiating Transcendental Functions Differentiation under the integral sign with transcendental kernels View
Consider the following function $f(x)$.
$$f(x) = \int_0^1 \frac{|t - x|}{1 + t^2}\,dt \qquad (0 \leq x \leq 1)$$
(1) Find the real number $\alpha$ satisfying $0 < \alpha < \dfrac{\pi}{4}$ such that $f'(\tan \alpha) = 0$.
(2) For the value of $\alpha$ found in (1), find the value of $\tan \alpha$.
(3) Find the maximum value and minimum value of the function $f(x)$ on the interval $0 \leq x \leq 1$. You may use the fact that $0.69 < \log 2 < 0.7$ if necessary.
Q3 Independent Events View
Consider a point P moving on the coordinate plane every second according to the following rules (i), (ii).
(i) Initially, P is at the point $(2, 1)$.
(ii) When P is at the point $(a, b)$ at some moment, one second later P is at:
  • with probability $\dfrac{1}{3}$, the point symmetric to $(a, b)$ with respect to the $x$-axis
  • with probability $\dfrac{1}{3}$, the point symmetric to $(a, b)$ with respect to the $y$-axis
  • with probability $\dfrac{1}{6}$, the point symmetric to $(a, b)$ with respect to the line $y = x$
  • with probability $\dfrac{1}{6}$, the point symmetric to $(a, b)$ with respect to the line $y = -x$

Answer the following questions. For question (1), you only need to state the conclusion.
(1) Find all possible coordinates of points that P can occupy.
(2) Let $n$ be a positive integer. Show that the probability that P is at the point $(2, 1)$ after $n$ seconds from the start equals the probability that P is at the point $(-2, -1)$ after $n$ seconds from the start.
(3) Let $n$ be a positive integer. Find the probability that P is at the point $(2, 1)$ after $n$ seconds from the start.
Q4 Circles Circle Equation Derivation View
Let $f(x) = -\dfrac{\sqrt{2}}{4}x^2 + 4\sqrt{2}$. For a real number $t$ satisfying $0 < t < 4$, let $C_t$ be the circle that passes through the point $(t, f(t))$ in the coordinate plane, has a common tangent line with the parabola $y = f(x)$ at this point, and has its center on the $x$-axis.
  1. [(1)] Let the center of circle $C_t$ have coordinates $(c(t), 0)$ and radius $r(t)$. Express $c(t)$ and $\{r(t)\}^2$ as polynomials in $t$.
  2. [(2)] Suppose the real number $a$ satisfies $0 < a < f(3)$. How many real numbers $t$ in the range $0 < t < 4$ are there such that the circle $C_t$ passes through the point $(3, a)$?
Q5 Volumes of Revolution Volume of Revolution about a Horizontal Axis (Evaluate) View
In coordinate space, take three points $A(1, 0, 0)$, $B(0, 1, 0)$, $C(0, 0, 1)$, and let $D$ be the midpoint of segment $AC$. Find the volume of the solid obtained by rotating the boundary and interior of triangle $ABD$ one full revolution about the $x$-axis.
Q6 Proof Proof Involving Combinatorial or Number-Theoretic Structure View
An integer greater than or equal to 2 that has no positive divisors other than 1 and itself is called a prime number. Answer the following questions.
  • [(1)] Let $f(x) = x^3 + 10x^2 + 20x$. Find all integers $n$ such that $f(n)$ is a prime number.
  • [(2)] Let $a$, $b$ be integer constants, and let $g(x) = x^3 + ax^2 + bx$. Show that the number of integers $n$ such that $g(n)$ is a prime number is at most 3.