USTH Admissions Exam 2025: Mathematics and Logic Overview

Which of the following functions is an odd function?

What is the total time Vu spends collecting batteries from all 8 floors, considering 15 minutes per floor and 20 seconds between consecutive floors, including the return trip from Floor 8 to Floor 1?

Given the sequence $u_n = \cos\frac{2^n\pi}{3}$ for $n=1,2,3...$, what is the value of $u_{10}$?

Evaluate the limit: $\lim_{x \to +\infty} (-x^3 + x^2 + 2)$.

Four friends Ty, Suu, Dan, and Mao made statements about a 200m race. If exactly three statements are true and one is false, who finished first?

Given positive real numbers $a, b$ satisfying $a^{3b} = 4$, find $P = 3a^{9b} - 6$.

For positive real $a \neq 2$, find the value of $\log_{\frac{a^2}{4}}\left(\frac{a^3}{8}\right)$.

On which interval is the function $y = x^3 - 3x^2 + 2022$ decreasing?

If $f(x)$ is defined and increasing on $(0; +\infty)$, which statement is true?

How many extreme points does the graph of $y = 2x^4 + 5x^2 - 3$ have?

Find the equation of the tangent line to $y = x(1-x)^2$ at $x=3$.

For $f(x) = k\sqrt[3]{x^2} + \sqrt[4]{x^3} + 1$ on $[0; +\infty)$, find $k$ such that $f'(1) = 1$.

The tangent to $y = (2m-1)x^4 - m^2 + m - \frac{1}{4}$ at $x=1$ passes through $A(0; 9)$. Find $m$.

For $y = x^3 - 3x^2 - 2x - 1$, how many tangent lines are perpendicular to the line $x + 7y - 1 = 0$?

What is the negation of the proposition: "The sensor detects an error in the device"?

Find an antiderivative of $f(x) = 1 + \cos 3x$.

Find an antiderivative of $f(x) = 1 + \frac{1}{5x-2}$.

Find an antiderivative of $f(x) = 7^x + \cos x$.

Find the family of antiderivatives for $f(x) = \frac{2x-1}{(x-2)^2}$.

Given $f$ differentiable on $\mathbb{R}$ with $f(0)=1$, $f(x)>0$ for all $x$, and $f'(x) + x f(x) = 0$, find $f(2)$.

Given $f$ differentiable on $[0;1]$ with $f(0)=0$, $f(1)=1-e$, $\int_0^1 [f'(x)]^2 dx = \frac{e^2 - 1}{2}$, and $\int_0^1 e^x f(x) dx = e - \frac{e^2 + 1}{2}$, find $I = \int_0^1 f(x) dx$.

Four units (ICT, GED, FAS, DQE) competed for prizes with only two true statements among these: 1) ICT did not get 1st and FAS got 3rd; 2) ICT got 1st and GED got 2nd; 3) GED did not get 2nd and FAS got Consolation; 4) None of GED, FAS, DQE got Consolation and neither FAS nor DQE got 1st. Which unit won 1st prize?

Which statements are correct? 1) The angle between two planes is always acute or right; 2) The angle between $(\alpha)$ and $(\beta)$ equals the angle between $(\alpha)$ and $(\gamma)$ if $(\beta)$ is parallel or coincides with $(\gamma)$; 3) If the angle between $(\alpha)$ and $(\beta)$ equals the angle between $(\alpha)$ and $(\gamma)$, then $(\beta)$ is parallel to $(\gamma)$.

Find the coordinates of point $A$ if $\vec{OA} = \vec{i} - 3\vec{k}$ in space Oxyz.

Find the length of vector $\vec{u} = 2\vec{i} - \vec{j} - 2\vec{k}$ in space Oxyz.

Find the radius $R$ of the sphere centered at $I(2; 1; 2)$ tangent to the plane $(P): 2x - y + 2z - 1 = 0$.

Find the equation of the line on plane $(P): x - y - z = 0$ that intersects lines $d_1: \frac{x-2}{1} = \frac{y-1}{-5} = \frac{z-1}{-1}$ and $d_2: \frac{x-1}{2} = \frac{y-1}{-1} = \frac{z}{1}$.

Given sphere $(S): (x-1)^2 + (y-2)^2 + (z-3)^2 = 25$ and points $A(1; 0; 2)$, $B(-1; 2; 2)$, find $a+b+c$ if plane $(P): ax + by + cz + 3 = 0$ passes through $A$ and $B$ such that the circle cut from $(S)$ by $(P)$ has smallest radius.

How many planes pass through point $B \notin (\alpha)$ and are parallel to plane $(\alpha)$?

What is the volume of a pyramid with base area $12\text{ cm}^2$ and height $3\text{ cm}$?