Daily Math · 2026-07-28

Question 1

For $f: (0, +\infty) 	o \mathbb{R}$, $f(x) = \dfrac{1}{x}$, which statement is true?

Accepted Answer

$f(1) = 1$, $f(2) = \dfrac{1}{2}$. As $x$ grows, $\dfrac{1}{x}$ shrinks — $f$ is strictly decreasing on $(0, +\infty)$.

Answer

$f$ is strictly increasing on $(0, +\infty)$

Answer

$f$ is strictly decreasing on $(0, +\infty)$

Answer

$f$ has a minimum on $(0, +\infty)$

Answer

$f$ is constant on $(0, +\infty)$

Question 2

The remainder of $P(X) = X^4 - 2X + 1$ divided by $X - 1$ is:

Accepted Answer

$P(1) = 1 - 2 + 1 = 0$. So $X - 1$ divides $P$ exactly.

Answer

$0$

Answer

$1$

Answer

$2$

Answer

$-1$

Question 3

The limit $\displaystyle\lim_{n 	o \infty} \dfrac{3n^2 + 5}{2n^2 - 1}$ equals:

Accepted Answer

$\dfrac{3n^2 + 5}{2n^2 - 1} = \dfrac{3 + 5/n^2}{2 - 1/n^2} 	o \dfrac{3}{2}$.

Answer

$0$

Answer

$1$

Answer

$\dfrac{3}{2}$

Answer

$\infty$

Question 4

For which value of $m \in \mathbb{R}$ is $X = 2$ a root of $P(X) = X^3 - mX + 4$?

Accepted Answer

$P(2) = 8 - 2m + 4 = 12 - 2m = 0 \Rightarrow m = 6$.

Answer

$-6$

Answer

$-4$

Answer

$6$

Answer

$8$

Question 5

On the power set $\mathcal{P}(X)$, the symmetric difference $A 	riangle B = (A \setminus B) \cup (B \setminus A)$ has identity:

Accepted Answer

$A 	riangle \emptyset = A$, so $\emptyset$ is the identity.

Answer

$\emptyset$

Answer

$X$

Answer

$\{x\}$ for some $x \in X$

Answer

no identity

Question 6

For $n \ge 2$, the sum of all $n$th roots of unity equals:

Accepted Answer

The polynomial $z^n - 1$ has zero coefficient for $z^{n-1}$, so by Viète's formula the sum of its roots is $0$. Equivalently, the geometric sum $\sum_{k=0}^{n-1} e^{2\pi i k / n} = \dfrac{1 - 1}{1 - e^{2\pi i / n}} = 0$ (for $n \ge 2$).

Answer

$0$

Answer

$1$

Answer

$n$

Answer

$\dfrac{1}{n}$

Question 7

Given the matrix $A = \begin{pmatrix} 3 & 1 \ 5 & 4 \end{pmatrix}$. Then $A^2$ is:

Accepted Answer

$A^2 = A \cdot A = \begin{pmatrix} 3 & 1 \ 5 & 4 \end{pmatrix} \begin{pmatrix} 3 & 1 \ 5 & 4 \end{pmatrix} = \begin{pmatrix} 3 \cdot 3 + 1 \cdot 5 & 3 \cdot 1 + 1 \cdot 4 \ 5 \cdot 3 + 4 \cdot 5 & 5 \cdot 1 + 4 \cdot 4 \end{pmatrix} = \begin{pmatrix} 14 & 7 \ 35 & 21 \end{pmatrix}$.

Answer

$\begin{pmatrix} 2 & 10 \ 30 & 19 \end{pmatrix}$

Answer

$\begin{pmatrix} 1 & 3 \ -1 & 4 \end{pmatrix}$

Answer

$\begin{pmatrix} 4 & 9 \ 10 & 11 \end{pmatrix}$

Answer

$\begin{pmatrix} 14 & 7 \ 35 & 21 \end{pmatrix}$

Answer

$\begin{pmatrix} 10 & 11 \ 14 & 15 \end{pmatrix}$

Answer

$\begin{pmatrix} 20 & 12 \ 35 & 20 \end{pmatrix}$

Question 8

The function $f(x) = e^{-x}$ has at $x 	o \infty$:

Accepted Answer

$\lim_{x 	o \infty} e^{-x} = 0$, so $y = 0$ is a horizontal asymptote at $+\infty$.

Answer

a horizontal asymptote at $y = 0$

Answer

a horizontal asymptote at $y = 1$

Answer

a vertical asymptote at $x = 0$

Answer

no asymptote

Question 9

If $x = \log_2 5$, then $4^x$ equals:

Accepted Answer

$4^x = 2^{2x} = (2^x)^2 = 5^2 = 25$.

Answer

$5$

Answer

$10$

Answer

$25$

Answer

$\log_2 25$

Question 10

By the Rouché–Capelli theorem, a system has solutions iff:

Accepted Answer

A linear system is consistent (has at least one solution) iff $\operatorname{rank}(A) = \operatorname{rank}([A | b])$ — the rank of the coefficient matrix equals the rank of the augmented matrix.

Answer

$\det(A) = 0$

Answer

$\operatorname{rank}(A) = \operatorname{rank}([A | b])$

Answer

$\operatorname{rank}(A) > \operatorname{rank}([A | b])$

Answer

The system is square

Problems & worked solutions

Problem 1 — Elementary Functions

Problem 2 — Polynomial Rings

Problem 3 — Limits of Sequences

Problem 4 — Polynomials in ℂ

Problem 5 — Binary Operations

Problem 6 — Complex Numbers

Problem 7 — Matrices

Problem 8 — Asymptotes

Problem 9 — Powers, Radicals, Logarithms

Problem 10 — Systems of Linear Equations