Daily Math · 2026-06-04

Question 1

A semigroup is a non-empty set with a binary operation that is:

Accepted Answer

A semigroup is a set with an associative binary operation.

Answer

commutative

Answer

associative

Answer

both

Answer

neither

Question 2

Find $\displaystyle\lim_{x	o0}\dfrac{\cos2x-\cos3x}{\sin5x\cdot\sin8x}$.

Accepted Answer

$\bullet$ Convert the numerator via sum-to-product:

$$\cos 2x-\cos 3x=2\sin	frac{5x}{2}\sin	frac{x}{2}$$

$\bullet$ Apply $\sin u\sim u$ four times. Numerator:

$$2\sin	frac{5x}{2}\sin	frac{x}{2}\sim 2\cdot	frac{5x}{2}\cdot	frac{x}{2}=	frac{5x^2}{2}$$

$\bullet$ Denominator:

$$\sin 5x\cdot\sin 8x\sim 5x\cdot 8x=40x^2$$

$\bullet$ Take the ratio:

$$\frac{5x^2/2}{40x^2}=	frac{1}{16}$$

Answer

$\dfrac{1}{40}$

Answer

$\dfrac{1}{16}$

Answer

$\dfrac{5}{16}$

Answer

$0$

Answer

$\dfrac{5}{40}$

Answer

$\dfrac{1}{8}$

Question 3

The set of complex numbers $z$ satisfying $|z - 1| = |z + 1|$ is:

Accepted Answer

$|z - 1| = |z - (-1)|$ describes the set of points equidistant from $1$ and $-1$. That's the perpendicular bisector of the segment between them, i.e. the imaginary axis $\operatorname{Re}(z) = 0$.

Answer

A circle centered at the origin

Answer

A circle centered at $1$

Answer

The imaginary axis

Answer

The real axis

Problems & worked solutions

Problem 1 — Semigroups & Monoids

Problem 2 — Limits

Problem 3 — Complex Numbers