Daily Math · 2026-06-13

Question 1

The numbers $5, 8, 11, 14, \ldots$ form an arithmetic progression. The common difference $r$ equals:

Accepted Answer

$r = a_2 - a_1 = 8 - 5 = 3$.

Answer

$2$

Answer

$3$

Answer

$4$

Answer

$5$

Question 2

The map $\varphi: (\mathbb{R}, +) 	o ((0, \infty), \cdot)$ defined by $\varphi(x) = e^x$ is:

Accepted Answer

$e^{a + b} = e^a e^b$ ✓ (homomorphism). $e^x$ is strictly increasing (injective) and surjective onto $(0, \infty)$. Hence an isomorphism. (Inverse: $\ln$.)

Answer

not a homomorphism

Answer

a non-injective homomorphism

Answer

injective but not surjective

Answer

an isomorphism

Question 3

On $[0, 2\pi)$, the equation $\cos 2x + \cos x = 0$ has exactly:

Accepted Answer

$2\cos^2 x + \cos x - 1 = 0 \Rightarrow (2\cos x - 1)(\cos x + 1) = 0$. $\cos x = 1/2$: 2 solutions ($\pi/3, 5\pi/3$). $\cos x = -1$: 1 solution ($\pi$). Total: $3$.

Answer

$1$ solution

Answer

$2$ solutions

Answer

$3$ solutions

Answer

$4$ solutions

Problems & worked solutions

Problem 1 — Arithmetic Sequences

Problem 2 — Homomorphisms

Problem 3 — Trigonometric Equations