Daily Math · 2026-07-05

Question 1

A group is a non-empty set with a binary operation satisfying:

Accepted Answer

Group axioms: associativity, identity element, every element has an inverse. Commutativity is the additional axiom for an **abelian** group.

Answer

only associativity

Answer

only existence of identity

Answer

associativity, identity, and inverses

Answer

only commutativity

Question 2

Evaluate $\displaystyle\int_0^{10}\dfrac{\sqrt[7]{x^3}}{x^2}\,dx$.

Accepted Answer

$\bullet$ Simplify the integrand:

$$\frac{\sqrt[7]{x^3}}{x^2}=\frac{x^{3/7}}{x^2}=x^{-11/7}$$

$\bullet$ The integral $\int_0^a x^p\,dx$ converges only when $p>-1$:

$$-	frac{11}{7}<-1$$

$\bullet$ Hence the integral diverges at the lower endpoint $x=0$:

$$\int_0^{10}x^{-11/7}\,dx=	ext{Diverges}$$

Answer

$\dfrac{7}{4}\cdot10^{4/7}$

Answer

$\dfrac{7}{4}$

Answer

$7$

Answer

Diverges

Answer

$\dfrac{4}{7}$

Answer

$10^{-11/7}$

Question 3

A stock falls $20\%$, then rises $20\%$. The net change is closest to:

Accepted Answer

After both moves, the value is $0.8 \cdot 1.2 = 0.96$ — a $4\%$ net loss.

Answer

$0\%$

Answer

$-2\%$

Answer

$-4\%$

Answer

$+4\%$

Problems & worked solutions