Daily math problems for September 27, 2026 — Algebra, Quadratic Function, Logarithms & more

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

$\mathrm{\Delta }=(-3{)}^2-4\cdot 2\cdot (-5)=9+40=49$.

${\log }_3(81)={\log }_3(3^4)=4$ and ${\log }_3(3)=1$. The difference is $4-1=3$.

With $r=2$ and $n=5$: ${\displaystyle \frac{2^6-1}{2-1}}={\displaystyle \frac{63}{1}}=63$. (Check: $1+2+4+8+16+32=63$.)

$x^2=3\cdot 27=81\Rightarrow x=9$ (positive ratio).

$f$ is invertible iff its slope $a-2\mathrm{\ne }0$. So $f$ fails to be invertible exactly when $a=2$ (it then becomes the constant $5$, neither injective nor surjective).

$\left(\genfrac{}{}{0px}{}{5}{3}\right)={\displaystyle \frac{5!}{3!\cdot 2!}}=10$.

${\mathbb{Z}}_n$ is a field iff every non-zero element is invertible, iff every element from $1$ to $n-1$ is coprime to $n$, iff $n$ is prime.

By parts: $\int x\ln xdx={\displaystyle \frac{x^2}{2}}\ln x-\int {\displaystyle \frac{x}{2}}dx={\displaystyle \frac{x^2}{2}}\ln x-{\displaystyle \frac{x^2}{4}}$. Evaluating on $[0,1]$ (with $x^2\ln x\to 0$ as $x\to 0^{+}$) gives $0-\frac{1}{4}-0=-\frac{1}{4}$.

The exponent ${\displaystyle \frac{6-3k}{2}}$ is an integer when $k$ is even: $k\in \{0,2,4,6\}$ — four values.