Daily math problems for August 25, 2026 — Descriptive Statistics & Sampling, Calculus, Algebra & more

${\displaystyle \frac{2+4+6+8+10}{5}}={\displaystyle \frac{30}{5}}=6$.

By the power rule, $\frac{d}{dx}{x}^3=3x^2$.

The permutation maps $1\to 2$, $2\to 3$, $3\to 1$, which is the $3$-cycle $(123)$.

$A$ is $2\times 3$, so $A^T$ is $3\times 2$.

$(1,0,0)\times (0,2,0)=(0\cdot 0-0\cdot 2,0\cdot 0-1\cdot 0,1\cdot 2-0\cdot 0)=(0,0,2)$.

$\bullet$ Expand $\sqrt{x^2+1}$ for large $x$:

$$\sqrt{x^2+1}=x\sqrt{1+\frac{1}{x^2}}\approx x+\frac{1}{2x}$$

$\bullet$ The expression behaves like:

$$(1+a)x+b+\frac{1}{2x}$$

$\bullet$ For a finite limit the $x$-coefficient must vanish:

$$1+a=0\Rightarrow a=-1$$

$\bullet$ Then the limit equals $b$, which must be $2$:

$$b=2$$

Max $=22$, min $=4$, so range $=22-4=18$.

$\bullet$ Eliminate the first column using $2R_2-R_1$ and $2R_3-R_1$:

$$2R_2-R_1=(0,3,2a-1,3)$$

$$2R_3-R_1=(0,-1,1,2b+1)$$

$\bullet$ For rank $2$, these two rows must be proportional. Matching the second components gives ratio $-3$:

$$2a-1=-3\cdot 1\Rightarrow a=-1$$

$$3=-3(2b+1)\Rightarrow b=-1$$

$\bullet$ So the answer is:

$$(a,b)=(-1,-1)$$

By Pascal: $\left(\genfrac{}{}{0px}{}{7}{3}\right)+\left(\genfrac{}{}{0px}{}{7}{4}\right)=\left(\genfrac{}{}{0px}{}{8}{4}\right)={\displaystyle \frac{8!}{4!4!}}=70$.

$\sqrt{x+5}=2+\sqrt{x-3}$. Square: $x+5=4+4\sqrt{x-3}+x-3\Rightarrow 4=4\sqrt{x-3}\Rightarrow \sqrt{x-3}=1\Rightarrow x=4$. Check: $\sqrt{9}-\sqrt{1}=3-1=2$ ✓.