Daily Math · 2026-05-23

Question 1

Let $A = \{1, 2, 3\}$, $B = \{2, 3, 4\}$, with universal set $U = \{1, 2, 3, 4, 5\}$. Compute $\overline{A \cap B}$.

Accepted Answer

$A \cap B = \{2, 3\}$. The complement in $U = \{1, 2, 3, 4, 5\}$ is $\{1, 4, 5\}$.

Answer

$\{2, 3\}$

Answer

$\{1, 4, 5\}$

Answer

$\{1, 2, 4\}$

Answer

$\{1, 2, 3, 4, 5\}$

Question 2

How many solutions does $2\cos^2 x = 1$ have on $[0, 2\pi)$?

Accepted Answer

$\cos^2 x = 	frac{1}{2} \Rightarrow \cos x = \pm 	frac{\sqrt{2}}{2}$. Solutions: $	frac{\pi}{4}, 	frac{3\pi}{4}, 	frac{5\pi}{4}, 	frac{7\pi}{4}$.

Answer

$2$

Answer

$3$

Answer

$4$

Answer

$6$

Question 3

$\displaystyle\lim_{n 	o \infty} \dfrac{\sqrt[n]{n!}}{n}$ equals:

Accepted Answer

By Stirling, $\sqrt[n]{n!} \sim \dfrac{n}{e}\,(2\pi n)^{1/(2n)}$. The factor $(2\pi n)^{1/(2n)} 	o 1$, so $\dfrac{\sqrt[n]{n!}}{n} 	o \dfrac{1}{e}$.

Answer

$	frac{1}{e}$

Answer

$1$

Answer

$e$

Answer

$0$

Problems & worked solutions