Daily Math · 2026-07-22

Question 1

Let $A(2\log_4 8;\; 4\log_{1337}i)$, $B(\log_2 32;\; P_2)$, $C(0!;\; \log_2 512)$. Find the area of triangle $ABC$.

Accepted Answer

$\bullet$ Compute coordinates of $A$. Note $\log_4 8 = 	frac{3}{2}$ and treat $\log_{1337}|i| = 0$:

$$A = (2 \cdot 	frac{3}{2},\, 0) = (3,\, 0)$$

$\bullet$ Coordinates of $B$ from $\log_2 32 = 5$ and $P_2 = 2$:

$$B = (5,\, 2)$$

$\bullet$ Coordinates of $C$ from $0! = 1$ and $\log_2 512 = 9$:

$$C = (1,\, 9)$$

$\bullet$ Apply the area formula:

$$\mathcal{A} = 	frac{1}{2}|x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A - y_B)|$$

$$= 	frac{1}{2}|3(-7) + 5(9) + 1(-2)|$$

$$= 	frac{1}{2} \cdot 22 = 11$$

Answer

$7$

Answer

$11$

Answer

$22$

Answer

$44$

Answer

$9$

Answer

$\dfrac{11}{2}$

Question 2

The quotient when $P(X) = X^3 - 4X^2 + 5X - 2$ is divided by $X - 2$ has degree:

Accepted Answer

$\deg(Q) = 3 - 1 = 2$.

Answer

$1$

Answer

$2$

Answer

$3$

Answer

$4$

Question 3

A triangle has vertices $A(0, 0)$, $B(6, 0)$, $C(0, 4)$. Its area equals:

Accepted Answer

Base $= 6$, height $= 4$. Area $= \dfrac{1}{2} \cdot 6 \cdot 4 = 12$.

Answer

$6$

Answer

$10$

Answer

$12$

Answer

$24$

Question 4

If $x_1, x_2$ are the roots of $x^2 - 5x + 4 = 0$, the value of $x_1^2 + x_2^2$ is:

Accepted Answer

By Viète: $x_1 + x_2 = 5$, $x_1 x_2 = 4$. Hence $x_1^2 + x_2^2 = 25 - 8 = 17$.

Answer

$9$

Answer

$13$

Answer

$17$

Answer

$25$

Question 5

The limit $\displaystyle\lim_{x 	o 0} \dfrac{(1 + x)^r - 1}{x}$ equals (for $r \in \mathbb{R}$):

Accepted Answer

$\displaystyle\lim_{x 	o 0} \dfrac{(1+x)^r - 1}{x} = r$. (This generalizes the bonomial-derivative-at-zero.)

Answer

$0$

Answer

$1$

Answer

$r$

Answer

$r + 1$

Question 6

For $I_n = \displaystyle\int_0^1 \dfrac{x^n}{1 + x} \, dx$, $\lim_{n 	o \infty} I_n$ equals:

Accepted Answer

$0 \le I_n \le \dfrac{1}{n + 1} 	o 0$, so $I_n 	o 0$ by the squeeze theorem.

Answer

$0$

Answer

$\dfrac{1}{2}$

Answer

$\ln 2$

Answer

$\infty$

Question 7

The total number of asymptotes (vertical + horizontal + slant) of $f(x) = \dfrac{x^2 + 1}{x^2 - 1}$ is:

Accepted Answer

Vertical: $x = \pm 1$ (two). Horizontal: $\lim f(x) = 1$ (one). Total: $3$. (No slant — the rational function has equal-degree numerator and denominator.)

Answer

$1$

Answer

$2$

Answer

$3$

Answer

$4$

Question 8

A ring homomorphism $\varphi: R 	o S$ must satisfy:

Accepted Answer

A ring homomorphism preserves both operations: addition AND multiplication.

Answer

only $\varphi(a + b) = \varphi(a) + \varphi(b)$

Answer

only $\varphi(ab) = \varphi(a)\varphi(b)$

Answer

both $\varphi(a + b) = \varphi(a) + \varphi(b)$ and $\varphi(ab) = \varphi(a)\varphi(b)$

Answer

$\varphi$ is bijective

Question 9

For $n \ge 2$, the number of even permutations in $S_n$ (i.e. $|A_n|$) is:

Accepted Answer

$|A_n| = \dfrac{n!}{2}$ (one of the standard properties of the alternating group).

Answer

$n!$

Answer

$\dfrac{n!}{n}$

Answer

$\dfrac{n!}{2}$

Answer

$(n-1)!$

Question 10

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 — Analytic Geometry

Problem 2 — Polynomials in ℂ

Problem 3 — Distances & Areas

Problem 4 — Quadratic Function

Problem 5 — Limits of Functions

Problem 6 — Recursive Integrals

Problem 7 — Asymptotes

Problem 8 — Homomorphisms

Problem 9 — Permutations & Symmetric Groups

Problem 10 — Trigonometric Equations