Daily Math · 2026-06-16

Question 1

In $	riangle ABC$, $AB = 5$, $BC = 12$, $AC = 13$. The area equals:

Accepted Answer

Since $5^2 + 12^2 = 13^2$, $	riangle ABC$ is right-angled at $B$. Area $= \dfrac{1}{2} \cdot 5 \cdot 12 = 30$.

Answer

$20$

Answer

$24$

Answer

$30$

Answer

$60$

Question 2

Find $\displaystyle\int_{-2}^{2}\dfrac{x^5+x^3+x}{x^4+x^2+1}\,dx+i^{2026}\cdot\log_2 1024$.

Accepted Answer

$\bullet$ Factor the numerator:

$$x^5+x^3+x=x(x^4+x^2+1)$$

$\bullet$ The integrand simplifies to $x$, which is odd, so by symmetry:

$$\int_{-2}^{2}x\,dx=0$$

$\bullet$ Decode the second term:

$$i^{2026}=i^{2}=-1$$

$$\log_2 1024=10$$

$\bullet$ Combine:

$$0+(-1)(10)=-10$$

Answer

$0$

Answer

$10$

Answer

$-1$

Answer

$-10$

Answer

$-2^{2026}$

Answer

Diverges

Question 3

How many solutions does $2\sin^2 x - 3\sin x + 1 = 0$ have on $[0, 2\pi)$?

Accepted Answer

$(2\sin x - 1)(\sin x - 1) = 0$. From $\sin x = 	frac{1}{2}$: $x = 	frac{\pi}{6}, 	frac{5\pi}{6}$. From $\sin x = 1$: $x = 	frac{\pi}{2}$. Total: $3$.

Answer

$2$

Answer

$3$

Answer

$4$

Answer

$5$

Problems & worked solutions

Problem 1 — Solving Triangles

Problem 2 — Integrals

Problem 3 — Trigonometric Equations