Daily Math · 2026-05-11

Question 1

In $\mathbb{Z}_6$, the value of $\hat{4} + \hat{5}$ is:

Accepted Answer

$4 + 5 = 9$, and $9 \bmod 6 = 3$. So the answer is $\hat{3}$.

Answer

$\hat{0}$

Answer

$\hat{3}$

Answer

$\hat{9}$

Answer

$\hat{5}$

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

Two boxes — Box A has $3$ red and $1$ blue; Box B has $1$ red and $3$ blue. A box is chosen at random and a ball is drawn. Given the ball is red, the probability it came from Box A is:

Accepted Answer

$P(A) = P(B) = \dfrac{1}{2}$, $P(R|A) = \dfrac{3}{4}$, $P(R|B) = \dfrac{1}{4}$.

$P(A|R) = \dfrac{\dfrac{3}{4} \cdot \dfrac{1}{2}}{\dfrac{3}{4} \cdot \dfrac{1}{2} + \dfrac{1}{4} \cdot \dfrac{1}{2}} = \dfrac{3/8}{4/8} = \dfrac{3}{4}$.

Answer

$\dfrac{1}{4}$

Answer

$\dfrac{1}{2}$

Answer

$\dfrac{3}{4}$

Answer

$\dfrac{3}{8}$

Problems & worked solutions

Problem 1 — Modular Arithmetic (ℤₙ)

Problem 2 — Integrals

Problem 3 — Probability