Daily Math · 2026-05-19

Question 1

The distance from the origin to the point $P(-3, 4)$ is:

Accepted Answer

$d = \sqrt{(-3)^2 + 4^2} = \sqrt{25} = 5$.

Answer

$1$

Answer

$5$

Answer

$7$

Answer

$25$

Question 2

Let $A=\begin{pmatrix}1&0&2&4\ i^2&1&3&\lg 100\ 0&1&3&\ln 1\ \sin\frac{\pi}{2}&i^{24}&C_4^3&7\end{pmatrix}$. Find $\det(A)$.

Accepted Answer

$\bullet$ Evaluate each special entry:

$$i^2 = -1,\quad \lg 100 = 2,\quad \ln 1 = 0$$

$$\sin	frac{\pi}{2} = 1,\quad i^{24} = 1,\quad C_4^3 = 4$$

$\bullet$ Assemble the numerical matrix:

$$A = \begin{pmatrix}1 & 0 & 2 & 4\ -1 & 1 & 3 & 2\ 0 & 1 & 3 & 0\ 1 & 1 & 4 & 7\end{pmatrix}$$

$\bullet$ Row-reduce. $R_2 \mathrel{+}= R_1$, $R_4 \mathrel{-}= R_1$:

$$\begin{pmatrix}1 & 0 & 2 & 4\ 0 & 1 & 5 & 6\ 0 & 1 & 3 & 0\ 0 & 1 & 2 & 3\end{pmatrix}$$

$\bullet$ $R_3 \mathrel{-}= R_2$, $R_4 \mathrel{-}= R_2$:

$$\begin{pmatrix}1 & 0 & 2 & 4\ 0 & 1 & 5 & 6\ 0 & 0 & -2 & -6\ 0 & 0 & -3 & -3\end{pmatrix}$$

$\bullet$ $R_4 \mathrel{-}= 	frac{3}{2} R_3$ gives upper-triangular with pivots $1, 1, -2, 6$:

$$\det A = 1 \cdot 1 \cdot (-2) \cdot 6 = -12$$

Answer

$-12$

Answer

$1$

Answer

$-3$

Answer

$6$

Answer

$-1$

Answer

$3$

Question 3

In a monoid $(M, *)$ with cancellation, if $a * b = a * c$ then:

Accepted Answer

The cancellation law states: $a * b = a * c \Rightarrow b = c$. (Even without inverses, this property may or may not hold; in a group it always does.)

Answer

$b = e$ (identity)

Answer

$a$ is invertible

Answer

$b = c$

Answer

$b * c = e$

Problems & worked solutions

Problem 1 — Distances & Areas

Problem 2 — Matrices

Problem 3 — Semigroups & Monoids