Daily Math · 2026-07-19

Question 1

A semigroup is a non-empty set with a binary operation that is:

Accepted Answer

A semigroup is a set with an associative binary operation.

Answer

commutative

Answer

associative

Answer

both

Answer

neither

Question 2

For $\vec{u} = 3\vec{i} + 4\vec{j}$, the magnitude $|\vec{u}|$ equals:

Accepted Answer

$|\vec{u}| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$.

Answer

$3$

Answer

$4$

Answer

$5$

Answer

$7$

Question 3

The number of subsets of $A = \{a, b, c, d\}$ is:

Accepted Answer

$|A| = 4$, so $A$ has $2^4 = 16$ subsets (including $\emptyset$ and $A$ itself).

Answer

$4$

Answer

$8$

Answer

$16$

Answer

$24$

Question 4

The Klein four-group $V_4$ has the property that:

Accepted Answer

In $V_4$, every non-identity element has order $2$. (This distinguishes it from $\mathbb{Z}_4$, which has an element of order $4$.)

Answer

it has an element of order $4$

Answer

it is non-abelian

Answer

every non-identity element has order $2$

Answer

it is isomorphic to $\mathbb{Z}_4$

Question 5

For which value of $a \in \mathbb{R}$ is $A = \begin{pmatrix} 1 & a \ 3 & 2 \end{pmatrix}$ symmetric?

Accepted Answer

We need $a_{12} = a_{21}$, i.e. $a = 3$.

Answer

$-3$

Answer

$0$

Answer

$3$

Answer

$1$

Question 6

Rotating $y = 2x + 1$ on $[0, 2]$ about the $x$-axis generates a frustum (truncated cone) of volume:

Accepted Answer

$V = \pi \int_0^2 (4x^2 + 4x + 1) \, dx = \pi\!\left[\dfrac{4x^3}{3} + 2x^2 + x\right]_0^2 = \pi\!\left(\dfrac{32}{3} + 8 + 2\right) = \pi \cdot \dfrac{32 + 30}{3} = \dfrac{62\pi}{3}$.

Answer

$5\pi$

Answer

$14\pi$

Answer

$\dfrac{62\pi}{3}$

Answer

$\dfrac{62}{3}$

Question 7

The maximum value of $f(x) = \sin x + \cos x$ over $\mathbb{R}$ is:

Accepted Answer

$\sin x + \cos x = \sqrt{2}\!\left(\sin x \cdot \dfrac{\sqrt{2}}{2} + \cos x \cdot \dfrac{\sqrt{2}}{2}\right) = \sqrt{2}\sin\!\left(x + \dfrac{\pi}{4}\right)$.

Since $\sin$ has maximum $1$, the maximum of $f$ is $\sqrt{2}$, attained at $x = \dfrac{\pi}{4}$.

Answer

$1$

Answer

$\sqrt{2} - 1$

Answer

$\sqrt{2}$

Answer

$2$

Question 8

$\displaystyle\sum_{k=1}^{99} \dfrac{1}{k(k+1)}$ equals:

Accepted Answer

The sum telescopes: $\sum_{k=1}^{99} \left(\dfrac{1}{k} - \dfrac{1}{k+1}\right) = 1 - \dfrac{1}{100} = \dfrac{99}{100}$.

Answer

$	frac{99}{100}$

Answer

$	frac{100}{99}$

Answer

$	frac{1}{100}$

Answer

$1$

Question 9

A homogeneous system $AX = 0$ with $A$ a $3 	imes 3$ matrix has non-trivial solutions iff:

Accepted Answer

A square homogeneous system has non-trivial (non-zero) solutions iff $\det(A) = 0$.

Answer

$\det(A) = 1$

Answer

$\det(A) > 0$

Answer

$\det(A) = 0$

Answer

always

Question 10

Determine $n \in \mathbb{N}^*$ for which the binomial expansion $(1 + x)^n$ has $11$ terms.

Accepted Answer

$n + 1 = 11 \Rightarrow n = 10$.

Answer

$9$

Answer

$10$

Answer

$11$

Answer

$12$

Problems & worked solutions

Problem 1 — Semigroups & Monoids

Problem 2 — Vectors in the Plane

Problem 3 — Sets of Real Numbers

Problem 4 — Groups

Problem 5 — Matrices

Problem 6 — Volumes of Revolution

Problem 7 — Trigonometric Identities

Problem 8 — Sequences

Problem 9 — Systems of Linear Equations

Problem 10 — Binomial Theorem