Daily Math · 2026-07-16

Question 1

Let $f, g: \mathbb{R} 	o \mathbb{R}$, $f(x) = 2x - 1$ and $g(x) = x + 3$. Compute $(f \circ g)(2)$.

Accepted Answer

$g(2) = 5$, then $f(5) = 2 \cdot 5 - 1 = 9$. Hence $(f \circ g)(2) = 9$.

Answer

$4$

Answer

$7$

Answer

$9$

Answer

$11$

Question 2

Let $A = \{1, 2, 3\}$, $B = \{2, 3, 4\}$, with universal set $U = \{1, 2, 3, 4, 5\}$. Compute $\overline{A \cap B}$.

Accepted Answer

$A \cap B = \{2, 3\}$. The complement in $U = \{1, 2, 3, 4, 5\}$ is $\{1, 4, 5\}$.

Answer

$\{2, 3\}$

Answer

$\{1, 4, 5\}$

Answer

$\{1, 2, 4\}$

Answer

$\{1, 2, 3, 4, 5\}$

Question 3

The system $\begin{cases} x + y = 1 \ 2x + 2y = 5 \end{cases}$ has:

Accepted Answer

Multiplying the first by $2$: $2x + 2y = 2 
e 5$. Hence the system is inconsistent — no solution.

Answer

a unique solution

Answer

no solution

Answer

infinitely many solutions

Answer

exactly two solutions

Question 4

Compute $f'(1)$ where $f(x) = \dfrac{\ln x}{x}$.

Accepted Answer

$f'(x) = \dfrac{(1/x) \cdot x - \ln x \cdot 1}{x^2} = \dfrac{1 - \ln x}{x^2}$. At $x = 1$: $\dfrac{1 - 0}{1} = 1$.

Answer

$0$

Answer

$1$

Answer

$-1$

Answer

$	frac{1}{2}$

Question 5

The solution set of $|x - 2| < 3$ is:

Accepted Answer

$|x - 2| < 3 \Leftrightarrow -3 < x - 2 < 3 \Leftrightarrow -1 < x < 5$. The solution set is $(-1, 5)$.

Answer

$(-3, 3)$

Answer

$(-1, 5)$

Answer

$(2, 5)$

Answer

$[-1, 5]$

Question 6

For $f$ continuous and non-negative on $[a, b]$ with $b > a$:

Accepted Answer

The integral of a non-negative continuous function over $[a, b]$ is non-negative (representing the area under the curve).

Answer

$\int_a^b f < 0$ always

Answer

$\int_a^b f = 0$ always

Answer

$\int_a^b f \ge 0$

Answer

$\int_a^b f$ depends only on $b$

Question 7

The number of real solutions of $X^4 - 5X^2 + 4 = 0$ is:

Accepted Answer

With $Y = X^2$: $Y^2 - 5Y + 4 = 0 \Rightarrow Y \in \{1, 4\}$. Then $X^2 = 1 \Rightarrow X = \pm 1$ and $X^2 = 4 \Rightarrow X = \pm 2$. Four real solutions.

Answer

$1$

Answer

$2$

Answer

$3$

Answer

$4$

Question 8

For $A = \begin{pmatrix} 0 & 1 \ 0 & 0 \end{pmatrix}$, the matrix $A^2$ equals:

Accepted Answer

$A^2 = \begin{pmatrix} 0 \cdot 0 + 1 \cdot 0 & 0 \cdot 1 + 1 \cdot 0 \ 0 \cdot 0 + 0 \cdot 0 & 0 \cdot 1 + 0 \cdot 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}$. ($A$ is nilpotent.)

Answer

$A$

Answer

$I_2$

Answer

$\begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}$

Answer

$\begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}$

Question 9

The maximum value of $f(x) = \sin x + \sqrt{3}\cos x$ is:

Accepted Answer

$R = \sqrt{1 + 3} = 2$, so $f(x) = 2\sin(x + \pi/3)$. Max value: $2$.

Answer

$1 + \sqrt{3}$

Answer

$2$

Answer

$\sqrt{3}$

Answer

$4$

Question 10

The factorization of $P(X) = X^3 + X^2 - 4X - 4$ over $\mathbb{R}$ is:

Accepted Answer

$P(-1) = 0$, so $(X + 1)$ is a factor. Polynomial division: $P(X) = (X + 1)(X^2 - 4) = (X + 1)(X - 2)(X + 2)$.

Answer

$(X - 1)(X^2 + 4)$

Answer

$(X + 1)(X - 2)(X + 2)$

Answer

$(X - 1)(X^2 - X - 4)$

Answer

$(X^3 + 1)(X - 4)$

Problems & worked solutions

Problem 1 — Functions — General Properties

Problem 2 — Sets of Real Numbers

Problem 3 — Systems of Linear Equations

Problem 4 — Differentiation

Problem 5 — Linear Function

Problem 6 — Definite Integrals

Problem 7 — Polynomials in ℂ

Problem 8 — Matrix Equations

Problem 9 — Trigonometric Equations

Problem 10 — Polynomial Rings