Daily Math · 2026-07-09

Question 1

The number of $3$-element subsets of a $5$-element set is:

Accepted Answer

$\binom{5}{3} = \dfrac{5!}{3! \cdot 2!} = 10$.

Answer

$5$

Answer

$10$

Answer

$15$

Answer

$60$

Question 2

For $I_n = \displaystyle\int_0^1 x^n e^x \, dx$, integration by parts gives the recurrence:

Accepted Answer

$I_n = x^n e^x\Big|_0^1 - n \int_0^1 x^{n-1} e^x \, dx = e - n I_{n-1}$.

Answer

$I_n = e + n I_{n-1}$

Answer

$I_n = e - n I_{n-1}$

Answer

$I_n = n I_{n-1} - e$

Answer

$I_n = (n - 1) I_{n-1}$

Question 3

A group homomorphism $\varphi: (G, \cdot) 	o (H, *)$ is a function such that for all $a, b \in G$:

Accepted Answer

A homomorphism preserves the operation: $\varphi(a \cdot b) = \varphi(a) * \varphi(b)$.

Answer

$\varphi(a + b) = \varphi(a) + \varphi(b)$

Answer

$\varphi(ab) = a * b$

Answer

$\varphi(a \cdot b) = \varphi(a) * \varphi(b)$

Answer

$\varphi$ is bijective

Question 4

On $[0, 2\pi)$, the equation $\sin 2x = \sin x$ has exactly:

Accepted Answer

$2\sin x \cos x - \sin x = 0 \Rightarrow \sin x (2\cos x - 1) = 0$. $\sin x = 0$: $x \in \{0, \pi\}$. $\cos x = 1/2$: $x \in \{\pi/3, 5\pi/3\}$. Total: $4$.

Answer

$1$ solution

Answer

$2$ solutions

Answer

$3$ solutions

Answer

$4$ solutions

Question 5

For an arithmetic progression with $a_1 = 5$ and $r = 3$, find the smallest $n$ such that $S_n \ge 500$.

Accepted Answer

$S_n = \dfrac{n(3n + 7)}{2}$. Compute: $S_{17} = \dfrac{17 \cdot 58}{2} = 493 < 500$ and $S_{18} = \dfrac{18 \cdot 61}{2} = 549 \ge 500$. Hence the smallest $n$ is $18$.

Answer

$16$

Answer

$17$

Answer

$18$

Answer

$19$

Question 6

For which values of $m\in\mathbb{R}$ is the matrix $A=\begin{pmatrix}m&1&0\1&m&1\0&1&m\end{pmatrix}$ singular?

Accepted Answer

$\bullet$ Expand along the first row:

$$\det A = m(m^2-1) - 1\cdot(m-0) + 0$$

$\bullet$ Simplify:

$$\det A = m^3 - 2m = m(m^2-2)$$

$\bullet$ Set to zero:

$$m(m^2-2)=0$$

$\bullet$ Solutions:

$$m\in\{-\sqrt{2},\,0,\,\sqrt{2}\}$$

Answer

$m=0$ only

Answer

$m\in\{-\sqrt{2},\,0,\,\sqrt{2}\}$

Answer

$m\in\{-1,\,0,\,1\}$

Answer

$m=\sqrt{2}$ only

Answer

$m\in\{-2,\,0,\,2\}$

Answer

no real $m$

Question 7

The system $\begin{cases} 2x + 3y = 7 \ x - y = 1 \end{cases}$ in matrix form $AX = B$ has $A$ equal to:

Accepted Answer

$A = \begin{pmatrix} 2 & 3 \ 1 & -1 \end{pmatrix}$ — coefficients of $x$ and $y$ in each equation.

Answer

$\begin{pmatrix} 2 & 3 \ 1 & -1 \end{pmatrix}$

Answer

$\begin{pmatrix} 2 & 1 \ 3 & -1 \end{pmatrix}$

Answer

$\begin{pmatrix} 7 \ 1 \end{pmatrix}$

Answer

$\begin{pmatrix} 2 & 3 \ 7 & 1 \end{pmatrix}$

Question 8

$\displaystyle\lim_{n 	o \infty} \left(1 + \dfrac{1}{n^2}ight)^{n}$ equals:

Accepted Answer

$n \ln\!\left(1 + 	frac{1}{n^2}ight) \sim n \cdot 	frac{1}{n^2} = 	frac{1}{n} 	o 0$, so the limit equals $e^0 = 1$.

Answer

$1$

Answer

$e$

Answer

$0$

Answer

$\infty$

Question 9

For which value of $a$ does $f(x) = x^3 - 3ax + 1$ have a local minimum at $x = 2$?

Accepted Answer

$f'(x) = 3x^2 - 3a$, so $f'(2) = 12 - 3a = 0 \Rightarrow a = 4$. Check: $f''(x) = 6x$, so $f''(2) = 12 > 0$, confirming a local minimum.

Answer

$a = 1$

Answer

$a = 2$

Answer

$a = 3$

Answer

$a = 4$

Question 10

$\displaystyle\lim_{n 	o \infty} \dfrac{\sqrt[n]{n!}}{n}$ equals:

Accepted Answer

By Stirling, $\sqrt[n]{n!} \sim \dfrac{n}{e}\,(2\pi n)^{1/(2n)}$. The factor $(2\pi n)^{1/(2n)} 	o 1$, so $\dfrac{\sqrt[n]{n!}}{n} 	o \dfrac{1}{e}$.

Answer

$	frac{1}{e}$

Answer

$1$

Answer

$e$

Answer

$0$

Problems & worked solutions

Problem 1 — Permutations & Combinations

Problem 2 — Recursive Integrals

Problem 3 — Homomorphisms

Problem 4 — Trigonometric Equations

Problem 5 — Arithmetic Sequences

Problem 6 — Matrices

Problem 7 — Matrix Equations

Problem 8 — Calculus

Problem 9 — Local Extrema

Problem 10 — Calculus