Daily Math · 2026-07-11

Question 1

The volume generated by rotating $f(x) = x$ on $[0, 3]$ about the $x$-axis is:

Accepted Answer

$V = \pi \int_0^3 x^2 \, dx = \pi \cdot \dfrac{27}{3} = 9\pi$. (Equivalently, $V_{	ext{cone}} = \dfrac{1}{3}\pi r^2 h = \dfrac{1}{3}\pi \cdot 9 \cdot 3 = 9\pi$.)

Answer

$3\pi$

Answer

$6\pi$

Answer

$9\pi$

Answer

$27\pi$

Question 2

For the same position $x(t) = t^3 - 6t^2 + 9t$, the acceleration at $t = 1$ is:

Accepted Answer

$a(t) = x''(t) = 6t - 12$. At $t = 1$: $a = -6$ m/s².

Answer

$-6$ m/s²

Answer

$0$ m/s²

Answer

$6$ m/s²

Answer

$9$ m/s²

Question 3

The points $A(1, 2)$, $B(3, 6)$, $C(5, 10)$ are:

Accepted Answer

Area $= \dfrac{1}{2}\,|1(6 - 10) + 3(10 - 2) + 5(2 - 6)| = \dfrac{1}{2}\,|-4 + 24 - 20| = 0$. Hence collinear.

Answer

collinear

Answer

vertices of a right triangle

Answer

vertices of an isosceles triangle

Answer

vertices of an equilateral triangle

Question 4

For $f: \mathbb{R} 	o \mathbb{R}$, $f(x) = x^3$, which statement is correct?

Accepted Answer

$f$ is strictly increasing on $\mathbb{R}$, hence injective. For every $y \in \mathbb{R}$, $x = \sqrt[3]{y}$ satisfies $f(x) = y$, hence surjective. So $f$ is bijective, with $f^{-1}(y) = \sqrt[3]{y}$.

Answer

$f$ is neither injective nor surjective

Answer

$f$ is injective but not surjective

Answer

$f$ is surjective but not injective

Answer

$f$ is bijective

Question 5

The limit $\displaystyle\lim_{x 	o 1} \dfrac{x^3 - 3x + 2}{x - 1}$ equals (apply Bezout / factoring):

Accepted Answer

$x^3 - 3x + 2 = (x - 1)(x^2 + x - 2) = (x - 1)(x - 1)(x + 2) = (x - 1)^2 (x + 2)$. So $\dfrac{(x-1)^2(x+2)}{x - 1} = (x - 1)(x + 2) 	o 0$ as $x 	o 1$.

Answer

$0$

Answer

$1$

Answer

$3$

Answer

$\infty$

Question 6

Solve: $\dfrac{\log_2(x+25)+\log_{\sqrt{2}}\!\left(\dfrac{1}{x-3}\right)}{\log_{100}(10^x)}=\dfrac{2}{7}$.

Accepted Answer

$\bullet$ Simplify each logarithm:

$$\log_{\sqrt{2}}\!\left(	frac{1}{x-3}ight) = -2\log_2(x-3)$$

$$\log_{100}(10^x) = 	frac{x}{2}$$

$\bullet$ Combine the numerator:

$$\log_2(x+25) - 2\log_2(x-3) = \log_2\frac{x+25}{(x-3)^2}$$

$\bullet$ The equation becomes:

$$\frac{\log_2\frac{x+25}{(x-3)^2}}{x/2} = \frac{2}{7}$$

$$\log_2\frac{x+25}{(x-3)^2} = \frac{x}{7}$$

$\bullet$ Test $x = 7$:

$$\log_2\frac{32}{16} = \log_2 2 = 1 = \frac{7}{7}$$

Answer

$4$

Answer

$5$

Answer

$7$

Answer

$9$

Answer

$6$

Answer

$11$

Question 7

A square of side 4 contains an astroid $x^{2/3}+y^{2/3}=a^{2/3}$ tangent to all four sides. Find the area enclosed by the astroid.

Accepted Answer

$\bullet$ The astroid meets the axes at $(\pm a, 0)$ and $(0, \pm a)$, so it fits in a square of side $2a$.

$\bullet$ Given side $= 4$:

$$2a = 4 \;\Rightarrow\; a = 2$$

$\bullet$ Apply the astroid area formula:

$$\mathcal{A} = \frac{3\pi a^2}{8} = \frac{3\pi \cdot 4}{8} = \frac{3\pi}{2}$$

Answer

$2\pi$

Answer

$\dfrac{3\pi}{2}$

Answer

$6\pi$

Answer

$4-\pi$

Answer

$\pi$

Answer

$\dfrac{3\pi}{8}$

Question 8

Evaluate $\displaystyle\lim_{x	o 0} \dfrac{1 - \cos(x)}{x^2}$.

Accepted Answer

Using $1-\cos x = 2\sin^2(x/2)$: $\displaystyle\lim_{x	o 0}\dfrac{2\sin^2(x/2)}{x^2} = \dfrac{1}{2}\lim_{x	o 0}\left(\dfrac{\sin(x/2)}{x/2}ight)^2 = \dfrac{1}{2}\cdot 1 = \dfrac{1}{2}$.

Answer

$0$

Answer

$\dfrac{1}{4}$

Answer

$\dfrac{1}{2}$

Answer

$1$

Question 9

For $\vec{u} = \vec{i} + \vec{j}$ and $\vec{v} = a\vec{i} - 2\vec{j}$, find $a \in \mathbb{R}$ so that $|\vec{u} + \vec{v}|^2 = |\vec{u}|^2 + |\vec{v}|^2$.

Accepted Answer

$\vec{u} \cdot \vec{v} = a + (-2) = a - 2 = 0 \Rightarrow a = 2$.

Answer

$-2$

Answer

$-1$

Answer

$2$

Answer

$4$

Question 10

By induction one can show that $n^3 - n$ is divisible by $6$ for every $n \in \mathbb{N}$. Compute $\dfrac{10^3 - 10}{6}$.

Accepted Answer

$10^3 - 10 = 990$, and $990 / 6 = 165$. The induction proof factors $n^3 - n = (n-1)n(n+1)$, which is the product of three consecutive integers and is therefore divisible by both $2$ and $3$, hence by $6$.

Answer

$165$

Answer

$166$

Answer

$100$

Answer

$200$

Problems & worked solutions

Problem 1 — Volumes of Revolution

Problem 2 — Applications of Derivatives

Problem 3 — Distances & Areas

Problem 4 — Functions — General Properties

Problem 5 — Limits of Functions

Problem 6 — Logs

Problem 7 — Integrals

Problem 8 — Notable Limits

Problem 9 — Vectors in the Plane

Problem 10 — Logic & Induction