Daily Math · 2026-07-13

Question 1

The sum of the roots of $x^2 - 7x + 10 = 0$ equals:

Accepted Answer

By Viète's formulas, $x_1+x_2 = -(-7) = 7$. (The product is $10$, so the roots are $2$ and $5$.)

Answer

$7$

Answer

$10$

Answer

$-7$

Answer

$3$

Question 2

The area of a triangle with sides $5$, $6$, $7$ is closest to:

Accepted Answer

$s = \dfrac{5 + 6 + 7}{2} = 9$. $A = \sqrt{9 \cdot 4 \cdot 3 \cdot 2} = \sqrt{216} = 6\sqrt{6} \approx 14.70$.

Answer

$10$

Answer

$12$

Answer

$14.7$

Answer

$21$

Question 3

The multiplicity of the root $X = 2$ in $P(X) = (X - 2)^3 (X + 1)^2$ is:

Accepted Answer

The factor $(X - 2)$ appears with exponent $3$, so the multiplicity is $3$.

Answer

$1$

Answer

$2$

Answer

$3$

Answer

$5$

Question 4

Let $f(x)=C_x^3+\log_2(x+1)$ and $A=\begin{pmatrix}f(3)&	an\frac{\pi}{4}&2^{\lim_{x	o\infty}\frac{x^{23}}{x!}}\ \frac{f(7)}{38}&\log_{\sqrt{3}}f(3)&i^4\ \lg 10^{-48}&P_4&\ln 1\end{pmatrix}$. Find $\det(A)$.

Accepted Answer

$\bullet$ Evaluate the function values:

$$f(3) = C_3^3 + \log_2 4 = 1 + 2 = 3$$

$$f(7) = C_7^3 + \log_2 8 = 35 + 3 = 38$$

$\bullet$ So $f(7)/38 = 1$ and $\log_{\sqrt{3}} f(3) = \log_{\sqrt{3}} 3 = 2$.

$\bullet$ Reduce the trig / limit / log / factorial entries:

$$	an	frac{\pi}{4} = 1$$

$$2^{\lim x^{23}/x!} = 2^0 = 1$$

$$i^4 = 1,\;\; \lg 10^{-48} = -48,\;\; P_4 = 4! = 24,\;\; \ln 1 = 0$$

$\bullet$ Assemble the matrix:

$$A = \begin{pmatrix}3 & 1 & 1\ 1 & 2 & 1\ -48 & 24 & 0\end{pmatrix}$$

$\bullet$ Expand along row 1:

$$\det A = 3(2\cdot 0 - 1\cdot 24) - 1(1\cdot 0 - 1\cdot(-48)) + 1(1\cdot 24 - 2\cdot(-48))$$

$$\det A = 3(-24) - 1(48) + 1(120)$$

$$\det A = -72 - 48 + 120 = 0$$

Answer

$0$

Answer

$24$

Answer

$-72$

Answer

$48$

Answer

$-24$

Answer

$72$

Question 5

The volume of the solid generated by rotating the region between $y = x$ and $y = x^2$ on $[0, 1]$ about the $x$-axis equals:

Accepted Answer

$V = \pi \int_0^1 (x^2 - x^4) \, dx = \pi\!\left[\dfrac{x^3}{3} - \dfrac{x^5}{5}\right]_0^1 = \pi\!\left(\dfrac{1}{3} - \dfrac{1}{5}\right) = \dfrac{2\pi}{15}$.

Answer

$\pi/30$

Answer

$\pi/15$

Answer

$2\pi/15$

Answer

$\pi/3$

Question 6

In $\mathbb{Z}_7$, the value of $\hat{3}^{\,6}$ is:

Accepted Answer

By Fermat, $3^{6} \equiv 1 \pmod 7$, so $\hat{3}^{\,6} = \hat{1}$.

Answer

$\hat{1}$

Answer

$\hat{2}$

Answer

$\hat{3}$

Answer

$\hat{6}$

Question 7

Let $f(x)=\dfrac{1}{\sqrt[3]{x^7}}+\ln\sqrt[3]{x^{10}}$. Find $f'(2)+f''(3)$.

Accepted Answer

$\bullet$ Rewrite in power form using $\ln\sqrt[3]{x^{10}}=	frac{10}{3}\ln x$:

$$f(x)=x^{-7/3}+	frac{10}{3}\ln x$$

$\bullet$ Differentiate once:

$$f'(x)=-	frac{7}{3}x^{-10/3}+	frac{10}{3x}$$

$\bullet$ Differentiate again:

$$f''(x)=	frac{70}{9}x^{-13/3}-	frac{10}{3x^2}$$

$\bullet$ Plug in $x=2$ in $f'$ and $x=3$ in $f''$:

$$f'(2)=-	frac{7}{3\cdot 2^{10/3}}+	frac{5}{3}$$

$$f''(3)=	frac{70}{9\cdot 3^{13/3}}-	frac{10}{27}$$

Answer

$-\dfrac{7}{3\cdot2^{10/3}}+\dfrac{5}{3}$

Answer

$-\dfrac{7}{3\cdot2^{10/3}}+\dfrac{5}{3}+\dfrac{70}{9\cdot3^{13/3}}-\dfrac{10}{27}$

Answer

$0$

Answer

$\dfrac{5}{3}-\dfrac{10}{27}$

Answer

$-\dfrac{7}{3\cdot2^{10/3}}+\dfrac{10}{3}$

Answer

$\dfrac{70}{9\cdot3^{13/3}}-\dfrac{10}{27}$

Question 8

For $P(x) = x^3 - 3x + m$, find the values of $m$ for which $P$ has three distinct real roots:

Accepted Answer

$P(-1) > 0$ and $P(1) < 0$: $2 + m > 0$ and $-2 + m < 0$, giving $-2 < m < 2$.

Answer

$m < 0$

Answer

$m = 0$

Answer

$-2 < m < 2$

Answer

$m > 2$

Question 9

Let $\vec{a}$ and $\vec{b}$ be two non-collinear vectors. Find $m \in \mathbb{R}$ so that $\vec{u} = 3\vec{a} - (m + 1)\vec{b}$ and $\vec{v} = (m - 1)\vec{a} - 5\vec{b}$ are collinear.

Accepted Answer

$3 \cdot (-5) = (m - 1)\,(-(m+1))$, i.e. $-15 = -(m-1)(m+1) = 1 - m^2$, so $m^2 = 16$ and $m \in \{-4, 4\}$.

Answer

$\{-4\}$

Answer

$\{4\}$

Answer

$\{-4, 4\}$

Answer

$\{-2, 8\}$

Question 10

$\displaystyle\int_1^e \ln x \, dx$ equals:

Accepted Answer

$\int \ln x \, dx = x \ln x - x + C$. Evaluate: $(e \cdot 1 - e) - (1 \cdot 0 - 1) = 0 - (-1) = 1$.

Answer

$1$

Answer

$e$

Answer

$e - 1$

Answer

$0$

Problems & worked solutions

Problem 1 — Algebra

Problem 2 — Solving Triangles

Problem 3 — Polynomial Rings

Problem 4 — Matrices

Problem 5 — Volumes of Revolution

Problem 6 — Modular Arithmetic (ℤₙ)

Problem 7 — Calculus

Problem 8 — Rolle's Sign Method

Problem 9 — Vectors in the Plane

Problem 10 — Definite Integrals