Daily Math · 2026-08-02

Question 1

The value of $\sin\dfrac{7\pi}{6}$ is:

Accepted Answer

$\sin\!\left(\pi + \dfrac{\pi}{6}\right) = -\sin\dfrac{\pi}{6} = -\dfrac{1}{2}$.

Answer

$\dfrac{1}{2}$

Answer

$-\dfrac{1}{2}$

Answer

$\dfrac{\sqrt{3}}{2}$

Answer

$-\dfrac{\sqrt{3}}{2}$

Question 2

A price goes from $\$80$ to $\$100$. The percentage increase is:

Accepted Answer

$\dfrac{100 - 80}{80} \cdot 100\% = \dfrac{20}{80} \cdot 100\% = 25\%$.

Answer

$20\%$

Answer

$22.5\%$

Answer

$25\%$

Answer

$80\%$

Question 3

For $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & -1 \ 5 & 2 \end{pmatrix}$, $A + B$ equals:

Accepted Answer

Add componentwise: $\begin{pmatrix} 1 & 1 \ 8 & 6 \end{pmatrix}$.

Answer

$\begin{pmatrix} 1 & 1 \ 8 & 6 \end{pmatrix}$

Answer

$\begin{pmatrix} 1 & 3 \ 8 & 2 \end{pmatrix}$

Answer

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

Answer

$\begin{pmatrix} 0 & -2 \ 15 & 8 \end{pmatrix}$

Question 4

Find $\displaystyle\lim_{x	o\infty}x^{1/\sqrt{x}}$.

Accepted Answer

$\bullet$ Rewrite as an exponential:

$$x^{1/\sqrt{x}}=e^{\ln x/\sqrt{x}}$$

$\bullet$ Apply L'Hôpital to the exponent:

$$\lim_{x	o\infty}\frac{\ln x}{\sqrt{x}}=\lim_{x	o\infty}\frac{2}{\sqrt{x}}=0$$

$\bullet$ Exponentiate:

$$L=e^0=1$$

Answer

$e$

Answer

$+\infty$

Answer

$1$

Answer

$0$

Answer

$e^2$

Answer

$\sqrt{e}$

Question 5

For the equation $2x^2 - 7x + 3 = 0$ with roots $x_1, x_2$, compute $x_1 + x_2$.

Accepted Answer

$x_1 + x_2 = -\dfrac{-7}{2} = \dfrac{7}{2}$. (And the product is $\dfrac{c}{a} = \dfrac{3}{2}$.)

Answer

$-7$

Answer

$-\dfrac{3}{2}$

Answer

$\dfrac{7}{2}$

Answer

$7$

Question 6

Find $\displaystyle\int xe^x\,dx$.

Accepted Answer

$\bullet$ Integration by parts with $u=x$, $dv=e^x\,dx$, so $du=dx$, $v=e^x$:

$$\int xe^x\,dx = xe^x - \int e^x\,dx$$

$\bullet$ Evaluate the remaining integral:

$$xe^x - e^x + C$$

$\bullet$ Factor:

$$(x-1)e^x + C$$

Answer

$xe^x + C$

Answer

$(x-1)e^x + C$

Answer

$(x+1)e^x + C$

Answer

$\dfrac{x^2 e^x}{2} + C$

Answer

$e^x(x^2-x) + C$

Answer

$xe^x - x + C$

Question 7

In $\mathbb{Z}_6$, which element has NO multiplicative inverse?

Accepted Answer

$\gcd(2, 6) = 2 
e 1$, so $\hat{2}$ has no multiplicative inverse in $\mathbb{Z}_6$. ($\hat{1}$ and $\hat{5}$ are coprime to $6$ and invertible.)

Answer

$\hat{1}$

Answer

$\hat{5}$

Answer

$\hat{2}$

Answer

(all are invertible)

Question 8

$\displaystyle\int x \cos x \, dx$ equals:

Accepted Answer

$u = x$, $du = dx$, $v = \sin x$. $\int x \cos x \, dx = x \sin x - \int \sin x \, dx = x \sin x + \cos x + C$.

Answer

$\dfrac{x^2}{2} \cos x + C$

Answer

$-x \sin x + \cos x + C$

Answer

$x \sin x + \cos x + C$

Answer

$\sin x - x \cos x + C$

Question 9

A bank advertises $12\%$ annual rate compounded monthly. The effective annual rate is closest to:

Accepted Answer

$\!\left(1 + \dfrac{0.12}{12}\right)^{12} - 1 = 1.01^{12} - 1 \approx 1.1268 - 1 = 0.1268 \approx 12.68\%$.

Answer

$12\%$

Answer

$12.68\%$

Answer

$13.5\%$

Answer

$14.4\%$

Question 10

The solution set of $x^2 > 9$ in $\mathbb{R}$ is:

Accepted Answer

$x^2 > 9 \Leftrightarrow |x| > 3 \Leftrightarrow x < -3$ or $x > 3$. Solution: $(-\infty, -3) \cup (3, +\infty)$.

Answer

$(-3, 3)$

Answer

$[-3, 3]$

Answer

$(-\infty, -3) \cup (3, +\infty)$

Answer

$(-\infty, -3] \cup [3, +\infty)$

Problems & worked solutions

Problem 1 — Trigonometric Identities

Problem 2 — Financial Mathematics

Problem 3 — Matrices

Problem 4 — Limits

Problem 5 — Quadratic Function

Problem 6 — Integrals

Problem 7 — Modular Arithmetic (ℤₙ)

Problem 8 — Antiderivatives

Problem 9 — Financial Mathematics

Problem 10 — Elementary Functions