Daily Math · 2026-07-12

Question 1

The limit $\displaystyle\lim_{n 	o \infty} \dfrac{n + 1}{n^2 + 3}$ equals:

Accepted Answer

$\dfrac{n + 1}{n^2 + 3} = \dfrac{1/n + 1/n^2}{1 + 3/n^2} 	o \dfrac{0}{1} = 0$.

Answer

$0$

Answer

$1$

Answer

$\dfrac{1}{3}$

Answer

$\infty$

Question 2

The remainder of $P(X) = X^5 + X^3 + X + 2$ divided by $X + 1$ is:

Accepted Answer

$P(-1) = -1 - 1 - 1 + 2 = -1$.

Answer

$-1$

Answer

$0$

Answer

$1$

Answer

$2$

Question 3

The general solution of $\sin x = \cos x$ is:

Accepted Answer

$\sin x = \cos x \iff 	an x = 1 \iff x = \pi/4 + k\pi$.

Answer

$x = \pi/2 + k\pi$

Answer

$x = \pi/4 + k\pi$

Answer

$x = \pi/4 + 2k\pi$

Answer

$x = \pi/2 + 2k\pi$

Question 4

Compute $\displaystyle\int_0^1 x e^x\, dx$.

Accepted Answer

$\int x e^x\,dx = x e^x - \int e^x\,dx = (x-1)e^x + C$. Evaluating from $0$ to $1$: $(0)\cdot e - (-1)\cdot 1 = 0 - (-1) = 1$.

Answer

$0$

Answer

$1$

Answer

$e - 1$

Answer

$e + 1$

Question 5

For $I_n = \displaystyle\int_0^1 \dfrac{x^n}{1 + x} \, dx$, the sum $I_n + I_{n+1}$ equals:

Accepted Answer

$I_n + I_{n+1} = \int_0^1 \dfrac{x^n + x^{n+1}}{1 + x} \, dx = \int_0^1 x^n \, dx = \dfrac{1}{n + 1}$.

Answer

$\dfrac{1}{n + 1}$

Answer

$1$

Answer

$\dfrac{1}{n}$

Answer

$\ln 2$

Question 6

Let $A(x)=\begin{pmatrix}x&2\0&x^2\end{pmatrix}$ and $f(A)=A^2+2A^t-A^{-1}$. Find $\det\!\left(f(A(1))\cdot f(A(-1))ight)$.

Accepted Answer

$\bullet$ Build the pieces for $A(1)=\begin{pmatrix}1&2\0&1\end{pmatrix}$:

$$A(1)^2=\begin{pmatrix}1&4\0&1\end{pmatrix}$$

$$2A(1)^t=\begin{pmatrix}2&0\4&2\end{pmatrix}$$

$$A(1)^{-1}=\begin{pmatrix}1&-2\0&1\end{pmatrix}$$

$\bullet$ Assemble $f(A(1))=A^2+2A^t-A^{-1}$ and take the determinant:

$$f(A(1))=\begin{pmatrix}2&6\4&2\end{pmatrix},\ \det=4-24=-20$$

$\bullet$ For $A(-1)=\begin{pmatrix}-1&2\0&1\end{pmatrix}$ we have $A(-1)^2=I$ and:

$$2A(-1)^t=\begin{pmatrix}-2&0\4&2\end{pmatrix},\ A(-1)^{-1}=\begin{pmatrix}-1&2\0&1\end{pmatrix}$$

$\bullet$ Assemble $f(A(-1))$:

$$f(A(-1))=\begin{pmatrix}0&-2\4&2\end{pmatrix},\ \det=0-(-8)=8$$

$\bullet$ Multiply the two determinants:

$$\det\!\bigl(f(A(1))\cdot f(A(-1))\bigr)=(-20)(8)=-160$$

Answer

$4$

Answer

$-160$

Answer

$16$

Answer

$-16$

Answer

$160$

Answer

$0$

Question 7

The polynomial $P(x) = x^3 + x + 1$ has exactly:

Accepted Answer

$P'(x) > 0$ everywhere, so $P$ is strictly increasing on $\mathbb{R}$. Combined with $\lim_{\pm\infty}P = \pm\infty$, $P$ has exactly one real root by IVT.

Answer

$1$ real root

Answer

$2$ real roots

Answer

$3$ real roots

Answer

$0$ real roots

Question 8

Among rectangles with perimeter $20$, the one with maximum area has dimensions:

Accepted Answer

$A(x) = 10x - x^2$. $A'(x) = 10 - 2x = 0 \Rightarrow x = 5$. The maximum-area rectangle is the square $5 	imes 5$, with area $25$.

Answer

$4 	imes 6$

Answer

$3 	imes 7$

Answer

$5 	imes 5$

Answer

$2 	imes 8$

Question 9

Show by induction that $1 \cdot 2 + 2 \cdot 3 + \cdots + n(n+1) = \dfrac{n(n+1)(n+2)}{3}$ for every $n \ge 1$. Using this, compute $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + 4 \cdot 5$.

Accepted Answer

$\dfrac{4 \cdot 5 \cdot 6}{3} = \dfrac{120}{3} = 40$. (Direct: $2 + 6 + 12 + 20 = 40$.)

Answer

$30$

Answer

$40$

Answer

$50$

Answer

$60$

Question 10

The value of $\cos	frac{2\pi}{5} + \cos	frac{4\pi}{5} + \cos	frac{6\pi}{5} + \cos	frac{8\pi}{5}$ is:

Accepted Answer

The fifth roots of unity sum to $0$. Subtracting the root $1$ leaves a sum of $-1$, and taking real parts gives $\cos	frac{2\pi}{5}+\cos	frac{4\pi}{5}+\cos	frac{6\pi}{5}+\cos	frac{8\pi}{5}=-1$.

Answer

$-1$

Answer

$0$

Answer

$1$

Answer

$-	frac{1}{2}$

Problems & worked solutions

Problem 1 — Limits of Sequences

Problem 2 — Polynomial Rings

Problem 3 — Trigonometric Equations

Problem 4 — Integration by Parts

Problem 5 — Recursive Integrals

Problem 6 — Matrices

Problem 7 — Rolle's Sign Method

Problem 8 — Applications of Derivatives

Problem 9 — Logic & Induction

Problem 10 — Trigonometry