Problem Bank

Given that

find the range of $\tan\theta$.

Let $a_i, b_i > 0$ for $1 \leqslant i \leqslant n+1$, and suppose $b_{i+1} - b_i \geqslant \delta > 0$ for $i = 1, 2, \dots, n$, where $\delta$ is a constant. If $\displaystyle\sum…

Functions $f(x), g(x)$ on $\mathbb{R}$ have derivatives $f'(x), g'(x)$, and

Determine, with proof…

Consider the circle $C:(x-2)^2+(y-2)^2=1$. Through a point $A$ on the $x$-axis, a line $l$ meets the circle $C$ at two points $M$ and $N$ with

Sequences $\{a_n\}$ and $\{b_n\}$ satisfy $a_1 = 1$, $b_1 = 2$, and

1. If additionally $b_n = 2a_n$ for all $n$, prove that for $n \geqslant …

The inequality

holds for all $x \in \mathbb{R}$. Find the range of $a$.

Let $F$ be a focus of the hyperbola $C \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ (with $a, b > 0$). A line through $F$ is perpendicular to one asymptote, meeting it at $A$, a…

The three complex roots of the cubic equation

correspond to three collinear points in the complex plane. Find the positive integer $k$.

The figure shows part of the graph of the function $f(x)=2\cos(\omega x+\varphi)$, where $\omega>0$; the graph att…

Evaluate

A function $f(x)$ on $(0, +\infty)$ has derivative $f'(x)$ satisfying

Determine, with proof, which of…

Let $f(x)=\ln(x+1)+a(x^2-x)$, where $a\in\mathbb R$.

1. Discuss the number of extreme points of $f(x)$, with justification. 2. If $f(x)\geqslant 0$ for all $x>0$, find the range o…