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\\ $Let $ P(z) = \sum_{k=0}^{n} a_k z^k $ be a monic non-constant and non-zero polynomial with real coefficients $ a_k . \\\\ $Show that if $ |P(i)| < 1 $ , then there exists $ a,b \in \mathbb{R} $ such that $ P(a+ib)=0 $ and $ (a^2+b^2+1)^2 < 4b^2 + 1

A student recently asked me a very interesting question: Evaluate the integral: \int_{0}^{\frac{\pi}{2}} \frac{dx}{1+\tan^n x} I think the end result is really cool. I absolutely love problems like these. The method I used is a bit unconventional (don't know how often it is used in the...

Hello all, This post is targeted at those who just graduated and will be undertaking Mathematics at a Tertiary level. The goal of this thread is to help prepare you with the transition from education at School, to education at University. Each University functions differently, so I will try...

Frequently Asked Questions What is this section is for? If you have any questions regarding UNIVERSITY level Mathematics, this is the place to ask them. What kinds of questions can I ask? ​ Anything OUTSIDE the HSC Syllabus. This is the Tertiary section after all. Tutorial/Homework...

As title stated. Reason why I'm doing this is because I've had many different students have various styles of writing, and I'm curious to see what the majority of users here use. This is mine:

Re: What have you eaten today? What kind of tea does everybody like? I personally prefer Jasmine tea or Chai. Note: Created a new thread for this topic.

Though really, the celebratory period should be ~3pm today. 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679 8214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196...

Here are two not-so-difficult problems I thought of a few days ago. It should be do-able for a keen 3U student. Problem #1: Consider the following *poorly drawn* diagram. We start with a right angled triangle of hypotenuse length n. The 'base' of this triangle is the hypotenuse of another...

Does anybody know what is exactly involved in a "Practice Class"? It is on top of a tutorial, but I cannot imagine what else you could have apart from a Lecture and a Tutorial. ie: For Physics, I understand that a tutorial and pracs should be separate, but not quite sure for Maths.

In case people haven't noticed, many Question 8's or Question 7's are either one of two things. 1. Elaborate and complicated proofs for very simple expresssions. 2. Proofs for the convergence of a particular series, and finding it's closed form. For example... 2010 HSC All of Question 8...

Time is running out - Muse

So I know most people here have drawn this thing at SOME point in their lives out of pure boredom. The other day I was just thinking about it and I thought 'Suppose I were to make the interval K approach 0, and thus have an infinitely large number of lines and intervals, what smooth...