# The Flying Colours Maths Blog: Latest posts

## Ask Uncle Colin: Auxiliary Equations With Repeated Roots

In this post, I swap liberally between d-notation and '-notation for derivatives. Deal with it. Dear Uncle Colin, Why do we have to treat second-order ODEs differently when the auxiliary equation has a repeated root? Something Or Other Defies Expectations Hi, SOODE, and thanks for your message! First, some background

## Eigenvalues

I remember, with a faint feeling of dread, having to calculate the eigenvalues of a matrix. It became routine in the end, but I was recently reminded of the pain when a student asked if there was a shortcut. For a 2-by-2 matrix? Yes. It is up to you, though,

## Ask Uncle Colin: A Fractional Kerfuffle

Dear Uncle Colin, I was trying to work out $\frac{\frac{3}{7+h}-\frac{3}{7}}{h}$, and I got it down to $\frac{\frac{3}{h}}{h}$ - but that's not the answer in the book! What have I done wrong? - Likely I've Mistreated It Terribly Hi, LIMIT, and thank you for your message! I'm afraid you're right, you

## Wrong, But Useful: Episode 55

This month, it's Gathering4Gardner special, largely recorded in Atlanta with Adam Atkinson and @dragon_dodo. We discuss: Our favourite talks of the event, including: The Juggler Problem The Taxman Problem Ramanujan Sums Optical illusions Doris Schattschneider's talk on Marjorie Rice, an amateur mathematician who made huge leaps in the study of

## The Involution of Polynomials

Last time out, I looked at a problem unearthed by @mathsjem - to find the cube root of a degree-six polynomial. This led (unsurprisingly) to a quadratic: $3 + 4x - 2x^2$. When checking whether this was indeed the answer, I hit a problem: is there a simple way to

## Ask Uncle Colin: Tangents to a circle

Dear Uncle Colin, I'm told that two lines through $(0,12)$ are tangent to the circle with equation $(x-6)^2 + (y-5)^2 = 17$ and I need to find their equations - but I'm getting in a muddle. Can you help? - Terribly Awkward Numbers, Getting Equations Not Trivial Hi, TANGENT, and

## The Evolution of Polynomials

It's always fascinating to see what's going on in textbooks of the olden days, and National Treasure @mathsjem recently found a beauty of its type. Look at those whences! Check out the subjunctives! It thrills the heart, doesn't it?1 What caught my attention, though, was evolution - in this context,

## Ask Uncle Colin: A Troublesome Triangle

Dear Uncle Colin, I couldn't make head nor tail of this geometry problem: "If $a:b=12:7$, $c=3$, and $B\hat{A}C = 2 B\hat{C}A$, find the length of the sides $a$ and $b$." - Totally Rubbish In Geometry Hi, TRIG, and thank you for your message! (And don't put yourself down like that,

## Using Units to Deal With Density

Glancing over sample papers for the new GCSE, I stumbled on this: Zahra mixes 150g of metal A and 150g of metal B to make 300g of an alloy. Metal A has a density of $19.3 \unit{g/cm^3}$. Metal B has a density of $8.9 \unit{g/cm^3}$. Work out the density of

## Ask Uncle Colin: A modulus power

Dear Uncle Colin, How would you work out $9^{41} \pmod{61}$? - Funky Exponential Result, Missed A Tutorial Hi, FERMAT, and thanks for your question! I think the answer is "ponderously"! There are only 61 possible answers (in fact, 60, because you know 61 is not a factor of $9^{41}$). I'd