Posted in ask uncle colin

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

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Posted in algebra

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,

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Posted in ask uncle colin, geometry

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,

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Posted in ninja maths, pirate maths

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

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Posted in ask uncle colin

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

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Posted in probability

A Christmas Pudding Puzzle I swear, this one came up in real life! My partner made a Christmas pudding for the most recent festive season. Delicious, it was. When it was about half-eaten, I went to microwave a portion. “Hang on,” she said: “there might be a coin in there.”

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Posted in ask uncle colin

Dear Uncle Colin, Why is $\arcsin\br{\sin\br{\frac {6}{7}\pi}}$ not $\frac{6}{7}\pi$? – A Reasonable Conclusion Seems Incorrect Numerically Hi, ARCSIN, and thanks for your message! On the face of it, it does seem like a reasonable conclusion: surely feeding the output of $\sin(x)$ into its inverse function should get you back where

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Posted in ninja lives

One of the most famous examples of stuckness – both for maths as a whole and for a mathematician in particular – is Fermat’s Last Theorem, which states that there is no solution to $a^n + b^n = c^n$ for whole numbers $a$, $b$, $c$ and $n$ unless $n$ is

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Posted in ask uncle colin

Dear Uncle Colin, I need to find the limit as $x$ approaches 1 of $\frac{x^{29}-1}{x-1}$. I tried factoring out $x^{28}$ but didn’t get anywhere. – Learning How Others Proceed In This Awful Limit Hi, LHOPITAL, and thanks for your message! Factoring out an $x^{28}$ is very unlikely to get you

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Posted in quadratics, there's more than one way to do it

Factorising a quadratic? It’s nice when it comes off, but there’s a lot of guesswork, and no guarantee it even factorises. Completing the square? Who has time for all that algebra? And as for the quadratic formula, or your clever calculator methods: honestly, what are you, an engineer? There is

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