A lovely old problem
Recently Ed Southall shared this problem from 1976:
I’m not entirely sure if it is from an A level or and O level paper. It covers topics that currently sit on the A level, but I think calculus was on the O level at some point. Edit: it’s O level I saw the question and couldn’t help but have a try at it.
First, I drew the diagram – of course:
I have the coordinates of P, and hence N so I needed to work out the coordinates of Q. To do this I differentiated to get the gradient of a tangent and followed to get the gradient of a tangent at P.
Next I found the equation, and hence the X intercept.
And then, because I’m am idiot, I decided to work out the Y coordinate I already knew and had used!
The word in brackets is duh…..
Now I had all three point.
It was a simple division to find the tangent ratio of the angle.
The next 2 parts were trivial:
And then I misread the question and assumed I’d been asked to find the shaded region (actually part d).
Because I decided calculators were probably not widely available in 1976 I did it without one:
I thought it was quite a lot of complicated simplifying, but then I saw part c and the nice answer it gives:
Which makes the simplifying in part d simpler:
I thought this was a lovely question and I found it enjoyable to do. It tests a number of skills together and although it is scaffolded it still requires a little bit of thinking. I hope to see some nice big questions like this on the new specification.
Edit: The front cover of the paper: