Monthly Archives: May 2015

How to check your answers in FP1

double


Correct bit-by-bit

Write neatly, give yourself plenty of space, always write each line below the previous line (leaving lots of blank space on the right of the page if necessary), and do big, clear diagrams with a ruler.

Try not to have your working run over from one page to the next. If a question is going to need long working, then start a new page with it. Be concise in your working (this isn’t GCSE: you don’t have to show every tiny step).

Then, if an answer looks wrong, or your working shows something different from the formula you are asked to prove, don’t cross out everything and start again. Continue reading

Fortnightly maths prize for 23 June

No-one attempted this one, which I guess is due to it being post-exam time rather than it being specially difficult.

Part 1:

image

To help you: 1+2+3+4+….+n=½n(n+1) for all n

Part 2:

Find the sum to n terms of the first n numbers multiplied pairwise

For n = 3 it is 1×2 + 1×3 + 2×3, which makes 11

Your job is to find a general formula which works for any n.

To help you: 12+22+32+….+n2=16n(n+1)(2n+1). Continue reading

Finding all the cube (or other) roots of a number

Finding all the cube (or other) roots of a number N:

Remember that in ℂ every number has three cube roots, four 4th roots, five 5th roots, 57 57th roots…. and on the Argand diagram (graph) they look like evenly-spaced spokes of a wheel, with one spoke along the x-axis.

Write

N = N cis 0 = N cis 2π = N cis 4π = N cis 6&6pi; +….

(as many different ways of writing N as you have roots)

If n is the positive real k’th root of N (example: 2 is the real fifth root of 32, 3 is the positive real fourth root of 81), then the roots are

n cis 0 (which is just n)

n cis 2π/k

n cis 4π/k

n cis 6π/k

and so on until you have k of them.

Draw the roots to check your working.

cuberoots

Example: the cube roots of 1 are 1, cis 2π/3, and cis 4π/3 (which is the same as cis −2π/3 )

If you want the roots in x+iy form, convert them using Rec( , ) on your calculator.

A lesson in why to draw pictures

kl-pics-worth-a-thousand-words

Everyone had trouble with question 6b(ii) of the FP1 “withdrawn” paper of June 2013. “A curve C is in the form of a parabola with equation y2=4x. P(p2, 2p) and Q(q2, 2q) are points on C where p>q. (a) Find an equation of the tangent to C at P. (b) The tangent at P and the tangent at Q are perpendicular and intersect at the point R(–1, 2). (i) Find the exact value of p and the exact value of q. (ii) Find the area of the triangle PQR“. Continue reading