A laser beam travels in a straight line from a source S S\,S to a detector DDD, passing through a filter FFF. The points SSS, FFF, and D D\,D have position vectors s\mathbf{s}s, f\mathbf{f}f, and d\mathbf{d}d respectively. Given that F F\,F lies between S S\,S and D D\,D such that the ratio of distances SF:SD=2:7SF : SD = 2 : 7SF:SD=2:7, show that: d=12(7f−5s)\mathbf{d} = \frac{1}{2}(7\mathbf{f} - 5\mathbf{s})d=21(7f−5s)
Given that n∈Zn \in \mathbb{Z}n∈Z, prove by contradiction that if n2 n^2\,n2 is a multiple of 3, then n n\,n is a multiple of 3.
Practise Edexcel A Level Maths 1.1 Proof by Contradiction with exam-style questions for A Level Maths. 100 questions, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.