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.
368 exam-style questions on Edexcel A Level Maths Algebraic Methods, covering 1.1 Proof by Contradiction, 1.2 Algebraic Fractions, 1.3 Partial Fractions, 1.4 Repeated Factors, and 1.5 Algebraic Division. Each one has a worked solution and a mark scheme showing where the marks go.