Relative to a fixed origin OOO, the points PPP, QQQ, and R R\,R have position vectors p\mathbf{p}p, q\mathbf{q}q, and r\mathbf{r}r respectively. The points PPP, QQQ, and R R\,R lie on a straight line such that Q Q\,Q lies between P P\,P and RRR. Given that PQ:PR=1:4PQ : PR = 1 : 4PQ:PR=1:4, show that:
r=4q−3p \mathbf{r} = 4\mathbf{q} - 3\mathbf{p} r=4q−3pGiven that n∈Zn \in \mathbb{Z}n∈Z, prove by contradiction that if n2 n^2\,n2 is a multiple of 5, then n n\,n is a multiple of 5.
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.