(i) Find the binomial expansion of (1+3p)−1(1 + 3p)^{-1}(1+3p)−1 up to and including the term in p2p^2p2.
(ii) Show that the first three terms in the binomial expansion of
12−5p \frac{1}{2 - 5p} 2−5p1form a geometric sequence and state the common ratio.
For an automated control system, the sensitivity factor S(p)S(p)S(p) depends on a small perturbation ppp, where
S(p)=55p(1+3p)(2−5p) S(p) = \frac{55p}{(1 + 3p)(2 - 5p)} S(p)=(1+3p)(2−5p)55pExpress S(p)S(p)S(p) in the form P2−5p+Q1+3p\frac{P}{2 - 5p} + \frac{Q}{1 + 3p}2−5pP+1+3pQ where PPP and QQQ are integers.
(i) Using your answers to parts (a) and (b), find the binomial expansion of
G(p)=22p(1+3p)(2−5p) G(p) = \frac{22p}{(1 + 3p)(2 - 5p)} G(p)=(1+3p)(2−5p)22pup to and including the term in p2p^2p2.
(ii) Determine the range of values of ppp for which the binomial expansion of G(p)G(p)G(p) is valid.
Practise AQA A Level Maths 1.5 B: Algebra and functions with exam-style questions for A Level Maths. 358 questions covering 1.5.1 Laws of indices, 1.5.2 Surds, 1.5.3 Quadratic functions, 1.5.4 Simultaneous equations, 1.5.5 Linear and quadratic inequalities, 1.5.6 Polynomials and rational expressions, 1.5.7 Graphs of functions and proportion, 1.5.8 Composite and inverse functions (A-level only), 1.5.9 Transformations of graphs, 1.5.10 Partial fractions (A-level only), and 1.5.11 Functions in modelling (A-level only), matched to the AQA A Level Maths (7357) 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.
