(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.
303 exam-style questions on OCR (MEI) A Level Maths 1.2 Algebra, covering 1.2.1 Algebraic vocabulary and notation, 1.2.2 Solve linear equations, 1.2.3 Change the subject of a formula, 1.2.4 Solve quadratic equations, 1.2.5 Discriminant of a quadratic, 1.2.6 Linear simultaneous equations, 1.2.7 One linear one quadratic simultaneous equations, 1.2.8 Points of intersection and solutions, 1.2.9 Linear inequalities, 1.2.10 Quadratic inequalities, 1.2.11 Expressing solutions of inequalities, 1.2.12 Use and manipulate surds, 1.2.13 Rationalise the denominator, 1.2.14 Laws of indices, 1.2.15 Negative, fractional and zero indices, 1.2.16 Proportional relationships, 1.2.17 Partial fractions (A-level only), and 1.2.18 Simplify rational expressions (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.