(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.
566 exam-style questions on OCR A Level Maths 1.2 Algebra and Functions, covering 1.2.1 Indices, 1.2.2 Surds, 1.2.3 Simultaneous equations, 1.2.4 Quadratic functions, 1.2.5 Completing the square, 1.2.6 Solving quadratic equations, 1.2.7 Inequalities, 1.2.8 Expressing solutions of inequalities, 1.2.9 Representing inequalities graphically, 1.2.10 Manipulating polynomials, 1.2.11 Simplifying rational expressions, 1.2.12 The modulus function (A-level only), 1.2.13 Graphs of functions, 1.2.14 Sketching curves from equations, 1.2.15 Sketching reciprocal curves, 1.2.16 Interpreting solutions graphically, 1.2.17 Intersection points of graphs, 1.2.18 Proportional relationships, 1.2.19 Graph of the modulus of a linear function (A-level only), 1.2.20 Solving modulus equations graphically (A-level only), 1.2.21 Definition of a function (A-level only), 1.2.22 Inverse and composite functions (A-level only), 1.2.23 Simple graph transformations, 1.2.24 Combinations of graph transformations (A-level only), 1.2.25 Partial fractions (A-level only), and 1.2.26 Models in context. Each one has a worked solution and a mark scheme showing where the marks go.