(i) Find the binomial expansion of (1+3x)−1(1 + 3x)^{-1}(1+3x)−1 up to and including the term in x2x^2x2.
(ii) Show that the first three terms in the binomial expansion of
15−2x \frac{1}{5 - 2x} 5−2x1form a geometric sequence and state the common ratio.
It is given that
85x(1+3x)(5−2x)≡P5−2x+Q1+3x \frac{85x}{(1 + 3x)(5 - 2x)} \equiv \frac{P}{5 - 2x} + \frac{Q}{1 + 3x} (1+3x)(5−2x)85x≡5−2xP+1+3xQwhere PPP and QQQ are integers. Find the value of PPP and the value of QQQ.
(i) In a structural model, the stress SSS on a beam at distance xxx from the support is given by S(x)=17x(1+3x)(5−2x)S(x) = \frac{17x}{(1 + 3x)(5 - 2x)}S(x)=(1+3x)(5−2x)17x. Using your answers to parts (a) and (b), find the binomial expansion of S(x)S(x)S(x) up to and including the term in x2x^2x2.
(ii) Determine the range of values of xxx for which the binomial expansion of S(x)S(x)S(x) 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.