Binomial expansion for rational n and series modelling – Year 13 A level maths test to download as PDF

Expand \(\dfrac1{(2+x)^3}\) in ascending powers of \(x\) up to and including the term in \(x^2\). State the values of \(x\) for which the expansion is valid. [4 marks]

(a) Expand \(\sqrt{1-4x}\) up to and including \(x^3\). [2 marks] (b) State the range of validity. [1 mark] (c) Use \(x=0.01\) to estimate \(\sqrt{0.96}\) to 6 decimal places. [1 mark]

\(h(x)=\dfrac{4-x}{(1-x)(2+x)}\). (a) Write \(h(x)\) in partial fractions. [2 marks] (b) Expand it in ascending powers of \(x\) up to and including \(x^3\). [1 mark] (c) State the range of validity. [1 mark]

In the expansion of \((1+ax)^n\) the coefficient of \(x\) is \(-3\) and the coefficient of \(x^2\) is \(\dfrac{27}4\). (a) Find \(a\) and \(n\). [2 marks] (b) Find the coefficient of \(x^3\) and the range of validity. [2 marks]

A pendulum has period \(T=2\pi\sqrt{\dfrac Lg}\). Because of a design change the effective gravity \(g\) is reduced by 4%, so the new period is \(T_0(0.96)^{-1/2}\) where \(T_0\) is the original period. (a) Expand \((1-x)^{-1/2}\) up to \(x^2\) and use \(x=0.04\) to estimate \((0.96)^{-1/2}\) to 5 decimal places. [3 marks] (b) By what percentage does the period increase, to 1 decimal place? [1 mark]