June 2022 Paper 3 Q7

OCR MEICurrent spec12 marksBinomial ExpansionIntegration

7 A student is trying to find the binomial expansion of \(\sqrt{1 - x^3}\).

She gets the first three terms as \(1 - \dfrac{x^3}{2} + \dfrac{x^6}{8}\).

She draws the graphs of the curves \(y = \sqrt{1 - x^3}\), \(y = 1 - \dfrac{x^3}{2}\) and \(y = 1 - \dfrac{x^3}{2} + \dfrac{x^6}{8}\) using software.

Graphs for x from about −2 to 3. All three curves pass through (0, 1) and are close together near there. y = √(1 − x³) is defined only for x ≤ 1 and meets the x-axis at x = 1. y = 1 − x³/2 crosses the x-axis between 1 and 2 and continues downwards. y = 1 − x³/2 + x⁶/8 has a minimum at about (1.3, 0.5) and then rises steeply. For negative x all three rise, the three-term curve most steeply.
(a) Explain why \(1 - \dfrac{x^3}{2} + \dfrac{x^6}{8} \geqslant 1 - \dfrac{x^3}{2}\) for all values of \(x\). [1]
(b) Explain why the graphs suggest that the student has made a mistake in the binomial expansion. [1]
(c) Find the first four terms in the binomial expansion of \(\sqrt{1 - x^3}\). [3]
(d) State the set of values of \(x\) for which the binomial expansion in part (c) is valid. [1]
(e) Sketch the curve \(y = 2.5\sqrt{1 - x^3}\) on the grid in the Printed Answer Booklet. [2]
(f) In this question you must show detailed reasoning.
The end of a bus shelter is modelled by the area between the curve \(y = 2.5\sqrt{1 - x^3}\), the lines \(x = -0.75\), \(x = 0.75\) and the \(x\)-axis. Lengths are in metres.
Calculate, using your answer to part (c), an approximation for the area of the end of the bus shelter as given by this model. [4]
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