June 2023 Paper 2 Q9

AQACurrent spec6 marksBinomial Expansion

(a) Find the first three terms, in ascending powers of \(x\), of the binomial expansion of\[(1 + x)^{-\frac{1}{2}}\] [2 marks]

(b) A student substitutes \(x = 2\) into the expansion of \((1 + x)^{-\frac{1}{2}}\) to find an approximation for \(\dfrac{1}{\sqrt{3}}\)

Explain the mistake in the student’s approach. [1 mark]

(c) By substituting \(x = -\dfrac{1}{4}\) in your expansion for \((1 + x)^{-\frac{1}{2}}\) find an approximation for \(\dfrac{1}{\sqrt{3}}\)

Give your answer to three significant figures. [3 marks]

Mark scheme (a)

Scheme	Marks	AO
Uses the binomial expansion to obtain either \(\left(-\dfrac{1}{2}\right)x\) or \(\dfrac{\left(-\dfrac{1}{2}\right)\left(-\dfrac{3}{2}\right)x^2}{2!}\) OE	M1	1.1a
Obtains \(1 - \dfrac{1}{2}x + \dfrac{3}{8}x^2\) Must have evaluated coefficients – allow equivalent fractions.	A1	1.1b
	(2)

Typical solution

\[(1 + x)^{-\frac{1}{2}} \approx 1 + \left(-\frac{1}{2}\right)x + \frac{\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)x^2}{2!}\]\[\approx 1 - \frac{1}{2}x + \frac{3}{8}x^2\]

Mark scheme (b)

Scheme	Marks	AO
Explains that the expansion is only valid for \(\|x\| \lt 1\) OE Accept that the expansion is not valid for \(\|x\| \gt 1\) Must include the word valid or invalid.	E1	2.3
	(1)

Typical solution

The expansion is valid for \(|x| \lt 1\)

Mark scheme (c)

Scheme	Marks	AO
Substitutes \(x = -\dfrac{1}{4}\) into their answer to part (a)	M1	1.1a
Obtains \(\dfrac{147}{128}\) AWRT 1.148 Condone 1.15 if a fully correct substituted expansion is seen.	A1	1.1b
Deduces the value 0.574 AWRT 0.574 or Deduces the value 0.580 AWRT 0.580	A1	2.2a
	(3)
	(6 marks)

Typical solution

\[1 - \frac{1}{2}\left(-\frac{1}{4}\right) + \frac{3}{8}\left(-\frac{1}{4}\right)^2 = \frac{147}{128}\]\[\left(\frac{3}{4}\right)^{-\frac{1}{2}} = \frac{2}{\sqrt{3}}\]\[\frac{2}{\sqrt{3}} = \frac{147}{128}\]\[\frac{1}{\sqrt{3}} = \frac{147}{256} \approx 0.574\]