October 2020 Paper 1 Q1
1.
There is no need to carry out the calculation. (2)
| Scheme | Marks | AO |
|---|---|---|
| \((1+8x)^{\frac{1}{2}} = 1 + \dfrac{1}{2} \times 8x + \dfrac{\frac{1}{2} \times -\frac{1}{2}}{2!} \times (8x)^2 + \dfrac{\frac{1}{2} \times -\frac{1}{2} \times -\frac{3}{2}}{3!} \times (8x)^3\) | M1 A1 | 1.1b 1.1b |
| \(= 1 + 4x - 8x^2 + 32x^3 + \ldots\) | A1 | 1.1b |
| (3) |
Notes
M1: Attempts the binomial expansion with \(n = \dfrac{1}{2}\) and obtains the correct structure for term 3 or term 4.
Award for the correct coefficient with the correct power of \(x\). Do not accept \({}^{n}\mathrm{C}_{r}\) notation for coefficients.
For example look for term 3 in the form \(\dfrac{\frac{1}{2} \times -\frac{1}{2}}{2!} \times (*x)^2\) or \(\dfrac{\frac{1}{2} \times -\frac{1}{2} \times -\frac{3}{2}}{3!} \times (*x)^3\)
A1: Correct (unsimplified) expression. May be implied by correct simplified expression
A1: \(1 + 4x - 8x^2 + 32x^3\)
Award if there are extra terms (even if incorrect).
Award if the terms are listed \(1,\ 4x,\ -8x^2,\ 32x^3\)
| Scheme | Marks | AO |
|---|---|---|
| Substitutes \(x = \dfrac{1}{32}\) into \((1+8x)^{\frac{1}{2}}\) to give \(\dfrac{\sqrt{5}}{2}\) | M1 | 1.1b |
| Explains that \(x = \dfrac{1}{32}\) is substituted into \(1 + 4x - 8x^2 + 32x^3\) and you multiply the result by 2 | A1ft | 2.4 |
| (2) | ||
| (5 marks) |
Notes
M1: Score for substituting \(x = \dfrac{1}{32}\) into \((1+8x)^{\frac{1}{2}}\) to obtain \(\dfrac{\sqrt{5}}{2}\) or equivalent such as \(\sqrt{\dfrac{5}{4}}\)
Alternatively award for substituting \(x = \dfrac{1}{32}\) into both sides and making a connection between the two sides by use of an = or \(\approx\).
E.g. \(\left(1 + \dfrac{8}{32}\right)^{\frac{1}{2}} = 1 + 4 \times \dfrac{1}{32} - 8 \times \left(\dfrac{1}{32}\right)^2 + 32 \times \left(\dfrac{1}{32}\right)^3\) following through on their expansion
Also implied by \(\dfrac{\sqrt{5}}{2} = \dfrac{1145}{1024}\) for a correct expansion
It is not enough to state substitute \(x = \dfrac{1}{32}\) into "the expansion" or just the rhs "\(1 + 4x - 8x^2 + 32x^3\)"
A1ft: Requires a full (and correct) explanation as to how the expansion can be used to estimate \(\sqrt{5}\)
E.g. Calculates \(1 + 4 \times \dfrac{1}{32} - 8 \times \left(\dfrac{1}{32}\right)^2 + 32 \times \left(\dfrac{1}{32}\right)^3\) and multiplies by 2.
This can be scored from an incorrect binomial expansion or a binomial expansion with more terms.
The explanation could be mathematical. So \(\dfrac{\sqrt{5}}{2} = \dfrac{1145}{1024} \to \sqrt{5} = \dfrac{1145}{512}\) is acceptable.
SC: For 1 mark, M1,A0 score for a statement such as "substitute \(x = \dfrac{1}{32}\) into both sides of part (a) and make \(\sqrt{5}\) the subject"