C4 January 2008 Q2

Notes

B1: Takes 8 outside the bracket to give any of \(\underline{(8)^{\frac{1}{3}}}\) or \(\underline{2}\).

M1: Expands \((1 + **x)^{\frac{1}{3}}\) to give a simplified or an un-simplified \(1 + (\tfrac{1}{3})(**x)\);

A1ft: A correct simplified or an un-simplified \(\{\ldots\ldots\}\) expansion with candidate’s followed through \((**x)\)

Award SC M1 if you see \(\tfrac{(\frac{1}{3})(-\frac{2}{3})}{2!}(**x)^2 + \tfrac{(\frac{1}{3})(-\frac{2}{3})(-\frac{5}{3})}{3!}(**x)^3\)

A1: Either \(2\{1 - \tfrac{1}{8}x \ldots\ldots\}\) or anything that cancels to \(2 - \tfrac{1}{4}x\);   A1: Simplified \(-\tfrac{1}{32}x^2 - \tfrac{5}{768}x^3\)

You would award B1M1A0 for\[= 2\left\{1 + (\tfrac{1}{3})(-\tfrac{3x}{8}) + \tfrac{(\frac{1}{3})(-\frac{2}{3})}{2!}(-\tfrac{3x}{8})^2 + \tfrac{(\frac{1}{3})(-\frac{2}{3})(-\frac{5}{3})}{3!}(-3x)^3 + \ldots\right\}\]because ** is not consistent.

If you see the constant term “2” in a candidate’s final binomial expansion, then you can award B1.

Aliter 2. (a) Way 2

B1: 2 or \((8)^{\frac{1}{3}}\) (See note ↓)

M1: Expands \((8 - 3x)^{\frac{1}{3}}\) to give an un-simplified or simplified \((8)^{\frac{1}{3}} + (\tfrac{1}{3})(8)^{-\frac{2}{3}}(**x)\);

A1ft: A correct un-simplified or simplified \(\{\ldots\ldots\}\) expansion with candidate’s followed through \((**x)\)

Award SC M1 if you see \(\tfrac{(\frac{1}{3})(-\frac{2}{3})}{2!}(8)^{-\frac{5}{3}}(**x)^2 + \tfrac{(\frac{1}{3})(-\frac{2}{3})(-\frac{5}{3})}{3!}(8)^{-\frac{8}{3}}(**x)^3\)

A1: Anything that cancels to \(2 - \tfrac{1}{4}x\); or \(2\{1 - \tfrac{1}{8}x \ldots\ldots\}\)   A1: Simplified \(-\tfrac{1}{32}x^2 - \tfrac{5}{768}x^3\)

Attempts using Maclaurin expansion should be escalated up to your team leader.

If you see the constant term “2” in a candidate’s final binomial expansion, then you can award B1.

Notes

M1: Attempt to substitute \(x = 0.1\) into a candidate’s binomial expansion.

A1: awrt 1.9746810

Be wary of calculator value of \((7.7)^{\frac{1}{3}} = 1.974680822\ldots\)