June 2024 Paper 2 Q10
10
(a) Determine the first three terms in ascending powers of \(x\) of the binomial expansion of \((8 + 3x)^{\frac{1}{3}}\). [4]
(b) State the range of values of \(x\) for which this expansion is valid. [1]
| Scheme | Marks | AO |
|---|---|---|
| \(8^{\frac{1}{3}}\) or 2 seen | B1 | 1.1 |
| \(1 + \left(\frac{1}{3}\right)\left(\frac{3x}{8}\right) + \left(\frac{1}{3}\right)\left(\frac{1}{3} - 1\right)\frac{\left(\frac{3x}{8}\right)^2}{2!} + \cdots\) | M1 | 1.1 |
| \(\left(1 + \frac{x}{8} - \frac{x^2}{64} + \cdots\right)\) | A1 | 1.1 |
| \(2 + \frac{x}{4} - \frac{x^2}{32}\) or \(2\left(1 + \frac{x}{8} - \frac{x^2}{64} + \cdots\right)\) isw | A1 | 1.1 |
| [4] |
Notes
M1: two of the first three terms correct; ignore terms in \(x^3\) and above; may be embedded; must see at least substitution for third term
A1: may be unsimplified; may be embedded
A1: all three terms correct; ignore exra terms
if M0 allow SCB1 for \(\left(1 + \frac{1}{2}x - \frac{1}{4}x^2\right)\) following the equivalent method with use of \(\frac{3x}{2}\); may see eg \(2 + x - \frac{1}{2}x^2\)
if M0 allow SCB2 for correct expansion not fully supported
if M1A0 allow SCB1 for correct expansion not fully supported
| Scheme | Marks | AO |
|---|---|---|
| \(|x| \lt \frac{8}{3}\) or \(-\frac{8}{3} \lt x \lt \frac{8}{3}\) | B1FT | 2.5 |
| [1] |
Notes
B1FT: allow \(|x| \leqslant \frac{8}{3}\) or \(-\frac{8}{3} \leqslant x \leqslant \frac{8}{3}\); mark the final answer
FT their \(\left(1 + \frac{a}{b}x\right)\)