June 2025 Paper 3 Q1
1
(a) Find the first three terms in the binomial expansion of \((1+3x)^{\frac{1}{2}}\). [3]
(b) State the range of values of \(x\) for which this expansion is valid. [1]
| Scheme | Marks | AO |
|---|---|---|
| \(\left(1 + \frac{1}{2}(3x) + \cdots\right.\) | M1 | 1.1 |
| \(\ldots\ldots\ldots\ldots + \dfrac{\frac{1}{2}\left(\frac{1}{2}-1\right)}{2!}(3x)^2)\) oe | M1 | 1.1 |
| \(1 + \dfrac{3x}{2} - \dfrac{9x^2}{8}\) oe | A1 | 1.1 |
| [3] |
Notes
M1: Use of binomial expansion as far as \(1 + \frac{1}{2}(3x)\)
soi by correct answer, nfww
M1: Full expansion up to quadratic term.
Condone e.g. \(3x^2\) but not \(x^2\)
soi by correct answer
A1: Allow exact decimals: \(1 + 1.5x - 1.125x^2\)
Ignore cubic and higher order terms.
Mark final answer
isw writing terms as a list after correct answer seen, but A0 if correct answer spoilt by further manipulation
| Scheme | Marks | AO |
|---|---|---|
| Valid for \(|x| < \frac{1}{3}\) | B1 | 1.1 |
| [1] |
Notes
B1: OR \(-\frac{1}{3} < x < \frac{1}{3}\) OR \(x < \frac{1}{3} \cap x > -\frac{1}{3}\)
Condone \(|x| \leqslant \frac{1}{3}\) or \(-\frac{1}{3} \leqslant x \leqslant \frac{1}{3}\) (as \(n > 0\))
Must be \(x\) not e.g. \(3x\)
Mark final answer