| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Multiply by polynomial |
| Difficulty | Standard +0.3 This is a straightforward application of the binomial expansion for negative/fractional powers followed by polynomial multiplication. Part (i) is routine recall of the formula with n=-1/3, and part (ii) requires only basic algebraic manipulation to multiply the expansion by (1-3x) and collect terms. The validity condition is standard. Slightly above average difficulty due to the fractional power and two-part structure, but still a textbook exercise with no novel insight required. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| B1: Valid for \(-\frac{1}{2} < x < \frac{1}{2}\) or \( | x | < \frac{1}{2}\) (must be strict inequality for \(+\frac{1}{2}\)). Independent of expansion. Accept, say, \(-\frac{1}{2} < x < \frac{1}{2}\) or \(-\frac{1}{2} \leq x \leq \frac{1}{2}\) (must be strict inequality for \(\pm\frac{1}{2}\)). |
# Question 1
## (i)
B1: $n = -1/3$. See below SC for those with $n = 1/3$
M1: All three correct unsimplified binomial coefficients (not $\binom{n}{r}$) soi. Condone absence of brackets only if it is clear from subsequent work that they were assumed.
B1: $(12x)^{1/3}$
B1: $\frac{3}{\sqrt[3]{12x}}$
B1: $\frac{1}{4}\left(\frac{1}{3}\right)(2x)^{\frac{1}{3}}\frac{1}{2!}(2x)^2 \ldots$
B1: $\frac{1 \cdot 2 \cdot 8}{3 \cdot 9} = 1\frac{2}{3}x - \frac{8}{9}x^2 \ldots$
B1: If there is an error in say the third coefficient of the expansion then M0B1B0 is possible.
B1: Valid for $-\frac{1}{2} < x < \frac{1}{2}$ or $|x| < \frac{1}{2}$ (must be strict inequality for $+\frac{1}{2}$). Independent of expansion. Accept, say, $-\frac{1}{2} < x < \frac{1}{2}$ or $-\frac{1}{2} \leq x \leq \frac{1}{2}$ (must be strict inequality for $\pm\frac{1}{2}$).
SC: For $n = \frac{1}{3}$ award B1 for $1 + \frac{2}{3}x$ and B1 for $\frac{4}{9}x^2$ (so max 2 out of the first 4 marks).
[5]
## (ii)
M1: Use of $(1-3x)\times\text{their}\left[1 + \frac{2}{3}x + \frac{8}{9}x^2 \ldots\right]$ and attempt at removal of brackets. Condone absence of brackets but must have two terms in $x$ and two terms in $x^2$.
A1ft: Correct simplified expansion following their expansion in (i). This mark is dependent on scoring both M marks in (i) and (ii).
A1: $1 + \frac{7}{3}x - \frac{10}{9}x^2 \ldots$ cao or B3 www in either part.
SC: Following either M0 or M1, B1 for either $a$ or $b$ correct.
[3]1 (i) Find the first three terms of the binomial expansion of $\frac { 1 } { \sqrt [ 3 ] { 1 - 2 x } }$. State the set of values of $x$ for which\\
the expansion is valid.\\
(ii) Hence find $a$ and $b$ such that $\frac { 1 - 3 x } { \sqrt [ 3 ] { 1 - 2 x } } = 1 + a x + b x ^ { 2 } + \ldots$.
\hfill \mbox{\textit{OCR MEI C4 Q1 [8]}}