| 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 generalized binomial theorem with standard validity condition |2x| < 1. Part (ii) requires only multiplying the result by (1+2x) and collecting terms—a mechanical process with no conceptual challenge beyond the standard technique. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((1-2x)^{-\frac{1}{2}} = 1 - \frac{1}{2}(-2x) + \frac{\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)}{2!}(-2x)^2+\ldots\) | M1 | Binomial expansion with \(p = -\frac{1}{2}\) |
| \(= 1+x+\frac{3}{2}x^2+\ldots\) | A1 | Correct expression |
| Valid for \(-1 < -2x < 1 \Rightarrow -\frac{1}{2} < x < \frac{1}{2}\) | A1, M1, A1 | cao |
| [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{1+2x}{\sqrt{1-2x}} = (1+2x)\left(1+x+\frac{3}{2}x^2+\ldots\right)\) | M1 | Substituting their \(1+x+\frac{3}{2}x^2+\ldots\) and expanding |
| \(= 1+x+\frac{3}{2}x^2+2x+2x^2+\ldots\) | A1ft | |
| \(= 1+3x+\frac{7}{2}x^2+\ldots\) | A1 | cao |
| [3] |
## Question 6(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(1-2x)^{-\frac{1}{2}} = 1 - \frac{1}{2}(-2x) + \frac{\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)}{2!}(-2x)^2+\ldots$ | M1 | Binomial expansion with $p = -\frac{1}{2}$ |
| $= 1+x+\frac{3}{2}x^2+\ldots$ | A1 | Correct expression |
| Valid for $-1 < -2x < 1 \Rightarrow -\frac{1}{2} < x < \frac{1}{2}$ | A1, M1, A1 | cao |
| **[5]** | | |
## Question 6(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1+2x}{\sqrt{1-2x}} = (1+2x)\left(1+x+\frac{3}{2}x^2+\ldots\right)$ | M1 | Substituting their $1+x+\frac{3}{2}x^2+\ldots$ and expanding |
| $= 1+x+\frac{3}{2}x^2+2x+2x^2+\ldots$ | A1ft | |
| $= 1+3x+\frac{7}{2}x^2+\ldots$ | A1 | cao |
| **[3]** | | |
---6 (i) Find the first three terms in the binomial expansion of $\frac { 1 } { \sqrt { 1 - 2 x } }$. State the set of values of $x$ for which the expansion is valid.\\
(ii) Hence find the first three terms in the series expansion of $\frac { 1 + 2 x } { \sqrt { 1 - 2 x } }$.
\hfill \mbox{\textit{OCR MEI C4 Q6 [8]}}