| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2008 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Non-zero terms only |
| Difficulty | Standard +0.3 This is a straightforward application of the binomial expansion for negative/fractional powers. Part (i) requires standard substitution into (1+u)^n with n=-1/2 and u=4x², then stating the validity condition |4x²|<1. Part (ii) is a simple multiplication of the result from (i) by (1-x²). While it requires careful algebraic manipulation, it follows a well-practiced procedure with no novel insight needed. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((1+4x^2)^{-\frac{1}{2}} = 1 - \frac{1}{2}\cdot4x^2 + \frac{(-\frac{1}{2})(-\frac{3}{2})}{2!}(4x^2)^2 + \ldots\) | M1 | binomial expansion with \(p = -\frac{1}{2}\) |
| \(= 1 - 2x^2 + \ldots\) | A1 | \(1 - 2x^2\ldots\) |
| \(+ 6x^4\) | A1 | \(+6x^4\) |
| Valid for \(-1 < 4x^2 < 1 \Rightarrow -\frac{1}{2} < x < \frac{1}{2}\) | M1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{1-x^2}{\sqrt{1+4x^2}} = (1-x^2)(1-2x^2+6x^4+\ldots)\) | M1 | substituting their \(1-2x^2+6x^4+\ldots\) and expanding |
| \(= 1-2x^2+6x^4-x^2+2x^4+\ldots\) | A1 | ft their expansion (of three terms) |
| \(= 1-3x^2+8x^4+\ldots\) | A1 | cao |
## Question 6(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(1+4x^2)^{-\frac{1}{2}} = 1 - \frac{1}{2}\cdot4x^2 + \frac{(-\frac{1}{2})(-\frac{3}{2})}{2!}(4x^2)^2 + \ldots$ | M1 | binomial expansion with $p = -\frac{1}{2}$ |
| $= 1 - 2x^2 + \ldots$ | A1 | $1 - 2x^2\ldots$ |
| $+ 6x^4$ | A1 | $+6x^4$ |
| Valid for $-1 < 4x^2 < 1 \Rightarrow -\frac{1}{2} < x < \frac{1}{2}$ | M1A1 | |
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## Question 6(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{1-x^2}{\sqrt{1+4x^2}} = (1-x^2)(1-2x^2+6x^4+\ldots)$ | M1 | substituting their $1-2x^2+6x^4+\ldots$ and expanding |
| $= 1-2x^2+6x^4-x^2+2x^4+\ldots$ | A1 | ft their expansion (of three terms) |
| $= 1-3x^2+8x^4+\ldots$ | A1 | cao |
---6 (i) Find the first three non-zero terms of the binomial series expansion of $\frac { 1 } { \sqrt { 1 + 4 x ^ { 2 } } }$, and state the set of values of $x$ for which the expansion is valid.\\
(ii) Hence find the first three non-zero terms of the series expansion of $\frac { 1 - x ^ { 2 } } { \sqrt { 1 + 4 x ^ { 2 } } }$.
\hfill \mbox{\textit{OCR MEI C4 2008 Q6 [8]}}