| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| 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 (1+x)^n with n=-1/2, followed by a simple multiplication. Part (i) requires standard technique (rewriting, expanding, stating validity), while part (ii) is routine algebraic manipulation. Slightly above average due to the two-part structure and need for careful coefficient work, but no novel insight required. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| \[(1+4x^2)^{-\frac{1}{2}} = 1 - \frac{1}{2} \cdot 4x^2 + \frac{\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)}{2!}(4x)^2 + \ldots\] | M1 | binomial expansion with \(p = -\frac{1}{2}\) |
| \[= -1 \quad 2 \quad 6\] | A1 | |
| \[\frac{x^2}{2} \quad x^4 \ldots\] | A1 | \(1 - 2x^2 \ldots\) |
| Valid for \(-1 < 4x^2 < 1 \Rightarrow -\frac{1}{2} < x < \frac{1}{2}\) | M1A1 [5] | \(+ \; x^4\) |
| Answer | Marks | Guidance |
|---|---|---|
| \[\frac{1-x^2}{\sqrt{1+4x^2}} = (1-x^2)(1 - 2x^2 + 6x^4 + \ldots)\] | M1 | substituting their \(1-2\), \(x^2 \; 6x^4 \ldots\) and expanding |
| \[= -1 \quad 2 \quad 6 \quad x^4 \quad x^2 \quad 2\] | A1 | ft their expansion (of three terms) |
| \[= 1 - 3x^2 + 8x^4 + \ldots\] | A1 [3] | cao |
## Question 9(i):
$$(1+4x^2)^{-\frac{1}{2}} = 1 - \frac{1}{2} \cdot 4x^2 + \frac{\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)}{2!}(4x)^2 + \ldots$$ | M1 | binomial expansion with $p = -\frac{1}{2}$ |
$$= -1 \quad 2 \quad 6$$ | A1 | |
$$\frac{x^2}{2} \quad x^4 \ldots$$ | A1 | $1 - 2x^2 \ldots$ |
Valid for $-1 < 4x^2 < 1 \Rightarrow -\frac{1}{2} < x < \frac{1}{2}$ | M1A1 [5] | $+ \; x^4$ |
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## Question 9(ii):
$$\frac{1-x^2}{\sqrt{1+4x^2}} = (1-x^2)(1 - 2x^2 + 6x^4 + \ldots)$$ | M1 | substituting their $1-2$, $x^2 \; 6x^4 \ldots$ and expanding |
$$= -1 \quad 2 \quad 6 \quad x^4 \quad x^2 \quad 2$$ | A1 | ft their expansion (of three terms) |
$$= 1 - 3x^2 + 8x^4 + \ldots$$ | A1 [3] | cao |
---9 (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 Q9 [8]}}