| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2014 |
| Session | June |
| Marks | 5 |
| 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) requires direct substitution into the binomial formula (standard C4 technique), and part (ii) involves multiplying the expansion by (x+3) and collecting terms—a routine 'hence' question with no conceptual challenges beyond careful algebraic manipulation. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1+(-\frac{1}{2})(-2x)+(-\frac{1}{2})(\frac{-3}{2})\frac{(\pm2x)^2}{2!}[+\ldots]\) | B1 | First two terms; allow recovery from omission of brackets |
| B1 | Third term; do not allow \(2x^2\) unless fully recovered in answer | |
| \(1+x+\frac{3}{2}x^2\) oe | B1 | |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use of \((x+3)\times\) their\((1+x+\frac{3}{2}x^2)\) | M1 | |
| Coefficient is \(5.5\) oe | A1 | or B2 www in either part; may be embedded (eg \(5.5x^2\) alone or in expansion) |
| [2] |
# Question 3(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1+(-\frac{1}{2})(-2x)+(-\frac{1}{2})(\frac{-3}{2})\frac{(\pm2x)^2}{2!}[+\ldots]$ | B1 | First two terms; allow recovery from omission of brackets |
| | B1 | Third term; do not allow $2x^2$ unless fully recovered in answer |
| $1+x+\frac{3}{2}x^2$ oe | B1 | |
| **[3]** | | |
# Question 3(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use of $(x+3)\times$ their$(1+x+\frac{3}{2}x^2)$ | M1 | |
| Coefficient is $5.5$ oe | A1 | or **B2** www in either part; may be embedded (eg $5.5x^2$ alone or in expansion) |
| **[2]** | | |3 (i) Find the first three terms in the expansion of $( 1 - 2 x ) ^ { - \frac { 1 } { 2 } }$ in ascending powers of $x$, where $| x | < \frac { 1 } { 2 }$.\\
(ii) Hence find the coefficient of $x ^ { 2 }$ in the expansion of $\frac { x + 3 } { \sqrt { 1 - 2 x } }$.
\hfill \mbox{\textit{OCR C4 2014 Q3 [5]}}