| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2015 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Factoring out constants first |
| Difficulty | Moderate -0.3 This is a straightforward application of the binomial expansion for fractional powers. Students must factor out 8^(2/3) first, then apply the standard formula with n=2/3 and substitute -9x/8. The validity condition |9x/8| < 1 is routine. While it requires careful algebraic manipulation, it's a standard C4 question with no novel problem-solving required. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(8^{2/3}=4\) | B1 | May be embedded |
| \(\left(1-\frac{9x}{8}\right)^{2/3}\) seen | M1 | \(8^{2/3}+(\frac{2}{3})8^{-1/3}(\pm9x)+\frac{\frac{2}{3}\times(\frac{2}{3}-1)}{2!}8^{-4/3}(\pm9x)^2\); ignore extra terms |
| \(1+\left(\frac{2}{3}\right)\left(\frac{\pm9x}{k}\right)+\frac{1}{2!}\left(\frac{2}{3}\right)\left(\frac{2}{3}-1\right)\left(\frac{\pm9x}{k}\right)^2\) where \(k\) is an integer greater than 1 | M1 | \(4+(\frac{2}{3})(\frac{1}{2})(\pm9x)+\frac{\frac{2}{3}\times(\frac{2}{3}-1)}{2!}(\frac{1}{16})(\pm9x)^2\) or better |
| \(4-3x-\frac{9}{16}x^2\) or \(4(1-\frac{3}{4}x-\frac{9}{64}x^2)\) cao | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
\(-\frac{8}{9}| x |
<\frac{8}{9}\) isw cao |
|
# Question 4:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $8^{2/3}=4$ | B1 | May be embedded |
| $\left(1-\frac{9x}{8}\right)^{2/3}$ seen | M1 | $8^{2/3}+(\frac{2}{3})8^{-1/3}(\pm9x)+\frac{\frac{2}{3}\times(\frac{2}{3}-1)}{2!}8^{-4/3}(\pm9x)^2$; ignore extra terms |
| $1+\left(\frac{2}{3}\right)\left(\frac{\pm9x}{k}\right)+\frac{1}{2!}\left(\frac{2}{3}\right)\left(\frac{2}{3}-1\right)\left(\frac{\pm9x}{k}\right)^2$ where $k$ is an integer greater than 1 | M1 | $4+(\frac{2}{3})(\frac{1}{2})(\pm9x)+\frac{\frac{2}{3}\times(\frac{2}{3}-1)}{2!}(\frac{1}{16})(\pm9x)^2$ or better |
| $4-3x-\frac{9}{16}x^2$ or $4(1-\frac{3}{4}x-\frac{9}{64}x^2)$ cao | A1 | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $-\frac{8}{9}<x<\frac{8}{9}$ or $|x|<\frac{8}{9}$ isw cao | B1 | |
---4 (i) Find the first three terms in the binomial expansion of $( 8 - 9 x ) ^ { \frac { 2 } { 3 } }$ in ascending powers of $x$.\\
(ii) State the set of values of $x$ for which this expansion is valid.
\hfill \mbox{\textit{OCR C4 2015 Q4 [5]}}