| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2010 |
| Session | January |
| Marks | 7 |
| 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 with fractional powers. Part (i) is routine recall of the formula, part (ii)(a) requires the standard technique of factoring out constants (8 = 2³), and part (ii)(b) tests understanding of validity conditions (|x| < 1/2). While it requires multiple steps, each is a standard textbook procedure with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| \((1+x)^{\frac{1}{3}} = 1 + \frac{1}{3}x + ...\) | B1 | |
| \(... - \frac{1}{9}x^2\) | B1 | \(-\frac{2}{18}x^2\) acceptable |
| Answer | Marks | Guidance |
|---|---|---|
| \((8+16x)^{\frac{1}{3}} = 8^{\frac{1}{3}}(1+2x)^{\frac{1}{3}}\) | B1 | not \(16^{\frac{1}{3}}(\frac{1}{2}+x)^{\frac{1}{3}}\) |
| \((1+2x)^{\frac{1}{3}}\) = their (i) expansion with \(2x\) replacing \(x\) | M1 | not dep on prev B1 |
| \(= 1 + \frac{2}{3}x - \frac{4}{9}x^2 + ...\) | \(\sqrt{}\)A1 | \(-\frac{8}{18}x^2\) acceptable |
| Required expansion \(= 2\) (expansion just found) | \(\sqrt{}\)B1 | accept equiv fractions |
| Answer | Marks | Guidance |
|---|---|---|
| \(-\frac{1}{2} < x < \frac{1}{2}\) or \( | x | < \frac{1}{2}\) |
# Question 5:
## Part (i):
| $(1+x)^{\frac{1}{3}} = 1 + \frac{1}{3}x + ...$ | B1 | |
|---|---|---|
| $... - \frac{1}{9}x^2$ | B1 | $-\frac{2}{18}x^2$ acceptable |
## Part (ii)(a):
| $(8+16x)^{\frac{1}{3}} = 8^{\frac{1}{3}}(1+2x)^{\frac{1}{3}}$ | B1 | not $16^{\frac{1}{3}}(\frac{1}{2}+x)^{\frac{1}{3}}$ |
|---|---|---|
| $(1+2x)^{\frac{1}{3}}$ = their (i) expansion with $2x$ replacing $x$ | M1 | not dep on prev B1 |
| $= 1 + \frac{2}{3}x - \frac{4}{9}x^2 + ...$ | $\sqrt{}$A1 | $-\frac{8}{18}x^2$ acceptable |
| Required expansion $= 2$ (expansion just found) | $\sqrt{}$B1 | accept equiv fractions |
## Part (ii)(b):
| $-\frac{1}{2} < x < \frac{1}{2}$ or $|x| < \frac{1}{2}$ | B1 | no equality |
|---|---|---|
---5
\begin{enumerate}[label=(\roman*)]
\item Expand $( 1 + x ) ^ { \frac { 1 } { 3 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.
\item (a) Hence, or otherwise, expand $( 8 + 16 x ) ^ { \frac { 1 } { 3 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.\\
(b) State the set of values of $x$ for which the expansion in part (ii) (a) is valid.
\end{enumerate}
\hfill \mbox{\textit{OCR C4 2010 Q5 [7]}}