| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Binomial with negative or fractional powers of x |
| Difficulty | Standard +0.3 This is a straightforward binomial expansion question requiring students to match coefficients to find k, then calculate A and the constant term. While it involves multiple steps and careful index manipulation, the method is standard and mechanical—apply the binomial theorem formula, equate coefficients, and identify the required term. Slightly above average difficulty due to the fractional power term and need for systematic working, but no novel insight required. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \((x + \frac{k}{x})^{15} = x^{15} + 15(x^{14})(\frac{k}{x}) + \binom{15}{2}(x^{13})(\frac{k}{x})^2 + \ldots\) | M1 A1 | |
| \(\therefore 15k = 30\) | M1 | |
| \(k = 2\) | A1 | |
| \(A = \frac{15 \times 14}{2} \times k^2 = 420\) | A1 | |
| (b) \((x + \frac{2}{x})^{15} = \ldots + \binom{15}{5}(x^{10})(\frac{2}{x})^5 + \ldots\) | M1 A1 | |
| term indep. of \(x = \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2} \times 32 = 96\,096\) | A1 | (8 marks) |
(a) $(x + \frac{k}{x})^{15} = x^{15} + 15(x^{14})(\frac{k}{x}) + \binom{15}{2}(x^{13})(\frac{k}{x})^2 + \ldots$ | M1 A1 |
$\therefore 15k = 30$ | M1 |
$k = 2$ | A1 |
$A = \frac{15 \times 14}{2} \times k^2 = 420$ | A1 |
(b) $(x + \frac{2}{x})^{15} = \ldots + \binom{15}{5}(x^{10})(\frac{2}{x})^5 + \ldots$ | M1 A1 |
term indep. of $x = \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2} \times 32 = 96\,096$ | A1 | (8 marks)4. The first three terms in the expansion in descending powers of $x$ of
$$\left( x + \frac { k } { x ^ { 2 } } \right) ^ { 15 } ,$$
where $k$ is a constant, are
$$x ^ { 15 } + 30 x ^ { 12 } + A x ^ { 9 } .$$
\begin{enumerate}[label=(\alph*)]
\item Find the values of $k$ and $A$.
\item Find the value of the term independent of $x$ in the expansion.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q4 [8]}}