| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Standard product of two binomials |
| Difficulty | Moderate -0.8 This is a straightforward multi-part binomial expansion question requiring routine application of the binomial theorem and basic algebraic manipulation. Part (a) is trivial expansion of a squared binomial, part (b) is standard binomial theorem application with fractional coefficients, and part (c) involves multiplying polynomials to find a specific coefficient—all mechanical processes with no problem-solving insight required, making it easier than average. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\left(1+\frac{3}{x}\right)^2 = 1+\frac{6}{x}+\frac{9}{x^2}\) | M1 | Attempts \(\left(1+\frac{3}{x}\right)^2 = A+\frac{B}{x}+\frac{C}{x^2}\) |
| Correct equation \(1+\frac{6}{x}+\frac{9}{x^2}\) | A1 | Fully correct |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\left(1+\frac{3}{4}x\right)^6 = 1+6\times\left(\frac{3}{4}x\right)+...\) | B1 | First two terms correct, may be unsimplified |
| \(1+6\times\left(\frac{3}{4}x\right)+\frac{6\times5}{2}\times\left(\frac{3}{4}x\right)^2+\frac{6\times5\times4}{3\times2}\times\left(\frac{3}{4}x\right)^3+...\) | M1 | Attempts binomial expansion; correct coefficient and power of \(x\) seen at least once in term 3 or 4 |
| Unsimplified expansion correct | A1 | Binomial expansion correct and unsimplified |
| \(= 1+\frac{9}{2}x+\frac{135}{16}x^2+\frac{135}{16}x^3+...\) | A1 | Binomial expansion correct and simplified |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\left(1+\frac{3}{x}\right)^2\left(1+\frac{3}{4}x\right)^6 = \left(1+\frac{6}{x}+\frac{9}{x^2}\right)\left(1+\frac{9}{2}x+\frac{135}{16}x^2+\frac{135}{16}x^3+...\right)\) | M1 | Combines all relevant terms for \(\left(1+\frac{A}{x}+\frac{B}{x^2}\right)\left(1+Cx+Dx^2+Ex^3+...\right)\) to find coefficient of \(x\) |
| Coefficient of \(x = \frac{9}{2}+6\times\frac{135}{16}+9\times\frac{135}{16} = \frac{2097}{16}\) | A1 | Fully correct |
## Question 7:
### Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| $\left(1+\frac{3}{x}\right)^2 = 1+\frac{6}{x}+\frac{9}{x^2}$ | M1 | Attempts $\left(1+\frac{3}{x}\right)^2 = A+\frac{B}{x}+\frac{C}{x^2}$ |
| Correct equation $1+\frac{6}{x}+\frac{9}{x^2}$ | A1 | Fully correct |
### Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| $\left(1+\frac{3}{4}x\right)^6 = 1+6\times\left(\frac{3}{4}x\right)+...$ | B1 | First two terms correct, may be unsimplified |
| $1+6\times\left(\frac{3}{4}x\right)+\frac{6\times5}{2}\times\left(\frac{3}{4}x\right)^2+\frac{6\times5\times4}{3\times2}\times\left(\frac{3}{4}x\right)^3+...$ | M1 | Attempts binomial expansion; correct coefficient and power of $x$ seen at least once in term 3 or 4 |
| Unsimplified expansion correct | A1 | Binomial expansion correct and unsimplified |
| $= 1+\frac{9}{2}x+\frac{135}{16}x^2+\frac{135}{16}x^3+...$ | A1 | Binomial expansion correct and simplified |
### Part (c):
| Working | Mark | Guidance |
|---------|------|----------|
| $\left(1+\frac{3}{x}\right)^2\left(1+\frac{3}{4}x\right)^6 = \left(1+\frac{6}{x}+\frac{9}{x^2}\right)\left(1+\frac{9}{2}x+\frac{135}{16}x^2+\frac{135}{16}x^3+...\right)$ | M1 | Combines all relevant terms for $\left(1+\frac{A}{x}+\frac{B}{x^2}\right)\left(1+Cx+Dx^2+Ex^3+...\right)$ to find coefficient of $x$ |
| Coefficient of $x = \frac{9}{2}+6\times\frac{135}{16}+9\times\frac{135}{16} = \frac{2097}{16}$ | A1 | Fully correct |
---\begin{enumerate}
\item (a) Expand $\left( 1 + \frac { 3 } { x } \right) ^ { 2 }$ simplifying each term.\\
(b) Use the binomial expansion to find, in ascending powers of $x$, the first four terms in the expansion of
\end{enumerate}
$$\left( 1 + \frac { 3 } { 4 } x \right) ^ { 6 }$$
simplifying each term.\\
(c) Hence find the coefficient of $x$ in the expansion of
$$\left( 1 + \frac { 3 } { x } \right) ^ { 2 } \left( 1 + \frac { 3 } { 4 } x \right) ^ { 6 }$$
\hfill \mbox{\textit{Edexcel AS Paper 1 Q7 [8]}}