| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with reciprocal term binomial |
| Difficulty | Moderate -0.8 This is a straightforward binomial expansion question requiring routine application of the binomial theorem and algebraic multiplication. Part (a) is trivial algebra, part (b) is direct formula application, and part (c) requires identifying which term products give x, but the 'hence' structure guides students through the method with no problem-solving insight needed. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 - \frac{1}{x} + \frac{1}{4x^2}\) | B1 | OE. Multiply or use binomial expansion. Allow unsimplified. |
Total: 1 mark
| Answer | Marks | Guidance |
|---|---|---|
| \(1 + 12x + 60x^2 + 160x^3\) | B2, 1, 0 | Withhold 1 mark for each error; B2, 1, 0. ISW if more than 4 terms in the expansion. |
Total: 2 marks
| Answer | Marks | Guidance |
|---|---|---|
| \(their(1 \times 12) + their(-1 \times 60) + their(\frac{1}{4} \times 160)\) | M1 | Attempts at least 2 products where each product contains one term from each expansion. |
| \([12 - 60 + 40 =] -8\) | A1 | Allow \(-8x\). |
Total: 2 marks
**Question 1:**
**Part (a):**
$1 - \frac{1}{x} + \frac{1}{4x^2}$ | **B1** | OE. Multiply or use binomial expansion. Allow unsimplified.
Total: 1 mark
---
**Part (b):**
$1 + 12x + 60x^2 + 160x^3$ | **B2, 1, 0** | Withhold 1 mark for each error; B2, 1, 0. ISW if more than 4 terms in the expansion.
Total: 2 marks
---
**Part (c):**
$their(1 \times 12) + their(-1 \times 60) + their(\frac{1}{4} \times 160)$ | **M1** | Attempts at least 2 products where each product contains one term from each expansion.
$[12 - 60 + 40 =] -8$ | **A1** | Allow $-8x$.
Total: 2 marks1
\begin{enumerate}[label=(\alph*)]
\item Expand $\left( 1 - \frac { 1 } { 2 x } \right) ^ { 2 }$.
\item Find the first four terms in the expansion, in ascending powers of $x$, of $( 1 + 2 x ) ^ { 6 }$.
\item Hence find the coefficient of $x$ in the expansion of $\left( 1 - \frac { 1 } { 2 x } \right) ^ { 2 } ( 1 + 2 x ) ^ { 6 }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q1 [5]}}