| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | March |
| Marks | 5 |
| 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 application of the binomial theorem requiring only direct substitution into the formula and basic algebraic multiplication. Parts (a) and (b) are routine recall, while part (c) involves multiplying two expansions to find a specific coefficient—a standard textbook exercise with no problem-solving insight required. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1 + 5x + 10x^2\) | B1 | |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1 - 12x + 60x^2\) | B2, 1, 0 | B2 all correct, B1 for two correct components |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((1 + 5x + 10x^2)(1 - 12x + 60x^2)\) leading to \(60 - 60 + 10\) | M1 | 3 products required |
| \(10\) | A1 | Allow \(10x^2\) |
| Total: 2 |
## Question 1:
**Part (a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1 + 5x + 10x^2$ | B1 | |
| | **Total: 1** | |
**Part (b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1 - 12x + 60x^2$ | B2, 1, 0 | B2 all correct, B1 for two correct components |
| | **Total: 2** | |
**Part (c):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(1 + 5x + 10x^2)(1 - 12x + 60x^2)$ leading to $60 - 60 + 10$ | M1 | 3 products required |
| $10$ | A1 | Allow $10x^2$ |
| | **Total: 2** | |
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\begin{enumerate}[label=(\alph*)]
\item Find the first three terms in the expansion, in ascending powers of $x$, of $( 1 + x ) ^ { 5 }$.
\item Find the first three terms in the expansion, in ascending powers of $x$, of $( 1 - 2 x ) ^ { 6 }$.
\item Hence find the coefficient of $x ^ { 2 }$ in the expansion of $( 1 + x ) ^ { 5 } ( 1 - 2 x ) ^ { 6 }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q1 [5]}}