| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| 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 binomial expansion question requiring direct application of the binomial theorem formula for parts (a) and (b), followed by a routine multiplication of polynomials in part (c). The question involves only positive integer powers, standard coefficients, and collecting like terms—all mechanical processes with no problem-solving insight required. It's easier than average but not trivial since it requires careful arithmetic across multiple steps. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(16 + 96x + 216x^2\) | B2, 1, 0 | ISW (higher powers of \(x\)). Terms may be in any order or presented as a list. |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1 - 10x + 40x^2\) | B2, 1, 0 | ISW (higher powers of \(x\)). Terms may be in any order or presented as a list. |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((16\times40)-(10\times96)+(1\times216)\) | M1 | *Their* 3 products which would give the term in \(x^2\) (FT *their* values). Look for \(640 - 960 + 216\). |
| \(-104\) | A1 | Condone \(-104x^2\). |
| Total: 2 |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $16 + 96x + 216x^2$ | B2, 1, 0 | ISW (higher powers of $x$). Terms may be in any order or presented as a list. |
| **Total: 2** | | |
---
## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1 - 10x + 40x^2$ | B2, 1, 0 | ISW (higher powers of $x$). Terms may be in any order or presented as a list. |
| **Total: 2** | | |
---
## Question 2(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(16\times40)-(10\times96)+(1\times216)$ | M1 | *Their* 3 products which would give the term in $x^2$ (FT *their* values). Look for $640 - 960 + 216$. |
| $-104$ | A1 | Condone $-104x^2$. |
| **Total: 2** | | |2
\begin{enumerate}[label=(\alph*)]
\item Find the first three terms in the expansion, in ascending powers of $x$, of $( 2 + 3 x ) ^ { 4 }$.
\item Find the first three terms in the expansion, in ascending powers of $x$, of $( 1 - 2 x ) ^ { 5 }$.
\item Hence find the coefficient of $x ^ { 2 }$ in the expansion of $( 2 + 3 x ) ^ { 4 } ( 1 - 2 x ) ^ { 5 }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2023 Q2 [6]}}