| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | November |
| 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 application of the binomial theorem requiring routine expansion of two binomials and multiplication of terms to find a specific coefficient. Part (c) involves simple algebraic manipulation of the results from (a) and (b), with no problem-solving insight needed beyond recognizing which terms multiply to give x². Easier than average due to small integer powers and guided structure. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 + 10x + 40x^2\) (may be part of a complete expansion) | B2, 1, 0 | \(1^5\) must be simplified to 1, allow if the '1' is seen in a more complete expansion but not the final answer. Mis-reads not condoned. |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 - 12x + 54x^2\) (may be part of a complete expansion) | B2, 1, 0 | \(1^4\) must be simplified to 1, allow if the '1' is seen in a more complete expansion but not the final answer. Mis-reads not condoned. |
| Answer | Marks | Guidance |
|---|---|---|
| \(54 - 120 + 40\) | M1 | Forming exactly 3 products correctly using their terms. |
| \(-26\) | A1 | Allow \(-26x^2\). If in a list with other terms it must be clear this is the required term otherwise A0. |
## Question 3:
### Part 3(a):
$1 + 10x + 40x^2$ (may be part of a complete expansion) | **B2, 1, 0** | $1^5$ must be simplified to 1, allow if the '1' is seen in a more complete expansion but not the final answer. Mis-reads not condoned.
### Part 3(b):
$1 - 12x + 54x^2$ (may be part of a complete expansion) | **B2, 1, 0** | $1^4$ must be simplified to 1, allow if the '1' is seen in a more complete expansion but not the final answer. Mis-reads not condoned.
### Part 3(c):
$54 - 120 + 40$ | **M1** | Forming exactly 3 products correctly using their terms.
$-26$ | **A1** | Allow $-26x^2$. If in a list with other terms it must be clear this is the required term otherwise A0.
---3
\begin{enumerate}[label=(\alph*)]
\item Find the first three terms in ascending powers of $x$ of the expansion of $( 1 + 2 x ) ^ { 5 }$.
\item Find the first three terms in ascending powers of $x$ of the expansion of $( 1 - 3 x ) ^ { 4 }$.
\item Hence find the coefficient of $x ^ { 2 }$ in the expansion of $( 1 + 2 x ) ^ { 5 } ( 1 - 3 x ) ^ { 4 }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q3 [6]}}