| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Binomial times quadratic coefficient |
| Difficulty | Moderate -0.8 Part (a) is straightforward application of binomial expansion formula requiring only the first three terms. Part (b) involves multiplying polynomials and collecting like terms, which is routine algebraic manipulation once part (a) is complete. This is a standard textbook exercise testing basic binomial theorem mechanics with minimal problem-solving demand. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1 + 18x + 135x^2\) | B2, 1, 0 | Accept 1, \(18x\), \(135x^2\) listed horizontally or vertically or \(1x^0 + 18x + 135x^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Coefficient of \(x^2\) is \(135 - 7 \times 18 + 1 = 10\) | M1 A1 | 3 products, allow \(10x^2\). If full expansion given, like terms must be collected for M1. |
**Question 1:**
**Part (a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1 + 18x + 135x^2$ | B2, 1, 0 | Accept 1, $18x$, $135x^2$ listed horizontally or vertically or $1x^0 + 18x + 135x^2$ |
**Part (b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Coefficient of $x^2$ is $135 - 7 \times 18 + 1 = 10$ | M1 A1 | 3 products, allow $10x^2$. If full expansion given, like terms must be collected for M1. |
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\begin{enumerate}[label=(\alph*)]
\item Expand $( 1 + 3 x ) ^ { 6 }$ in ascending powers of $x$ up to, and including, the term in $x ^ { 2 }$.
\item Hence find the coefficient of $x ^ { 2 }$ in the expansion of $\left( 1 - 7 x + x ^ { 2 } \right) ( 1 + 3 x ) ^ { 6 }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2023 Q1 [4]}}