| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with unknown constant to determine |
| Difficulty | Moderate -0.3 This is a straightforward binomial expansion question requiring students to expand two binomials, multiply the results, and solve a quadratic equation to find the constant. While it involves multiple steps (two expansions, multiplication, equation setup), each step uses routine techniques with no novel insight required. It's slightly easier than average because the binomial theorem with positive integer powers is one of the most practiced topics, though the algebraic manipulation keeps it from being trivial. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1 + 10x + 40x^2\) | B1 | Ignore any additional terms (ISW). Allow \(x^0\) or \(1\cdot x^0\) for the first term, allow lists |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\{1\}\{-6ax\}\{+15a^2x^2\}\) | B2, 1, 0 | Ignore any additional terms (ISW). Allow \(x^0\) or \(1\cdot x^0\) for the first term, allow lists |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(15a^2 - 60a + 40 = -5\) | M1 A1 | Correct 3 products from *their* expansions for M1. Condone inclusion of \(x^2\) for M1 |
| \([15](a-1)(a-3) = 0\) OE | DM1 | OE. Rearranging and solving a quadratic |
| \(a = 1\) and \(3\) | A1 | Special case: If M1 A1 DM0 scored then SC B1 can be awarded for correct answers |
| 4 |
## Question 4(a)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1 + 10x + 40x^2$ | B1 | Ignore any additional terms (ISW). Allow $x^0$ or $1\cdot x^0$ for the first term, allow lists |
| | **1** | |
## Question 4(a)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\{1\}\{-6ax\}\{+15a^2x^2\}$ | B2, 1, 0 | Ignore any additional terms (ISW). Allow $x^0$ or $1\cdot x^0$ for the first term, allow lists |
| | **2** | |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $15a^2 - 60a + 40 = -5$ | M1 A1 | Correct 3 products from *their* expansions for M1. Condone inclusion of $x^2$ for M1 |
| $[15](a-1)(a-3) = 0$ OE | DM1 | OE. Rearranging and solving a quadratic |
| $a = 1$ and $3$ | A1 | **Special case:** If M1 A1 DM0 scored then **SC B1** can be awarded for correct answers |
| | **4** | |4
\begin{enumerate}[label=(\alph*)]
\item Expand the following in ascending powers of $x$ up to and including the term in $x ^ { 2 }$.
\begin{enumerate}[label=(\roman*)]
\item $( 1 + 2 x ) ^ { 5 }$.
\item $( 1 - a x ) ^ { 6 }$, where $a$ is a constant.\\
In the expansion of $( 1 + 2 x ) ^ { 5 } ( 1 - a x ) ^ { 6 }$, the coefficient of $x ^ { 2 }$ is - 5 .
\end{enumerate}\item Find the possible values of $a$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2023 Q4 [7]}}