| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with unknown constant to determine |
| Difficulty | Moderate -0.3 Part (a) is straightforward binomial expansion requiring recall of the formula. Part (b) involves multiplying expansions and equating coefficients to solve a quadratic, which is a standard technique but requires careful algebraic manipulation across multiple steps. This is slightly easier than average as it's a routine multi-part question with no novel insight required. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1 + 6ax + 15a^2x^2\) | B1 | Terms must be evaluated. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| their \(15a^2 \pm (3 \times \text{their } 6a)\) | \*M1 | Expect \(15a^2 - 18a\) |
| \(15a^2 - 18a = -3\) | A1 | |
| \((3)(a-1)(5a-1) [= 0]\) | DM1 | Dependent on 3-term quadratic. Or solve using formula or completing the square. |
| \(a = 1, \frac{1}{5}\) | A1 | WWW. If DM0 awarded SC B1 if both answers correct. |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1 + 6ax + 15a^2x^2$ | B1 | Terms must be evaluated. |
## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| their $15a^2 \pm (3 \times \text{their } 6a)$ | \*M1 | Expect $15a^2 - 18a$ |
| $15a^2 - 18a = -3$ | A1 | |
| $(3)(a-1)(5a-1) [= 0]$ | DM1 | Dependent on 3-term quadratic. Or solve using formula or completing the square. |
| $a = 1, \frac{1}{5}$ | A1 | WWW. If DM0 awarded **SC B1** if both answers correct. |
---2
\begin{enumerate}[label=(\alph*)]
\item Find the first three terms, in ascending powers of $x$, in the expansion of $( 1 + a x ) ^ { 6 }$.
\item Given that the coefficient of $x ^ { 2 }$ in the expansion of $( 1 - 3 x ) ( 1 + a x ) ^ { 6 }$ is - 3 , find the possible values of the constant $a$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q2 [5]}}