| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| 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 Part (a) is straightforward binomial expansion requiring direct application of the formula for three terms. Part (b) requires multiplying two expansions and collecting terms, which is a standard textbook exercise with clear methodology. The question involves routine algebraic manipulation with no conceptual challenges or novel problem-solving required. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(243\) | B1 | |
| \(-810x\) | B1 | |
| \(+1080x^2\) | B1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((4+x)^2 = 16 + 8x + x^2\) | B1 | |
| Coefficient of \(x^2\) is \(16 \times 1080 + 8 \times (-810) + 243\) | M1 | Allow if at least 2 pairs used correctly |
| \(11043\) | A1 | Allow \(11043x^2\) |
| 3 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $243$ | B1 | |
| $-810x$ | B1 | |
| $+1080x^2$ | B1 | |
| | **3** | |
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(4+x)^2 = 16 + 8x + x^2$ | B1 | |
| Coefficient of $x^2$ is $16 \times 1080 + 8 \times (-810) + 243$ | M1 | Allow if at least 2 pairs used correctly |
| $11043$ | A1 | Allow $11043x^2$ |
| | **3** | |3
\begin{enumerate}[label=(\alph*)]
\item Find the first three terms in the expansion of $( 3 - 2 x ) ^ { 5 }$ in ascending powers of $x$.
\item Hence find the coefficient of $x ^ { 2 }$ in the expansion of $( 4 + x ) ^ { 2 } ( 3 - 2 x ) ^ { 5 }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q3 [6]}}