| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Substitution into binomial expansion |
| Difficulty | Moderate -0.8 This is a straightforward application of the binomial theorem with n=5, followed by a simple substitution of a=(x+x²) and collecting terms. Part (a) is pure recall of the binomial expansion formula, and part (b) requires only careful algebraic manipulation with no problem-solving insight needed. Easier than average A-level questions. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| 4(a): \(1 + 5a + 10a^2 + 10a^3 + ...\) | B1 | |
| 4(b): \(1 + 5(x+x^2) + 10(x+x^2)^2 + 10(x+x^2)^3 + ...\) | M1 | SOI |
| \(1 + 5(x+x^2) + 10(x^2 + 2x^3 + ...) + 10(x^3 + ...) + ...\) | A1 | SOI |
| \(1 + 5x + 15x^2 + 30x^3 + ...\) | A1 |
## Question 4:
**4(a):** $1 + 5a + 10a^2 + 10a^3 + ...$ | B1 |
**4(b):** $1 + 5(x+x^2) + 10(x+x^2)^2 + 10(x+x^2)^3 + ...$ | M1 | SOI
$1 + 5(x+x^2) + 10(x^2 + 2x^3 + ...) + 10(x^3 + ...) + ...$ | A1 | SOI
$1 + 5x + 15x^2 + 30x^3 + ...$ | A1 |
---4
\begin{enumerate}[label=(\alph*)]
\item Expand $( 1 + a ) ^ { 5 }$ in ascending powers of $a$ up to and including the term in $a ^ { 3 }$.
\item Hence expand $\left[ 1 + \left( x + x ^ { 2 } \right) \right] ^ { 5 }$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying your answer.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2020 Q4 [4]}}