| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Substitution into binomial expansion |
| Difficulty | Moderate -0.3 Part (i) is a straightforward binomial expansion with small integer power requiring only formula application. Part (ii) involves substitution and collecting terms, which adds a modest problem-solving element, but the overall question remains routine for A-level with clear signposting and limited algebraic complexity. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
Question 1:
(i) $(2-y)^5 = 32 - 80y + 80y^2$
B2,1
[2]
(ii) $(2-(2x-x^2))^5$ where $y = 2x - x^2$
M1
$80 + 320 = 400$
M1
A1
[3]
**Guidance:**
$-1$ for each error. Accept $\frac{5}{2}$.
Allow for $y = 2x + x^2$
Needs to consider exactly 2 terms.
$400x^2$ — accept if $\frac{400x^2}{...}$ is part of it, accept full expansion.1 (i) Find the first 3 terms in the expansion of $( 2 - y ) ^ { 5 }$ in ascending powers of $y$.\\
(ii) Use the result in part (i) to find the coefficient of $x ^ { 2 }$ in the expansion of $\left( 2 - \left( 2 x - x ^ { 2 } \right) \right) ^ { 5 }$.
\hfill \mbox{\textit{CAIE P1 2011 Q1 [5]}}