| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2018 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with unknown constant to determine |
| Difficulty | Moderate -0.8 Part (i) is straightforward binomial expansion requiring only the first three terms using Pascal's triangle or the binomial formula. Part (ii) involves multiplying polynomials and collecting the x² coefficient, then solving a simple linear equation for a. This is a routine textbook exercise testing basic binomial expansion and algebraic manipulation with no problem-solving insight required, making it easier than average. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| \((1-2x)^5 = 1 - 10x + 40x^2\) (no penalty for extra terms) | B2,1 | Loses a mark for each incorrect term. Treat \(-32x^5 + 80x^4 - 80x^3\) as MR \(-1\) |
| Answer | Marks | Guidance |
|---|---|---|
| 3 terms in \(x^2 \rightarrow 40 - 10a + 2\) | M1 A1FT | Selects 3 terms in \(x^2\). FT from (i) |
| Equate with \(12 \rightarrow a = 3\) | A1 | CAO |
## Question 1:
### Part (i)
$(1-2x)^5 = 1 - 10x + 40x^2$ (no penalty for extra terms) | **B2,1** | Loses a mark for each incorrect term. Treat $-32x^5 + 80x^4 - 80x^3$ as MR $-1$
---
### Part (ii)
$\rightarrow (1 + ax + 2x^2)(1 - 10x + 40x^2)$
3 terms in $x^2 \rightarrow 40 - 10a + 2$ | **M1 A1FT** | Selects 3 terms in $x^2$. FT from **(i)**
Equate with $12 \rightarrow a = 3$ | **A1** | CAO
---1 (i) Find the first three terms in the expansion, in ascending powers of $x$, of $( 1 - 2 x ) ^ { 5 }$.\\
(ii) Given that the coefficient of $x ^ { 2 }$ in the expansion of $\left( 1 + a x + 2 x ^ { 2 } \right) ( 1 - 2 x ) ^ { 5 }$ is 12 , find the value of the constant $a$.\\
\hfill \mbox{\textit{CAIE P1 2018 Q1 [5]}}