| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with unknown constant to determine |
| Difficulty | Moderate -0.8 This is a straightforward binomial expansion question requiring routine application of the binomial theorem formula, followed by simple algebraic manipulation to find 'a'. The first part is pure recall, and the second part involves multiplying out one term and solving a linear equation. No problem-solving insight required, just mechanical application of a standard technique. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((2+ax)^5 = 32 + 80ax + 80a^2x^2\) | \(3\times\) B1 [3] | B1 for each term |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\times(1+2x)\) | M1 | Realises need to consider 2 terms |
| \(240 = 80a^2 + 160a\) | DM1A1 | Solution of 3-term quadratic |
| \(\rightarrow a = 1\) or \(a = -3\) | [3] |
## Question 4:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(2+ax)^5 = 32 + 80ax + 80a^2x^2$ | $3\times$ B1 [3] | B1 for each term |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\times(1+2x)$ | M1 | Realises need to consider 2 terms |
| $240 = 80a^2 + 160a$ | DM1A1 | Solution of 3-term quadratic |
| $\rightarrow a = 1$ or $a = -3$ | [3] | |
---4 (i) Find the first three terms in the expansion of $( 2 + a x ) ^ { 5 }$ in ascending powers of $x$.\\
(ii) Given that the coefficient of $x ^ { 2 }$ in the expansion of $( 1 + 2 x ) ( 2 + a x ) ^ { 5 }$ is 240 , find the possible values of $a$.
\hfill \mbox{\textit{CAIE P1 2013 Q4 [6]}}