| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Arithmetic/geometric progression coefficients |
| Difficulty | Moderate -0.3 This is a straightforward application of the binomial expansion formula for negative indices. Part (i) requires equating two coefficients and solving a simple equation, while part (ii) involves routine substitution and simplification. The question tests standard technique with minimal problem-solving, making it slightly easier than average. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | ||
| Obtain correct unsimplified terms in \(x\) and \(x^3\) | B1 + B1 | |
| Equate coefficients and solve for \(a\) | M1 | |
| Obtain final answer \(a = \frac{1}{\sqrt{2}}\), or exact equivalent | A1 | [4] |
| (ii) | ||
| Use correct method and value of \(a\) to find the first two terms of the expansion \((1+\alpha x)^{-2}\) | M1 | |
| Obtain \(1 - \sqrt{2}x\), or equivalent | A1 \(\checkmark\) | |
| Obtain term \(-\frac{2}{5}x^2\) | A1 \(\checkmark\) | [3] |
**(i)** | |
Obtain correct unsimplified terms in $x$ and $x^3$ | B1 + B1 |
Equate coefficients and solve for $a$ | M1 |
Obtain final answer $a = \frac{1}{\sqrt{2}}$, or exact equivalent | A1 | [4]
**(ii)** | |
Use correct method and value of $a$ to find the first two terms of the expansion $(1+\alpha x)^{-2}$ | M1 |
Obtain $1 - \sqrt{2}x$, or equivalent | A1 $\checkmark$ |
Obtain term $-\frac{2}{5}x^2$ | A1 $\checkmark$ | [3]
[Symbolic coefficients, e.g. $\binom{-2}{1}a$, are not sufficient for the first B marks]
[The f.t. is solely on the value of $a$.]4 When $( 1 + a x ) ^ { - 2 }$, where $a$ is a positive constant, is expanded in ascending powers of $x$, the coefficients of $x$ and $x ^ { 3 }$ are equal.\\
(i) Find the exact value of $a$.\\
(ii) When $a$ has this value, obtain the expansion up to and including the term in $x ^ { 2 }$, simplifying the coefficients.
\hfill \mbox{\textit{CAIE P3 2012 Q4 [7]}}