| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2009 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Finding unknown constant from coefficient |
| Difficulty | Standard +0.3 This question requires applying the binomial expansion for fractional powers and collecting coefficients, but follows a standard pattern. Students must expand (1+ax)^(2/3) using the generalised binomial theorem, multiply by (1+2x), and set the x coefficient to zero to find a. Part (ii) is routine calculation once a is known. The algebraic manipulation is straightforward with no novel insight required. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State correct first two terms of the expansion of \((1 + ax)^{\frac{2}{3}}\), i.e. \(1 + \frac{2}{3}ax\) | B1 | Symbolic binomial coefficients e.g. \(\binom{2/3}{1}\) not acceptable |
| Form an expression for the coefficient of \(x\) in the expansion of \((1 + 2x)(1 + ax)^{\frac{2}{3}}\) and equate it to zero | M1 | |
| Obtain \(a = -3\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Obtain correct unsimplified terms in \(x^2\) and \(x^3\) in the expansion of \((1 - 3x)^{\frac{2}{3}}\) or \((1 + ax)^{\frac{2}{3}}\) | B1\(\sqrt{}\) + B1\(\sqrt{}\) | Symbolic binomial coefficients not acceptable for B marks |
| Carry out multiplication by \(1 + 2x\) obtaining two terms in \(x^3\) | M1 | |
| Obtain final answer \(-\frac{10}{3}x^3\), or equivalent | A1 |
## Question 5:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State correct first two terms of the expansion of $(1 + ax)^{\frac{2}{3}}$, i.e. $1 + \frac{2}{3}ax$ | B1 | Symbolic binomial coefficients e.g. $\binom{2/3}{1}$ not acceptable |
| Form an expression for the coefficient of $x$ in the expansion of $(1 + 2x)(1 + ax)^{\frac{2}{3}}$ and equate it to zero | M1 | |
| Obtain $a = -3$ | A1 | |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain correct unsimplified terms in $x^2$ and $x^3$ in the expansion of $(1 - 3x)^{\frac{2}{3}}$ or $(1 + ax)^{\frac{2}{3}}$ | B1$\sqrt{}$ + B1$\sqrt{}$ | Symbolic binomial coefficients not acceptable for B marks |
| Carry out multiplication by $1 + 2x$ obtaining two terms in $x^3$ | M1 | |
| Obtain final answer $-\frac{10}{3}x^3$, or equivalent | A1 | |
---5 When $( 1 + 2 x ) ( 1 + a x ) ^ { \frac { 2 } { 3 } }$, where $a$ is a constant, is expanded in ascending powers of $x$, the coefficient of the term in $x$ is zero.\\
(i) Find the value of $a$.\\
(ii) When $a$ has this value, find the term in $x ^ { 3 }$ in the expansion of $( 1 + 2 x ) ( 1 + a x ) ^ { \frac { 2 } { 3 } }$, simplifying the coefficient.
\hfill \mbox{\textit{CAIE P3 2009 Q5 [7]}}