| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem and Partial Fractions |
| Type | Partial fractions then binomial expansion |
| Difficulty | Standard +0.3 This is a standard two-part question combining partial fractions with binomial expansion. Part (i) requires decomposing into the form A/(3-x) + B/(1+2x) + C/(1+2x)², which is routine. Part (ii) involves expanding three simple terms using the binomial theorem and collecting coefficients—straightforward application of techniques with no novel insight required. Slightly easier than average due to the mechanical nature and clear structure. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Either State or imply partial fractions are of form \(\frac{A}{3-x} + \frac{B}{1+2x} + \frac{C}{(1+2x)^2}\) | B1 | |
| Use any relevant method to obtain a constant | M1 | |
| Obtain \(A = 1\) | A1 | |
| Obtain \(B = \frac{3}{2}\) | A1 | |
| Obtain \(C = -\frac{1}{2}\) | A1 | [5] |
| Or State or imply partial fractions are of form \(\frac{A}{3-x} + \frac{Dx+E}{(1+2x)^2}\) | B1 | |
| Use any relevant method to obtain a constant | M1 | |
| Obtain \(A = 1\) | A1 | |
| Obtain \(D = 3\) | A1 | |
| Obtain \(E = 1\) | A1 | [5] |
| (ii) Obtain first two terms of one of the expansion of \((3-x)^{-1} \cdot \left(1 - \frac{1}{3}x\right)^{-1}\) | M1 | |
| \((1+2x)^{-1}\) and \((1+2x)^{-2}\) | M1 | |
| Obtain correct unsimplified expansion up to term in \(x^2\) of each partial fraction, following in each case the value of \(A, B, C\) | A1✓ A1✓ A1✓ | |
| Obtain answer \(\frac{4}{3} - \frac{8}{9}x + \frac{1}{27}x^2\) | A1 | [5] |
(i) Either **State or imply partial fractions are of form $\frac{A}{3-x} + \frac{B}{1+2x} + \frac{C}{(1+2x)^2}$** | B1 |
Use any relevant method to obtain a constant | M1 |
Obtain $A = 1$ | A1 |
Obtain $B = \frac{3}{2}$ | A1 |
Obtain $C = -\frac{1}{2}$ | A1 | [5]
Or **State or imply partial fractions are of form $\frac{A}{3-x} + \frac{Dx+E}{(1+2x)^2}$** | B1 |
Use any relevant method to obtain a constant | M1 |
Obtain $A = 1$ | A1 |
Obtain $D = 3$ | A1 |
Obtain $E = 1$ | A1 | [5]
(ii) Obtain first two terms of one of the expansion of $(3-x)^{-1} \cdot \left(1 - \frac{1}{3}x\right)^{-1}$ | M1 |
$(1+2x)^{-1}$ and $(1+2x)^{-2}$ | M1 |
Obtain correct unsimplified expansion up to term in $x^2$ of each partial fraction, following in each case the value of $A, B, C$ | A1✓ A1✓ A1✓ |
Obtain answer $\frac{4}{3} - \frac{8}{9}x + \frac{1}{27}x^2$ | A1 | [5]
[If $A, D, E$ approach used in part (i), give M1A1✓A1✓ for expansions, M1 for multiplying out fully and A1 for final answer]9 (i) Express $\frac { 4 + 12 x + x ^ { 2 } } { ( 3 - x ) ( 1 + 2 x ) ^ { 2 } }$ in partial fractions.\\
(ii) Hence obtain the expansion of $\frac { 4 + 12 x + x ^ { 2 } } { ( 3 - x ) ( 1 + 2 x ) ^ { 2 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.
\hfill \mbox{\textit{CAIE P3 2014 Q9 [10]}}