| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2007 |
| Session | November |
| 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 decomposition with binomial expansion. Part (i) is routine A-level algebra requiring solving for three constants. Part (ii) applies standard binomial expansions to each fraction. While it requires careful algebraic manipulation and combining terms, it follows a well-practiced procedure with no novel insight needed, making it slightly easier than average. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04d Angles: between planes and between line and plane |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply the form \(\frac{A}{1-x} + \frac{B}{1+2x} + \frac{C}{2+x}\) | B1 | |
| Use any relevant method to determine a constant | M1 | |
| Obtain \(A = 1, B = 2\) and \(C = -4\) | A1 + A1 + A1 | [5] |
| (ii) Use correct method to obtain the first two terms of the expansion of \((1-x)^{-1}, (1+2x)^{-1}, (2+x)^{-1}\), or \((1+\frac{1}{2}x)^{-1}\) | M1 | |
| Obtain complete unsimplified expansions up to \(x^2\) of each partial fraction | A1∇ + A1∇ + A1∇ | |
| Combine expansions and obtain answer \(1 - 2x + \frac{17}{2}x^2\) | A1 | [5] |
**(i)** State or imply the form $\frac{A}{1-x} + \frac{B}{1+2x} + \frac{C}{2+x}$ | B1 |
Use any relevant method to determine a constant | M1 |
Obtain $A = 1, B = 2$ and $C = -4$ | A1 + A1 + A1 | [5]
**(ii)** Use correct method to obtain the first two terms of the expansion of $(1-x)^{-1}, (1+2x)^{-1}, (2+x)^{-1}$, or $(1+\frac{1}{2}x)^{-1}$ | M1 |
Obtain complete unsimplified expansions up to $x^2$ of each partial fraction | A1∇ + A1∇ + A1∇ |
Combine expansions and obtain answer $1 - 2x + \frac{17}{2}x^2$ | A1 | [5]
[Binomial coefficients such as $\begin{pmatrix}-1\\2\end{pmatrix}$ are not sufficient for the M1. The f.t. is on $A, B, C$.]
[Apply this scheme to attempts to expand $(2 - x + 8x^2)(1-x)^{-1}(1+2x)^{-1}(2+x)^{-1}$, giving M1A1A1A1 for the expansions, and A1 for the final answer.]
[Allow Maclaurin, giving M1A1∇A1∇ for $f(0) = 1$ and $f'(0) = -2$, A1∇ for $f''(0) = 17$ and A1 for the final answer (f.t. is on $A, B, C$).]9 (i) Express $\frac { 2 - x + 8 x ^ { 2 } } { ( 1 - x ) ( 1 + 2 x ) ( 2 + x ) }$ in partial fractions.\\
(ii) Hence obtain the expansion of $\frac { 2 - x + 8 x ^ { 2 } } { ( 1 - x ) ( 1 + 2 x ) ( 2 + x ) }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.
\hfill \mbox{\textit{CAIE P3 2007 Q9 [10]}}