| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2003 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem and Partial Fractions |
| Type | Partial fractions with verification |
| Difficulty | Standard +0.3 This is a straightforward two-part question combining standard partial fractions decomposition with binomial expansion. Part (i) is routine A-level technique, and part (ii) requires expanding each fraction to order x² then collecting terms—methodical but not conceptually demanding. Slightly easier than average due to the verification nature of part (ii) rather than requiring independent derivation. |
| Spec | 1.02y Partial fractions: decompose rational functions4.08f Integrate using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) EITHER State or imply \(f(x) = \frac{A}{2x+1} + \frac{B}{x-2} + \frac{C}{(x-2)^2}\) | B1 | State or obtain \(A = 1\) |
| OR State or imply \(f(x) = \frac{A}{2x+1} + \frac{Dx+E}{(x-2)^2}\) | B1 | State or obtain \(A = 1\) |
| (ii) EITHER Use correct method to obtain the first two terms of the expansion of \((1 + 2x)^{-1}\) or \((x-2)^{-1}\) or \((x-2)^{-2}\) or \((1-\frac{1}{2}x)^{-1}\) or \((1-\frac{1}{2}x)^2\) | M1 | Obtain any correct sum of unsimplified expansions up to the terms in \(x^2\) (deduct A1 for each incorrect expansion) |
| Answer | Marks | Guidance |
|---|---|---|
| OR Differentiate and evaluate \(f(0)\) and \(f'(0)\) | M1 | Obtain \(f(0) = 1\) and \(f'(0) = -1\) |
**(i)** **EITHER** State or imply $f(x) = \frac{A}{2x+1} + \frac{B}{x-2} + \frac{C}{(x-2)^2}$ | B1 | State or obtain $A = 1$ | B1 | State or obtain $C = 8$ | B1 | Use any relevant method to find B | M1 | Obtain value $B = 4$ | A1 |
**OR** State or imply $f(x) = \frac{A}{2x+1} + \frac{Dx+E}{(x-2)^2}$ | B1 | State or obtain $A = 1$ | B1 | Use any relevant method to find D or E | M1 | Obtain value $D = 4$ | A1 | Obtain value $E = 0$ | A1 | **[5]**
**(ii)** **EITHER** Use correct method to obtain the first two terms of the expansion of $(1 + 2x)^{-1}$ or $(x-2)^{-1}$ or $(x-2)^{-2}$ or $(1-\frac{1}{2}x)^{-1}$ or $(1-\frac{1}{2}x)^2$ | M1 | Obtain any correct sum of unsimplified expansions up to the terms in $x^2$ (deduct A1 for each incorrect expansion) | A2√ | Obtain the given answer correctly | A1 |
[Unexpanded binomial coefficients involving -1 or -2, e.g. $\binom{-2}{1}$ are not sufficient for the M1.]
[f.t. is on A, B, C, D, E.]
[Apply this scheme to attempts to expand $(9x^2 + 4)(1+2x)^{-1}(x-2)^{-2}$, giving M1A2 for a correct product of expansions and A1 for multiplying out and reaching the given answer correctly.]
[Allow attempts to multiply out $(1 + 2x)(x-2)^2(1 - x + 5x^2)$, giving B1 for reduction to a product of two expressions correct up to their terms in $x^2$, M1 for attempting to multiply out as far as terms in $x^2$, A1 for a correct expansion, and A1 for obtaining $9x^2 + 4$ correctly.]
**OR** Differentiate and evaluate $f(0)$ and $f'(0)$ | M1 | Obtain $f(0) = 1$ and $f'(0) = -1$ | A1 | Differentiate and obtain $f''(0) = 10$ | A1 | Form the Maclaurin expansion and obtain the given answer correctly | A1 | **[4]**6 Let $\mathrm { f } ( x ) = \frac { 9 x ^ { 2 } + 4 } { ( 2 x + 1 ) ( x - 2 ) ^ { 2 } }$.\\
(i) Express $\mathrm { f } ( x )$ in partial fractions.\\
(ii) Show that, when $x$ is sufficiently small for $x ^ { 3 }$ and higher powers to be neglected,
$$f ( x ) = 1 - x + 5 x ^ { 2 }$$
\hfill \mbox{\textit{CAIE P3 2003 Q6 [9]}}