| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2004 |
| 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 routine partial fractions decomposition with binomial expansion. Part (i) is standard A-level technique, and part (ii) requires expanding three simple binomial terms and collecting coefficients—mechanical work with no novel insight required. Slightly easier than average due to the verification nature of part (ii) rather than requiring independent derivation. |
| Spec | 1.02y Partial fractions: decompose rational functions4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (i) State or imply \(f(x) \equiv \frac{A}{x-1} + \frac{B}{x-2} + \frac{C}{x+1}\) | B1 | |
| EITHER: Use any relevant method to obtain a constant | M1 | |
| Obtain one of the values: \(A = -1\), \(B = 4\) and \(C = -2\) | A1 | |
| Obtain the remaining two values | A1 | |
| OR: Obtain one value by inspection | B1 | |
| State a second value | B1 | |
| State the third value | B1 | Total: 4 |
| [Apply same scheme to alternative form \(\frac{A}{x-2} + \frac{Bx+C}{x^2-1}\) which has \(A = 4\), \(B = -3\), \(C = 1\)] | ||
| (ii) Use correct method to obtain first two terms of expansion of \((x-1)^{-1}\) or \((x-2)^{-1}\) or \((x+1)^{-1}\) | M1 | Binomial coefficients involving \(-1\) e.g. \(\binom{-1}{1}\) not sufficient for M1. f.t. on \(A, B, C\). |
| Obtain any correct unsimplified expansion of the partial fractions up to terms in \(x^3\) (deduct A1 for each incorrect expansion) | A1\(\sqrt{}\) + A1\(\sqrt{}\) + A1\(\sqrt{}\) | |
| Obtain the given answer correctly | A1 | Total: 5 |
## Question 9:
| Answer/Working | Mark | Guidance |
|---|---|---|
| **(i)** State or imply $f(x) \equiv \frac{A}{x-1} + \frac{B}{x-2} + \frac{C}{x+1}$ | B1 | |
| **EITHER:** Use any relevant method to obtain a constant | M1 | |
| Obtain one of the values: $A = -1$, $B = 4$ and $C = -2$ | A1 | |
| Obtain the remaining two values | A1 | |
| **OR:** Obtain one value by inspection | B1 | |
| State a second value | B1 | |
| State the third value | B1 | **Total: 4** |
| [Apply same scheme to alternative form $\frac{A}{x-2} + \frac{Bx+C}{x^2-1}$ which has $A = 4$, $B = -3$, $C = 1$] | | |
| **(ii)** Use correct method to obtain first two terms of expansion of $(x-1)^{-1}$ or $(x-2)^{-1}$ or $(x+1)^{-1}$ | M1 | Binomial coefficients involving $-1$ e.g. $\binom{-1}{1}$ not sufficient for M1. f.t. on $A, B, C$. |
| Obtain any correct unsimplified expansion of the partial fractions up to terms in $x^3$ (deduct A1 for each incorrect expansion) | A1$\sqrt{}$ + A1$\sqrt{}$ + A1$\sqrt{}$ | |
| Obtain the given answer correctly | A1 | **Total: 5** |
---9 Let $\mathrm { f } ( x ) = \frac { x ^ { 2 } + 7 x - 6 } { ( x - 1 ) ( x - 2 ) ( x + 1 ) }$.\\
(i) Express $\mathrm { f } ( x )$ in partial fractions.\\
(ii) Show that, when $x$ is sufficiently small for $x ^ { 4 }$ and higher powers to be neglected,
$$f ( x ) = - 3 + 2 x - \frac { 3 } { 2 } x ^ { 2 } + \frac { 11 } { 4 } x ^ { 3 } .$$
\hfill \mbox{\textit{CAIE P3 2004 Q9 [9]}}