| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2002 |
| Session | November |
| 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 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 functions1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Content | Mark | Guidance |
| (i) State or imply \(f(x) = \frac{A}{(2-x)} + \frac{Bx+C}{(x^2+1)}\) | B1* | |
| State or obtain \(A = 4\) | B1(dep*) | |
| Use any relevant method to find \(B\) or \(C\) | M1 | |
| Obtain both \(B = 4\) and \(C = 1\) | A1 | Max 4 marks |
| (ii) EITHER: Use correct method to obtain the first two terms of the expansion of \((1-\frac{1}{2}x)^{-1}\), or \((1+x^2)^{-1}\), or \((2-x)^{-1}\) | M1* | |
| Obtain unsimplified expansions of the fractions e.g. \(\frac{4}{2}(1 + \frac{1}{2}x + \frac{1}{4}x^2 + \frac{1}{8}x^3)\); | A1✓/A1✓ | |
| Carry out multiplication of expansion of \((1+x^2)^{-1}\) by \((4x+1)\) | M1(dep*) | |
| Obtain given answer correctly | A1 | [Binomial coefficients involving \(-1\), such as \(\begin{pmatrix}-1\\1\end{pmatrix}\), are not sufficient for the first M1.] |
| OR: Differentiate and evaluate \(f(0)\) and \(f'(0)\) | M1 | |
| Obtain \(f(0) = 3\) and \(f'(0) = 5\) | A1✓ | |
| Differentiate and obtain \(f''(0)\) and form the Maclaurin expansion up to the term in \(x^3\) | A1✓ | |
| Simplify coefficients and obtain given answer correctly | M1 | |
| A1 | [f.t. is on A, B, C.] | |
| [SR: B or C omitted from the form of partial fractions. In part (i) give the first B1, and M1 for the use of a relevant method to obtain A, B, or C, but no further marks. In part (ii) only the first M1 and A1✓/A1✓ are available if an attempt is based on this form of partial fractions.] |
| Content | Mark | Guidance |
|---------|------|----------|
| **(i)** State or imply $f(x) = \frac{A}{(2-x)} + \frac{Bx+C}{(x^2+1)}$ | B1* | |
| State or obtain $A = 4$ | B1(dep*) | |
| Use any relevant method to find $B$ or $C$ | M1 | |
| Obtain both $B = 4$ and $C = 1$ | A1 | Max 4 marks |
| **(ii)** **EITHER:** Use correct method to obtain the first two terms of the expansion of $(1-\frac{1}{2}x)^{-1}$, or $(1+x^2)^{-1}$, or $(2-x)^{-1}$ | M1* | |
| Obtain unsimplified expansions of the fractions e.g. $\frac{4}{2}(1 + \frac{1}{2}x + \frac{1}{4}x^2 + \frac{1}{8}x^3)$; | A1✓/A1✓ | |
| Carry out multiplication of expansion of $(1+x^2)^{-1}$ by $(4x+1)$ | M1(dep*) | |
| Obtain given answer correctly | A1 | [Binomial coefficients involving $-1$, such as $\begin{pmatrix}-1\\1\end{pmatrix}$, are not sufficient for the first M1.] | Max 5 marks |
| **OR:** Differentiate and evaluate $f(0)$ and $f'(0)$ | M1 | |
| Obtain $f(0) = 3$ and $f'(0) = 5$ | A1✓ | |
| Differentiate and obtain $f''(0)$ and form the Maclaurin expansion up to the term in $x^3$ | A1✓ | |
| Simplify coefficients and obtain given answer correctly | M1 | |
| | A1 | [f.t. is on A, B, C.] |
| | | [SR: B or C omitted from the form of partial fractions. In part (i) give the first B1, and M1 for the use of a relevant method to obtain A, B, or C, but no further marks. In part (ii) only the first M1 and A1✓/A1✓ are available if an attempt is based on this form of partial fractions.] |
---6 Let $f ( x ) = \frac { 6 + 7 x } { ( 2 - x ) \left( 1 + x ^ { 2 } \right) }$.\\
(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,
$$f ( x ) = 3 + 5 x - \frac { 1 } { 2 } x ^ { 2 } - \frac { 15 } { 4 } x ^ { 3 }$$
\hfill \mbox{\textit{CAIE P3 2002 Q6 [9]}}