| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| 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 with binomial expansion. The partial fractions decomposition is straightforward (linear and quadratic factors), and the binomial expansions required are routine applications of (1+x)^{-1} and (1+2x²)^{-1}. While it requires multiple steps and careful algebra, both techniques are core A-level skills with no novel insight needed. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks |
|---|---|
| (i) State or imply the form \(\frac{A}{1+x} + \frac{Bx + C}{1 + 2x^2}\) | B1 |
| Use any relevant method to evaluate a constant | M1 |
| Obtain one of \(A = -1\), \(B = 2\), \(C = 1\) | A1 |
| Obtain a second value | A1 |
| Obtain the third value | A1 |
| Answer | Marks |
|---|---|
| (ii) Use correct method to obtain the first two terms of the expansion of \((1 + x)^{-1}\) or \((1 + 2x^2)^{-1}\) | M1 |
| Obtain correct expansion of each partial fraction as far as necessary | A1√ + A1√ |
| Multiply out fully by \(Bx + C\), where \(BC \neq 0\) | M1 |
| Obtain answer \(3x - 3x^2 - 3x^3\) | A1 |
**(i)** State or imply the form $\frac{A}{1+x} + \frac{Bx + C}{1 + 2x^2}$ | B1 |
Use any relevant method to evaluate a constant | M1 |
Obtain one of $A = -1$, $B = 2$, $C = 1$ | A1 |
Obtain a second value | A1 |
Obtain the third value | A1 |
[5 marks total]
**(ii)** Use correct method to obtain the first two terms of the expansion of $(1 + x)^{-1}$ or $(1 + 2x^2)^{-1}$ | M1 |
Obtain correct expansion of each partial fraction as far as necessary | A1√ + A1√ |
Multiply out fully by $Bx + C$, where $BC \neq 0$ | M1 |
Obtain answer $3x - 3x^2 - 3x^3$ | A1 |
[5 marks total]
[Symbolic binomial coefficients, e.g., $\binom{-1}{1}$ are not sufficient for the first M1. The f.t. is on $A, B, C$.]
[If $B$ or $C$ omitted from the form of fractions, give B0M1A0A0 in (i); M1A1A1√ in (ii), max 4/10.]
[If a constant $D$ is added to the correct form, give M1A1A1A1 and B1 if and only if $D = 0$ is stated.]
[If an extra term $D/(1 + 2x^2)$ is added, give B1M1A1A1, and A1 if $C + D = 1$ is resolved to 1/(1 + 2x^2).]
[In the case of an attempt to expand $3x(1 + x)^{-1}(1 + 2x^2)^{-1}$, give M1A1A1 for the expansions up to the term in $x^3$, M1 for multiplying out fully, and A1 for the final answer.]
[For the identity $3x = (1 + x + 2x^2 + 2x^3)(a + bx + cx^2 + dx^3)$ give M1A1; then M1A1 for using a relevant method to find two of $a = 0, b = 3, c = -3$ and $d = -3$; and then A1 for the final answer in series form.]8 Let $\mathrm { f } ( x ) = \frac { 3 x } { ( 1 + x ) \left( 1 + 2 x ^ { 2 } \right) }$.\\
(i) Express $\mathrm { f } ( x )$ in partial fractions.\\
(ii) Hence obtain the expansion of $\mathrm { f } ( x )$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.
\hfill \mbox{\textit{CAIE P3 2010 Q8 [10]}}