| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| 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 with one linear and one irreducible quadratic factor is routine, and the binomial expansions of (1+x/2)^(-1) and (1+x²/4)^(-1) are straightforward applications of the formula. Slightly above average difficulty due to the quadratic denominator requiring careful coefficient matching, but still a textbook exercise with no novel insight required. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply the form \(\frac{A}{x+2} + \frac{Bx+C}{x^2+4}\) | B1 | |
| Use correct method to determine a constant | M1 | |
| Obtain one of \(A=2\), \(B=1\), \(C=-1\) | A1 | |
| Obtain a second value | A1 | |
| Obtain a third value | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use correct method to find first two terms of expansion of \((x+2)^{-1}\), i.e. \((1+\frac{1}{2}x)^{-1}\), \((4+x^2)^{-1}\) or \((1+\frac{1}{4}x^2)^{-1}\) | M1 | |
| Obtain correct unsimplified expansions up to term in \(x^2\) of each partial fraction | A1\(\checkmark\) + A1\(\checkmark\) | |
| Multiply out fully by \(Bx+C\), where \(BC \neq 0\) | M1 | |
| Obtain final answer \(\frac{3}{4} - \frac{1}{4}x + \frac{5}{16}x^2\), or equivalent | A1 | [5] |
| Symbolic binomial coefficients, e.g. \(\binom{-1}{1}\) are not sufficient for the M1. The f.t. is on \(A\), \(B\), \(C\). | ||
| In the case of an attempt to expand \((3x^2+x+6)(x+2)^{-1}(x^2+4)^{-1}\), give M1A1A1 for expansions, M1 for multiplying out fully, A1 for final answer. |
## Question 8:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply the form $\frac{A}{x+2} + \frac{Bx+C}{x^2+4}$ | B1 | |
| Use correct method to determine a constant | M1 | |
| Obtain one of $A=2$, $B=1$, $C=-1$ | A1 | |
| Obtain a second value | A1 | |
| Obtain a third value | A1 | [5] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use correct method to find first two terms of expansion of $(x+2)^{-1}$, i.e. $(1+\frac{1}{2}x)^{-1}$, $(4+x^2)^{-1}$ or $(1+\frac{1}{4}x^2)^{-1}$ | M1 | |
| Obtain correct unsimplified expansions up to term in $x^2$ of each partial fraction | A1$\checkmark$ + A1$\checkmark$ | |
| Multiply out fully by $Bx+C$, where $BC \neq 0$ | M1 | |
| Obtain final answer $\frac{3}{4} - \frac{1}{4}x + \frac{5}{16}x^2$, or equivalent | A1 | [5] |
| Symbolic binomial coefficients, e.g. $\binom{-1}{1}$ are not sufficient for the M1. The f.t. is on $A$, $B$, $C$. | | |
| In the case of an attempt to expand $(3x^2+x+6)(x+2)^{-1}(x^2+4)^{-1}$, give M1A1A1 for expansions, M1 for multiplying out fully, A1 for final answer. | | |
---8 Let $\mathrm { f } ( x ) = \frac { 3 x ^ { 2 } + x + 6 } { ( x + 2 ) \left( x ^ { 2 } + 4 \right) }$.\\
(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 $x ^ { 2 }$.
\hfill \mbox{\textit{CAIE P3 2016 Q8 [10]}}