| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | June |
| 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 repeated linear factors and binomial expansion. While it requires careful algebraic manipulation and the binomial theorem for negative/fractional powers, both techniques are routine for P3/C4 level with no novel problem-solving required. The repeated factor adds slight complexity but follows textbook methods. |
| 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 \(\frac{A}{x+1} + \frac{B}{x-3} + \frac{C}{(x-3)^2}\) | B1 | |
| Use correct method to determine a constant | M1 | |
| Obtain one of \(A = 1\), \(B = 3\), \(C = 12\) | A1 | |
| Obtain a second value | A1 | |
| Obtain a third value | A1 | |
| Total | [5] | [Mark the form \(\frac{A}{x+1} + \frac{Dx+E}{(x-3)^2}\) where \(A=1, D=3, E=3\), B1M1A1A1A1 as above.] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use correct method to find first two terms of expansion of \((x+1)^{-1}\), \((x-3)^{-1}\), \((1-\frac{1}{3}x)^{-1}\); \((x-3)^{-2}\) or \((1-\frac{1}{3}x)^{-2}\) | M1 | |
| Obtain correct unsimplified expansions up to \(x^2\) term of each partial fraction | \(A1^\checkmark + A1^\checkmark + A1^\checkmark\) | Follow-through marks |
| Obtain final answer \(\frac{4}{3} - \frac{4}{9}x + \frac{4}{3}x^2\), or equivalent | A1 | |
| Total | [5] |
## Question 8:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply the form $\frac{A}{x+1} + \frac{B}{x-3} + \frac{C}{(x-3)^2}$ | B1 | |
| Use correct method to determine a constant | M1 | |
| Obtain one of $A = 1$, $B = 3$, $C = 12$ | A1 | |
| Obtain a second value | A1 | |
| Obtain a third value | A1 | |
| **Total** | **[5]** | [Mark the form $\frac{A}{x+1} + \frac{Dx+E}{(x-3)^2}$ where $A=1, D=3, E=3$, B1M1A1A1A1 as above.] |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use correct method to find first two terms of expansion of $(x+1)^{-1}$, $(x-3)^{-1}$, $(1-\frac{1}{3}x)^{-1}$; $(x-3)^{-2}$ or $(1-\frac{1}{3}x)^{-2}$ | M1 | |
| Obtain correct unsimplified expansions up to $x^2$ term of each partial fraction | $A1^\checkmark + A1^\checkmark + A1^\checkmark$ | Follow-through marks |
| Obtain final answer $\frac{4}{3} - \frac{4}{9}x + \frac{4}{3}x^2$, or equivalent | A1 | |
| **Total** | **[5]** | |8 Let $\mathrm { f } ( x ) = \frac { 4 x ^ { 2 } + 12 } { ( x + 1 ) ( x - 3 ) ^ { 2 } }$.\\
(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]}}