| 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 repeated factor adds slight complexity, both techniques are routine for P3/C4 level with no novel problem-solving required. Slightly easier than average due to being a textbook-style exercise. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply the form \(\frac{A}{x+3} + \frac{B}{x-1} + \frac{C}{(x-1)^2}\) | B1 | |
| Use a correct method to determine a constant | M1 | |
| Obtain one of the values \(A = -3, B = 1, C = 2\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) Use a correct method to find the first two terms of the expansion of \((x+3)^{-1}, (1 + \frac{1}{3}x)^{-1}\), \((x-1)^{-1}, (1-x)^{-1}, (x-1)^{-2}\), or \((1-x)^{-2}\) | M1 | |
| Obtain correct unsimplified expressions up to the term in \(x^2\) of each partial fraction \(A1^{\checkmark} + A1^{\checkmark} + A1^{\checkmark}\) | ||
| Obtain final answer \(\frac{19}{3}x + \frac{44}{9}x^2\), or equivalent | A1 | [5] |
**(i)** State or imply the form $\frac{A}{x+3} + \frac{B}{x-1} + \frac{C}{(x-1)^2}$ | B1 |
Use a correct method to determine a constant | M1 |
Obtain one of the values $A = -3, B = 1, C = 2$ | A1 |
Obtain a second value | A1 |
Obtain the third value | A1 | [5]
*[Mark the form $\frac{A}{x+3} + \frac{Dx+E}{(x-1)^2}$, where $A = -3, D = 1, E = 1$, B1M1A1A1A1 as above.]*
**(ii)** Use a correct method to find the first two terms of the expansion of $(x+3)^{-1}, (1 + \frac{1}{3}x)^{-1}$, $(x-1)^{-1}, (1-x)^{-1}, (x-1)^{-2}$, or $(1-x)^{-2}$ | M1 |
Obtain correct unsimplified expressions up to the term in $x^2$ of each partial fraction $A1^{\checkmark} + A1^{\checkmark} + A1^{\checkmark}$ |
Obtain final answer $\frac{19}{3}x + \frac{44}{9}x^2$, or equivalent | A1 | [5]10 Let $\mathrm { f } ( x ) = \frac { 10 x - 2 x ^ { 2 } } { ( x + 3 ) ( x - 1 ) ^ { 2 } }$.\\
(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 Q10 [10]}}