| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem and Partial Fractions |
| Type | Partial fractions with quadratic factor |
| Difficulty | Standard +0.3 This is a standard two-part question combining partial fractions with binomial expansion. Part (i) is routine A-level technique (linear and irreducible quadratic factors), and part (ii) requires expanding each fraction using the binomial theorem—straightforward application of learned methods with no novel insight required. Slightly easier than average due to clear structure and standard techniques. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<14.05c Partial fractions: extended to quadratic denominators |
| Answer | Marks |
|---|---|
| 8(i) | A Bx+C |
| Answer | Marks |
|---|---|
| 3x+2 x2 +5 | B1 |
| Use a relevant method to determine a constant | M1 |
| Obtain one of the values A = 2, B = 1, C = −3 | A1 |
| Obtain a second value | A1 |
| Obtain the third value | A1 |
| Total: | 5 |
| Answer | Marks |
|---|---|
| 8(ii) | Use correct method to find the first two terms of the expansion of (3x+2) −1 , (1+3 x) −1 , |
| Answer | Marks |
|---|---|
| 2 | M1 |
| Answer | Marks |
|---|---|
| The FT is on A, B, C. from part (i) | A1FT + |
| Answer | Marks |
|---|---|
| Multiply out up to the term inx by Bx +C, where BC ≠ 0 | M1 |
| Answer | Marks |
|---|---|
| 5 10 100 | A1 |
| Total: | 5 |
Question 8:
--- 8(i) ---
8(i) | A Bx+C
State or imply the form +
3x+2 x2 +5 | B1
Use a relevant method to determine a constant | M1
Obtain one of the values A = 2, B = 1, C = −3 | A1
Obtain a second value | A1
Obtain the third value | A1
Total: | 5
--- 8(ii) ---
8(ii) | Use correct method to find the first two terms of the expansion of (3x+2) −1 , (1+3 x) −1 ,
2
(5+x 2 ) −1 or (1+1 x 2 ) −1
5
−1
[Symbolic coefficients, e.g. are not sufficient]
2 | M1
x2
Obtain correct unsimplified expansions up to the term in of each partial fraction.
The FT is on A, B, C. from part (i) | A1FT +
A1FT
2
Multiply out up to the term inx by Bx +C, where BC ≠ 0 | M1
Obtain final answer 2 − 13 x+ 237 x 2 , or equivalent
5 10 100 | A1
Total: | 5Let $\mathrm{f}(x) = \frac{5x^2 - 7x + 4}{(3x + 2)(x^2 + 5)}$.
\begin{enumerate}[label=(\roman*)]
\item Express $\mathrm{f}(x)$ in partial fractions. [5]
\item Hence obtain the expansion of $\mathrm{f}(x)$ in ascending powers of $x$, up to and including the term in $x^2$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2017 Q8 [10]}}