| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem and Partial Fractions |
| Type | Improper fraction partial fractions |
| Difficulty | Standard +0.3 This is a straightforward two-part question combining partial fractions and integration. Part (i) is routine algebraic manipulation with a polynomial division followed by standard partial fraction decomposition. Part (ii) requires integrating the resulting form and evaluating definite integrals of logarithmic functions—all standard techniques for P3/C4 level with no novel problem-solving required. The 'show that' format provides a target to verify, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.06f Laws of logarithms: addition, subtraction, power rules1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| 8(i) | Use a relevant method to determine a constant | M1 |
| Obtain one of the values A = 2, B = 2, C = –1 | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 |
| Answer | Marks |
|---|---|
| 8(ii) | 1 |
| Answer | Marks |
|---|---|
| omission) [The FT is on A, B and C] | B2 FT |
| Answer | Marks |
|---|---|
| where ab ≠ 0 | *M1 |
| Use at least one law of logarithms correctly | DM1 |
| Obtain the given answer after full and correct working | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 8:
--- 8(i) ---
8(i) | Use a relevant method to determine a constant | M1
Obtain one of the values A = 2, B = 2, C = –1 | A1
Obtain a second value | A1
Obtain the third value | A1
4
--- 8(ii) ---
8(ii) | 1
Integrate and obtain terms 2x+2ln(x+2)− ln(2x−1) (deduct B1 for each error or
2
omission) [The FT is on A, B and C] | B2 FT
Substitute limits correctly in an integral containing termsaln(x+2)and bln(2x−1),
where ab ≠ 0 | *M1
Use at least one law of logarithms correctly | DM1
Obtain the given answer after full and correct working | A1
5
Question | Answer | MarksLet $\text{f}(x) = \frac{4x^2 + 9x - 8}{(x + 2)(2x - 1)}$.
\begin{enumerate}[label=(\roman*)]
\item Express $\text{f}(x)$ in the form $A + \frac{B}{x + 2} + \frac{C}{2x - 1}$. [4]
\item Hence show that $\int_1^4 \text{f}(x) \, dx = 6 + \frac{1}{2} \ln\left(\frac{16}{7}\right)$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2017 Q8 [9]}}