| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2019 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Partial fractions then method of differences |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question combining routine asymptote finding (polynomial division) with standard method of differences. The asymptote part requires basic algebraic manipulation, and the series summation uses textbook partial fractions technique. While it's Further Maths content, both parts are mechanical applications of standard methods with no novel insight required, making it slightly easier than average overall. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x^2 + 1 = (ax+b)(2x+1) + c\) | M1 | Uses that \(2x+1\) is the quotient |
| \(\Rightarrow a = \frac{1}{2},\; b = -\frac{1}{4}\) | A1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x = \frac{1}{2}\) | B1 FT |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Graph with intersection \((0,-4)\) given and asymptotes drawn | B1 | Intersection \((0,-4)\) given and asymptotes drawn |
| Left branch correct | B1 | Left branch correct |
| Right branch correct | B1 FT | Right branch correct. Deduct at most one mark for poor forms at infinity |
## Question 4:
### Part 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2 + 1 = (ax+b)(2x+1) + c$ | M1 | Uses that $2x+1$ is the quotient |
| $\Rightarrow a = \frac{1}{2},\; b = -\frac{1}{4}$ | A1 A1 | |
**Total: 3 marks**
### Part 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x = \frac{1}{2}$ | B1 FT | |
**Total: 1 mark**
### Part 4(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Graph with intersection $(0,-4)$ given and asymptotes drawn | B1 | Intersection $(0,-4)$ given and asymptotes drawn |
| Left branch correct | B1 | Left branch correct |
| Right branch correct | B1 FT | Right branch correct. Deduct at most one mark for poor forms at infinity |
**Total: 3 marks**
---4 The line $y = 2 x + 1$ is an asymptote of the curve $C$ with equation
$$y = \frac { x ^ { 2 } + 1 } { a x + b }$$
(i) Find the values of the constants $a$ and $b$.\\
(ii) State the equation of the other asymptote of $C$.\\
(iii) Sketch C. [Your sketch should indicate the coordinates of any points of intersection with the $y$-axis. You do not need to find the coordinates of any stationary points.]\\
$5 \quad$ Let $S _ { N } = \sum _ { r = 1 } ^ { N } ( 5 r + 1 ) ( 5 r + 6 )$ and $T _ { N } = \sum _ { r = 1 } ^ { N } \frac { 1 } { ( 5 r + 1 ) ( 5 r + 6 ) }$.\\
\hfill \mbox{\textit{CAIE FP1 2019 Q4 [7]}}