| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Range restriction with discriminant (quadratic denominator) |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring polynomial division for the asymptote, algebraic manipulation to establish range bounds (involving rearranging to a quadratic in x and using discriminant ≥ 0), differentiation using quotient rule, and synthesis into a sketch. While each technique is standard, the range proof requires insight into discriminant methods and the question demands integration of multiple concepts, placing it moderately above average difficulty. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02z Models in context: use functions in modelling1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks |
|---|---|
| 9(i) | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| x2 +x+1 | M1 | Alt method: Finding limit |
| As x→±∞, y→5∴y=5 CAO | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks |
|---|---|
| 9(ii) | yx2 +yx+y=5x2 +5x+1 |
| Answer | Marks | Guidance |
|---|---|---|
| ⇒ y−5 x2 y−5 y−1 | B1 | Forms quadratic equation in x. |
| Answer | Marks | Guidance |
|---|---|---|
| For real x, y−5 −4 y−5 y−1 (cid:46)0 (condone >) | M1 | Uses discriminant |
| ⇒ ( y−5 )( 3y+1 ) (cid:45)0 | M1 | Factorising |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | A1 | Explaining strict upper inequality (AG) |
| Answer | Marks | Guidance |
|---|---|---|
| 9(iii) | ( ) ( ) | |
| y′=0 ⇒ x2 +x+1 ( 10x+5 ) − 5x2 +5x+1 ( 2x+1 )=0 | M1 | Differentiates and equates to 0. |
| Answer | Marks |
|---|---|
| 2 3 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Quu | estion | AAnswer |
Question 9:
--- 9(i) ---
9(i) | 4
y=5−
x2 +x+1 | M1 | Alt method: Finding limit
As x→±∞, y→5∴y=5 CAO | A1
2
Question | Answer | Marks | Guidance
--- 9(ii) ---
9(ii) | yx2 +yx+y=5x2 +5x+1
( ) +( ) x+( )=0
⇒ y−5 x2 y−5 y−1 | B1 | Forms quadratic equation in x.
( )2 ( )( )
For real x, y−5 −4 y−5 y−1 (cid:46)0 (condone >) | M1 | Uses discriminant
⇒ ( y−5 )( 3y+1 ) (cid:45)0 | M1 | Factorising
1
⇒− (cid:45)y<5, because y = 5 is an asymptote (www)
3 | A1 | Explaining strict upper inequality (AG)
4
--- 9(iii) ---
9(iii) | ( ) ( )
y′=0 ⇒ x2 +x+1 ( 10x+5 ) − 5x2 +5x+1 ( 2x+1 )=0 | M1 | Differentiates and equates to 0.
⇒4 ( 2x+1 )=0⇒x=− 1 , y=− 1
2 3 | A1
2
Quu | estion | AAnswer | Marks | Guii | danceThe curve $C$ has equation
$$y = \frac{5x^2 + 5x + 1}{x^2 + x + 1}.$$
\begin{enumerate}[label=(\roman*)]
\item Find the equation of the asymptote of $C$. [2]
\item Show that, for all real values of $x$, $-\frac{1}{5} \leqslant y < 5$. [4]
\item Find the coordinates of any stationary points of $C$. [2]
\item Sketch $C$, stating the coordinates of any intersections with the $y$-axis. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP1 2018 Q9 [10]}}