CAIE FP1 2012 June — Question 6 9 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2012
SessionJune
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCurve Sketching
TypeSketch rational with quadratic numerator
DifficultyStandard +0.3 This is a standard Further Maths curve sketching question requiring asymptote identification (vertical at x=2, oblique via polynomial division), stationary points via quotient rule differentiation, and a sketch. While it involves multiple techniques, each step follows routine procedures with no novel insight required, making it slightly easier than average.
Spec1.02n Sketch curves: simple equations including polynomials1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.02y Partial fractions: decompose rational functions1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives
6 The curve \(C\) has equation \(y = \frac { x ^ { 2 } } { x - 2 }\). Find the equations of the asymptotes of \(C\). Find the coordinates of the turning points on \(C\). Draw a sketch of \(C\).
Question 6:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Vertical asymptote is \(x=2\)B1
\(y = x+2+\frac{4}{x-2}\); oblique asymptote is \(y=x+2\)M1, A1 Part Mark: 3
\(y' = 1 - \frac{4}{(x-2)^2} = 0 \Rightarrow (x-2)^2=4\)M1 Differentiates and equates to zero
\(x = 0, 4\)A1
Turning points are \((0,0)\) and \((4,8)\)A1 Part Mark: 3
Axes and both asymptotes correctB1
Upper branch correctB1
Lower branch correctB1 Part Mark: 3; Total: [9]; *Deduct at most 1 mark for poor forms at infinity* 
## Question 6:

| Answer/Working | Marks | Guidance |
|---|---|---|
| Vertical asymptote is $x=2$ | B1 | |
| $y = x+2+\frac{4}{x-2}$; oblique asymptote is $y=x+2$ | M1, A1 | Part Mark: 3 |
| $y' = 1 - \frac{4}{(x-2)^2} = 0 \Rightarrow (x-2)^2=4$ | M1 | Differentiates and equates to zero |
| $x = 0, 4$ | A1 | |
| Turning points are $(0,0)$ and $(4,8)$ | A1 | Part Mark: 3 |
| Axes and both asymptotes correct | B1 | |
| Upper branch correct | B1 | |
| Lower branch correct | B1 | Part Mark: 3; Total: **[9]**; *Deduct at most 1 mark for poor forms at infinity* |

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6 The curve $C$ has equation $y = \frac { x ^ { 2 } } { x - 2 }$. Find the equations of the asymptotes of $C$.

Find the coordinates of the turning points on $C$.

Draw a sketch of $C$.

\hfill \mbox{\textit{CAIE FP1 2012 Q6 [9]}}
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Q1 5 Q2 5 Q3 8 Q4 8 Q5 9 Q6 9 Q7 10 Q8 10 Q9 11 Q10 11 Q11 EITHER Q11 OR