CAIE FP1 2016 June — Question 7 10 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2016
SessionJune
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCurve Sketching
TypeRange restriction with excluded interval (linear/mixed denominator)
DifficultyStandard +0.8 This Further Maths question requires finding asymptotes (standard), proving a range restriction (requires algebraic manipulation and discriminant analysis), and sketching with turning points (calculus with quotient rule). The range restriction proof is non-routine and requires insight into when the rearranged equation has real solutions, elevating this above typical curve sketching.
Spec1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials1.07n Stationary points: find maxima, minima using derivatives
7 A curve \(C\) has equation \(y = \frac { x ^ { 2 } } { x - 2 }\). Find the equations of the asymptotes of \(C\). Show that there are no points on \(C\) for which \(0 < y < 8\). Sketch \(C\), giving the coordinates of the turning points.
Question 7:
AnswerMarks Guidance
Working/AnswerMarks Guidance
Vertical asymptote is \(x = 2\)B1
\(y = x + 2 + \frac{4}{x-2} \Rightarrow\) Oblique asymptote is \(y = x+2\)M1A1 [3]
\(y = \frac{x^2}{x-2} \Rightarrow x^2 - yx + 2y = 0\)B1
Quadratic has no real roots (i.e. no points on \(C\)) if \(\Delta < 0 \Rightarrow y^2 - 8y < 0\)M1
\(\Rightarrow y(y-8) < 0 \Rightarrow 0 < y < 8\) (AG) Correct inequalityM1A1 [4]
Axes and asymptotes; each branch showing \((0,0)\) and \((4,8)\); (deduct at most 1 mark for poor forms at infinity and/or missing coordinates)B1√, B1B1 [3] 
## Question 7:

| Working/Answer | Marks | Guidance |
|---|---|---|
| Vertical asymptote is $x = 2$ | B1 | |
| $y = x + 2 + \frac{4}{x-2} \Rightarrow$ Oblique asymptote is $y = x+2$ | M1A1 [3] | |
| $y = \frac{x^2}{x-2} \Rightarrow x^2 - yx + 2y = 0$ | B1 | |
| Quadratic has no real roots (i.e. no points on $C$) if $\Delta < 0 \Rightarrow y^2 - 8y < 0$ | M1 | |
| $\Rightarrow y(y-8) < 0 \Rightarrow 0 < y < 8$ **(AG)** Correct inequality | M1A1 [4] | |
| Axes and asymptotes; each branch showing $(0,0)$ and $(4,8)$; (deduct at most 1 mark for poor forms at infinity and/or missing coordinates) | B1√, B1B1 [3] | |
7 A curve $C$ has equation $y = \frac { x ^ { 2 } } { x - 2 }$. Find the equations of the asymptotes of $C$.

Show that there are no points on $C$ for which $0 < y < 8$.

Sketch $C$, giving the coordinates of the turning points.

\hfill \mbox{\textit{CAIE FP1 2016 Q7 [10]}}
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Q1 4 Q2 6 Q3 6 Q4 8 Q5 9 Q6 9 Q7 10 Q8 11 Q9 11 Q10 12 Q11 EITHER Q11 OR Q15