Question 4 - A-Level Maths

CAIE FP1 2014 November — Question 4 7 marks

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2014
Session	November
Marks	7
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Polynomial Division & Manipulation
Type	Proving Excluded Range of Rational Function
Difficulty	Standard +0.8 This question requires polynomial division to find asymptotes, then solving a quadratic inequality to determine the range restriction. While the techniques are standard Further Maths content, the second part requires algebraic manipulation and careful analysis of when the resulting quadratic has no real solutions, which goes beyond routine application and requires some problem-solving insight.
Spec	1.02n Sketch curves: simple equations including polynomials 1.02y Partial fractions: decompose rational functions

4 A curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } + x - 1 } { x - 1 }\). Find the equations of the asymptotes of \(C\). Show that there is no point on \(C\) for which \(1 < y < 9\).

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Question 4:

Answer	Marks	Guidance
Working/Answer	Marks	Guidance
Vertical asymptote is \(x=1\)	B1
\(y=2x+3+2(x-1)^{-1}\Rightarrow y=2x+3\) is the oblique asymptote	M1A1 (3)
\(2x^2+(1-y)x+(y-1)=0\) has real roots
\(\Leftrightarrow (1-y)^2-8(y-1)\geq 0\)
\(\Leftrightarrow y^2-10y+9\geq 0\)
\(\Leftrightarrow (y-1)(y-9)\geq 0\)	M1A1
Hence \((y-1)(y-9)<0\Rightarrow\) no real roots	M1
i.e. \(1	A1 (4)	AG — thumbnail sketch or similar required

## Question 4:

| Working/Answer | Marks | Guidance |
|---|---|---|
| Vertical asymptote is $x=1$ | B1 | |
| $y=2x+3+2(x-1)^{-1}\Rightarrow y=2x+3$ is the oblique asymptote | M1A1 (3) | |
| $2x^2+(1-y)x+(y-1)=0$ has real roots | | |
| $\Leftrightarrow (1-y)^2-8(y-1)\geq 0$ | | |
| $\Leftrightarrow y^2-10y+9\geq 0$ | | |
| $\Leftrightarrow (y-1)(y-9)\geq 0$ | M1A1 | |
| Hence $(y-1)(y-9)<0\Rightarrow$ no real roots | M1 | |
| i.e. $1<y<9\Rightarrow$ no points on $C$ | A1 (4) | AG — thumbnail sketch or similar required |

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4 A curve $C$ has equation $y = \frac { 2 x ^ { 2 } + x - 1 } { x - 1 }$. Find the equations of the asymptotes of $C$.

Show that there is no point on $C$ for which $1 < y < 9$.

\hfill \mbox{\textit{CAIE FP1 2014 Q4 [7]}}

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Q1 5 Q2 6 Q3 7 Q4 7 Q5 8 Q6 9 Q7 10 Q8 11 Q9 11 Q10 12 Q11 EITHER Q11 OR