Question 8 - A-Level Maths

CAIE FP1 2015 November — Question 8

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2015
Session	November
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Polynomial Division & Manipulation
Type	Stationary Points of Rational Functions
Difficulty	Standard +0.8 This question requires finding stationary points by differentiating a rational function (using quotient rule), solving a quadratic inequality to find the range of k values, then analyzing asymptotes and sketching. It combines calculus, algebraic manipulation, and curve sketching—more demanding than standard A-level but typical for Further Maths FP1.
Spec	1.02n Sketch curves: simple equations including polynomials 1.07n Stationary points: find maxima, minima using derivatives 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)

8 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } + k x } { x + 1 }\), where \(k\) is a constant. Find the set of values of \(k\) for which \(C\) has no stationary points. For the case \(k = 4\), find the equations of the asymptotes of \(C\) and sketch \(C\), indicating the coordinates of the points where \(C\) intersects the coordinate axes.

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Question 8:

Answer	Marks	Guidance
Working/Answer	Marks	Guidance
\(y' = 0 \Rightarrow (x+1)(4x+k) - (2x^2+kx)\times 1 = 0\)	M1
\(\Rightarrow 4x^2 + (4+k)x + k - 2x^2 - kx = 0 \Rightarrow 2x^2 + 4x + k = 0\)	A1
\(B^2 - 4AC < 0 \Rightarrow\) no stationary points \(\Rightarrow 16 - 8k < 0\)	M1A1
\(\Rightarrow k > 2\) for no stationary points	A1
When \(k = 4\): Vertical asymptote: \(x = -1\)	B1
Oblique asymptote: \(y = 2x + 2 - \frac{2}{x+1} \Rightarrow y = 2x + 2\)	M1A1
Axes and asymptotes drawn	B1
Each branch drawn correctly	B1B1
Total: 11 marks	[11]

## Question 8:

| Working/Answer | Marks | Guidance |
|---|---|---|
| $y' = 0 \Rightarrow (x+1)(4x+k) - (2x^2+kx)\times 1 = 0$ | M1 | |
| $\Rightarrow 4x^2 + (4+k)x + k - 2x^2 - kx = 0 \Rightarrow 2x^2 + 4x + k = 0$ | A1 | |
| $B^2 - 4AC < 0 \Rightarrow$ no stationary points $\Rightarrow 16 - 8k < 0$ | M1A1 | |
| $\Rightarrow k > 2$ for no stationary points | A1 | |
| When $k = 4$: Vertical asymptote: $x = -1$ | B1 | |
| Oblique asymptote: $y = 2x + 2 - \frac{2}{x+1} \Rightarrow y = 2x + 2$ | M1A1 | |
| Axes and asymptotes drawn | B1 | |
| Each branch drawn correctly | B1B1 | |
| **Total: 11 marks** | [11] | |

Show LaTeX source

8 The curve $C$ has equation $y = \frac { 2 x ^ { 2 } + k x } { x + 1 }$, where $k$ is a constant. Find the set of values of $k$ for which $C$ has no stationary points.

For the case $k = 4$, find the equations of the asymptotes of $C$ and sketch $C$, indicating the coordinates of the points where $C$ intersects the coordinate axes.

\hfill \mbox{\textit{CAIE FP1 2015 Q8}}

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