CAIE A-Level Further Maths: Rational Functions & Graphs

[Maximum number: 15]

The curve C has equation y=f(x), where $\mathrm{f}(x)=\frac{x^{2}+2}{x^{2}-x-2}$ .

(a)

Find the equations of the asymptotes of C.

[ 2 ]

$x=-1, \quad x=2$

B1

Vertical asymptotes.

y=1

B1

Horizontal asymptote.

2

(b)

Find the coordinates of any stationary points on C, giving your answers correct to 1 decimal place.

[ 4 ]

$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\left(x^{2}-x-2\right)(2 x)-\left(x^{2}+2\right)(2 x-1)}{\left(x^{2}-x-2\right)^{2}}$

M1*

Finds $\frac{\mathrm{d} y}{\mathrm{~d} x}$ .

$x^{2}+8 x-2=0$

DM1

Sets equal to 0 and forms equation.

(-8.2,0.9), (0.2, -0.9).

A1 A1

Condone
$\left(-4-3 \sqrt{2}, \frac{2}{3} \sqrt{2}\right),\left(-4+3 \sqrt{2},-\frac{2}{3} \sqrt{2}\right) .$

4

(c)

Sketch C, stating the coordinates of any intersections with the axes.

[ 3 ]

(d)

Sketch the curve with equation $y=\frac{1}{\mathrm{f}(x)}$ .

[ 2 ]

(e)

Find the set of values for which $\frac{1}{\mathrm{f}(x)}<\mathrm{f}(x)$ .

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[ 4 ]

$\begin{aligned} \frac{x^{2}+2}{x^{2}-x-2}=1 \text { or } \frac{x^{2}+2}{x^{2}-x-2}=-1 x+4=0 \quad \text { or } \quad 2 x^{2}-x=0 \end{aligned}$

M2

Finds critical points, award M1 for each case.

$x=-4 \quad$ or $x=0, \quad x=\frac{1}{2}$

A1

$-4<x<-1, \quad 0<x<\frac{1}{2}, x>2$

B1

Must have three distinct regions. Condone $\leqslant-1$
and $\geqslant 2$ .

4

CAIE A-Level Further Math 1.2 Rational functions & graphs Question Bank

Exam points

Covered syllabus points

Covered syllabus points

1.2 Rational functions and graphs question 1

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)