CAIE A-Level Further Math AS 1.2 Rational Functions and Graphs Questions

$x=-1, \quad x=2$
Vertical asymptotes.

y=1
Horizontal asymptote.

2

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

M1*

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

$x^{2}+8 x-2=0$
Sets equal to 0 and forms equation.

(-8.2,0.9), (0.2, -0.9).
Condone
$$\left(-4-3 \sqrt{2}, \frac{2}{3} \sqrt{2}\right),\left(-4+3 \sqrt{2},-\frac{2}{3} \sqrt{2}\right) .$$

4

Axes and all three asymptotes.
Correct shape and position, crossing horizontal
asymptote.

States (0,-1) coordinates of intersection with axes,
may be seen on diagram.

3

B1 FT

FT from sketch in (c)

B1

All correct.

2

$$\begin{aligned} \frac{x^{2}+2}{x^{2}-x-2}=1 \text { or } \frac{x^{2}+2}{x^{2}-x-2}=-1 x+4=0 \quad \text { or } \quad 2 x^{2}-x=0 \end{aligned}$$
Finds critical points, award M1 for each case.

$x=-4 \quad$ or $x=0, \quad x=\frac{1}{2}$$-4<x<-1, \quad 0<x<\frac{1}{2}, x>2$
Must have three distinct regions. Condone $\leqslant-1$
and $\geqslant 2$.

4

x=-1
Vertical asymptote.

$y=\frac{(x+1)(x-1)+1}{x+1}$
Oblique asymptote.

y=x-1
3

$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{x^{2}+2 x}{(x+1)^{2}}=0$
Sets $\frac{\mathrm{d} y}{\mathrm{~d} x}=0$.

(0,0), (-2,-4)
2

Axes and asymptotes.
Left branch correct.
Right branch correct.

3

( $-2,-\frac{1}{4}$ )
B1 for each correct coordinate. SC B1
for ( $-2,-\frac{1}{4}$ ) and ( 0,0 ).

2

Left branch correct.

Right branch correct.

$$\begin{aligned} \frac{x^{2}}{x+1}=1 \text { or } \frac{x^{2}}{x+1}=-1 x^{2}-x-1=0 \end{aligned}$$
Finds critical points, award M1 for
each case.

$x=\frac{1}{2}-\frac{1}{2} \sqrt{5}$ or $x=\frac{1}{2}+\frac{1}{2} \sqrt{5}$$x<-1, \quad \frac{1}{2}-\frac{1}{2} \sqrt{5}<x<\frac{1}{2}+\frac{1}{2} \sqrt{5}, \quad x \neq 0$
Condone missing $x \neq 0$.

6