18N.2.AHL.TZ0.H_8

pestleMathematicsAAHLPaper 218N· sl-2-8-reciprocal-and-simple-rational-functions-equations-of-asymptotessource ↗

Consider the function $f\left( x \right) = \frac{{ax + 1}}{{bx + c}}$ , $x \ne - \frac{c}{b}$ , where $a$ , $b$ , $c \in \mathbb{Z}$ .

The following graph shows the curve $y = {\left( {f\left( x \right)} \right)^2}$ . It has asymptotes at $x = p$ and $y = q$ and meets the $x$ -axis at A.

On the following axes, sketch the two possible graphs of $y = f\left( x \right)$ giving the equations of any asymptotes in terms of $p$ and $q$ .

[4]

Given that $p = \frac{4}{3}$ , $q = \frac{4}{9}$ and A has coordinates $\left( { - \frac{1}{2},\,\,0} \right)$ , determine the possible sets of values for $a$ , $b$ and $c$ .

[4]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

either graph passing through (or touching) A A1

correct shape and vertical asymptote with correct equation for either graph A1

correct horizontal asymptote with correct equation for either graph A1

two completely correct sketches A1

[4 marks]

$a\left( { - \frac{1}{2}} \right) + 1 = 0 \Rightarrow a = 2$ A1

from horizontal asymptote, ${\left( {\frac{a}{b}} \right)^2} = \frac{4}{9}$ (M1)

$\frac{a}{b} = \pm \frac{2}{3} \Rightarrow b = \pm 3$ A1

from vertical asymptote, $b\left( {\frac{4}{3}} \right) + c = 0$

$b$ = 3, $c$ = −4 or $b$ = −3, $c$ = 4 A1

[4 marks]

Examiners’ report

[N/A]