19M.2.SL.TZ1.T_4

pestleMathematicsAISLPaper 219M· sl-2-2-functions-notation-domain-range-and-inverse-as-reflectionsource ↗

Consider the function $f\left( x \right) = {x^3} - 5{x^2} + 6x - 3 + \frac{1}{x}$ , $x > 0$

The function $f\left( x \right) = {x^3} - 5{x^2} + 6x - 3 + \frac{1}{x}$ , $x > 0$ , models the path of a river, as shown on the following map, where both axes represent distance and are measured in kilometres. On the same map, the location of a highway is defined by the function $g\left( x \right) = 0.5{\left( 3 \right)^{ - x}} + 1$ .

The origin, O(0, 0) , is the location of the centre of a town called Orangeton.

A straight footpath, $P$ , is built to connect the centre of Orangeton to the river at the point where $x = \frac{1}{2}$ .

Bridges are located where the highway crosses the river.

A straight road is built from the centre of Orangeton, due north, to connect the town to the highway.

State the domain of $P$ .

[2]

b.ii.

Find the distance from the centre of Orangeton to the point at which the road meets the highway.

[2]

This straight road crosses the highway and then carries on due north.

State whether the straight road will ever cross the river. Justify your answer.

[2]

Markscheme / solution

0 < $x$ < $\frac{1}{2}$ (A1)(A1)

Note: Award (A1) for both endpoints correct, (A1) for correct mathematical notation indicating an interval with two endpoints. Accept weak inequalities. Award at most (A1)(A0) for incorrect notation such as 0 − 0.5 or a written description of the domain with correct endpoints. Award at most (A1)(A0) for 0 < $y$ < $\frac{1}{2}$ .

[2 marks]

b.ii.

$g\left( 0 \right) = 0.5{\left( 3 \right)^0} + 1$ (M1)

1.5 (km) (A1)(G2)

[2 marks]

domain given as $x > 0$ (but equation of road is $x = 0$ ) (R1)

(equation of road is $x = 0$ ) the function of the river is asymptotic to $x = 0$ (R1)

so it does not meet the river (A1)

Note: Award the (R1) for a correct mathematical statement about the equation of the river (and the equation of the road). Justification must be based on mathematical reasoning. Do not award (R0)(A1).

[2 marks]

Examiners’ report

[N/A]

b.ii.

[N/A]