EXM.2.SL.TZ0.3

pestleMathematicsAISLPaper 2EXM· sl-2-6-modelling-skillssource ↗

Urvashi wants to model the height of a moving object. She collects the following data showing the height, $h$ metres, of the object at time $t$ seconds.

She believes the height can be modeled by a quadratic function, $h\left( t \right) = a{t^2} + bt + c$ , where $a{\text{,}}\,\,b{\text{,}}\,\,c \in \mathbb{R}$ .

Hence find

Show that $4a + 2b + c = 34$ .

[1]

Write down two more equations for $a$ , $b$ and $c$ .

[3]

Solve this system of three equations to find the value of $a$ , $b$ and $c$ .

[4]

when the height of the object is zero.

[3]

d.i.

the maximum height of the object.

[2]

d.ii.

Markscheme / solution

$t = 2{\text{,}}\,\,h = 34 \Rightarrow \,\,34 = a{2^2} + 2b + c$ M1

$\Rightarrow {\text{ }}34 = 4a + 2b + c$ AG

[1 mark]

attempt to substitute either (5, 38) or (7, 24) M1

$25a + 5b + c = 38$ A1

$49a + 7b + c = 24$ A1

[3 marks]

$a = - \frac{5}{3}{\text{,}}\,\,b = 13{\text{,}}\,\,c = \frac{{44}}{3}$ M1A1A1A1

[3 marks]

$- \frac{5}{3}{t^2} + 13t + \frac{{44}}{3} = 0$ M1

$t = 8.8$ seconds M1A1

[3 marks]

d.i.

attempt to find maximum height, e.g. sketch of graph M1

$h = 40.0$ metres A1

[2 marks]

d.ii.

Examiners’ report

[N/A]

d.i.

[N/A]

d.ii.