IB Questionbank

User interface language: English | Español

Date	May Example question	Marks available	3	Reference code	EXM.2.SL.TZ0.3
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Write down	Question number	3	Adapted from	N/A

Question

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

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

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

Syllabus sections

Topic 2—Functions » SL 2.6—Modelling skills

Show 75 related questions

22M.2.SL.TZ2.3g:
State one criticism of this interpretation.
22M.2.SL.TZ2.4a.iii:
$d$ .
22M.2.SL.TZ2.4b:
Calculate the number of revolutions of the Ferris wheel per ride.
22M.2.SL.TZ2.4e.i:
Identify which parameter will change.
22M.2.SL.TZ2.5c.i:
Find the value of $k$ .
22M.1.AHL.TZ1.12b:
Find the value of $b$ .
22M.1.SL.TZ2.9b.ii:
Write down three simultaneous equations for $a, b$ and $c$ .
22M.2.SL.TZ1.1b.i:
the amplitude.
22M.2.SL.TZ1.1b.ii:
the period.
22M.2.SL.TZ1.1b.iii:
the equation of the principal axis.
EXM.3.AHL.TZ0.9e:
Give two reasons why the prediction in part (b)(ii) might be lower than 14.
EXM.3.AHL.TZ0.9f.ii:
$c$ .
EXM.3.AHL.TZ0.9b.i:
the number of new people infected on day 6.
EXM.3.AHL.TZ0.9a:
Use an exponential regression to find the value of $a$ and of $b$ , correct to 4 decimal places.
22M.1.AHL.TZ1.12c:
Find the average number of earthquakes in a year with a magnitude of at least $7.2$ .
22M.1.SL.TZ2.9b.iii:
Hence, or otherwise, find the values of $a, b$ and $c$ .
22M.2.SL.TZ1.1c:
Hence or otherwise find the equation of this model in the form:

$f (t) = a \cos (b t) + d$
22M.2.SL.TZ2.2e:
Comment on the appropriateness of modelling this scenario with a geometric sequence.
22M.2.SL.TZ2.4a.ii:
$b$ .
22M.2.SL.TZ2.4e.ii:
Find the new value of the parameter identified in part (e)(i).
22M.2.AHL.TZ2.7f:
Write down one reason, with reference to the context, to support this decision.
21N.2.SL.TZ0.3f.ii:
The wind speed increases. The blades rotate at twice the speed, but still at a constant rate.

At any given instant, find the probability that Tim can see point $C$ from his window. Justify your answer.
22M.1.SL.TZ1.5a:
Write down three equations that express the information given above.
21N.2.SL.TZ0.3e.ii:
Find the time, in seconds, that point $C$ is above a height of $100 m$ , during each complete rotation.
22M.1.SL.TZ2.4a:
Calculate the pH of the coffee.
22M.1.SL.TZ2.4b:
Determine whether the unknown liquid is more or less acidic than the coffee. Justify your answer mathematically.
22M.2.SL.TZ2.4a.i:
$a$ .
22M.1.SL.TZ2.9b.i:
Write down the value of $d$ .
22M.1.SL.TZ2.9c:
Use the model to determine the total length of time, in years, for which a graduate is expected to be in debt after graduating from university.
22M.1.SL.TZ1.11a:
Find the value of $a$ .
22M.1.SL.TZ1.11b:
Find the value of $b$ .
22M.1.SL.TZ1.11c:
Given $0 < M < 8$ , find the range for $N$ .
22M.1.SL.TZ1.3a:
Write down the height of the ball above the ground at the instant it is hit by the bat.
22M.1.SL.TZ1.3b:
Find the value of $t$ when the ball hits the ground.
22M.1.SL.TZ1.5b:
Find the number of each type of ticket sold.
22M.1.AHL.TZ1.12a:
Find the value of $a$ .
21N.2.SL.TZ0.3c.i:
Write down the amplitude of the function.
21N.2.SL.TZ0.3e.i:
Find the height of $C$ above the ground when $t = 2$ .
21N.2.AHL.TZ0.2d:
Sketch the function $h (t)$ for $0 \leq t \leq 5$ , clearly labelling the coordinates of the maximum and minimum points.
EXM.3.AHL.TZ0.9b.ii:
the day when the total number of people infected will be greater than 1000.
EXM.3.AHL.TZ0.9f.i:
$L$ .
EXM.3.AHL.TZ0.9f.iii:
$k$ .
EXM.3.AHL.TZ0.9c:
Use your answer to part (a) to show that the model predicts 16.7 people will be infected on the first day.
SPM.2.SL.TZ0.5a.ii:
Find the values of $p$ and $q$ .
EXM.2.SL.TZ0.3c:
Solve this system of three equations to find the value of $a$ , $b$ and $c$ .
EXM.2.SL.TZ0.3d.i:
when the height of the object is zero.
SPM.2.SL.TZ0.5d:
Hence, identify why Model A may not be appropriate at lower speeds.
SPM.2.SL.TZ0.5f:
Calculate the percentage error in the estimate in part (e).
SPM.2.SL.TZ0.5c:
Using the values in the table and your answer to part (b), sketch the graph of $y = d\left( s \right)$ for 0 ≤ $s$ ≤ 10 and −10 ≤ $d$ ≤ 60, clearly showing the vertex.
EXM.3.AHL.TZ0.9h:
Use the logistic model to find the day when the rate of increase of people infected is greatest.
EXM.3.AHL.TZ0.9d.ii:
Perform a ${\chi ^2}$ goodness of fit test at the 5% significance level. You should clearly state your hypotheses, the p-value, and your conclusion.
EXM.3.AHL.TZ0.9d.i:
Explain why the number of degrees of freedom is 2.
SPM.2.SL.TZ0.5b:
Find the coordinates of the vertex of the graph of $y = d\left( s \right)$ .
EXM.3.AHL.TZ0.9g:
Hence predict the total number of people infected by this disease after several months.
SPM.2.SL.TZ0.5e:
Use Model B to calculate an estimate for the braking distance at a speed of $20\,{\text{m}}\,{{\text{s}}^{ - 1}}$ .
EXM.2.SL.TZ0.3d.ii:
the maximum height of the object.
SPM.2.SL.TZ0.5a.i:
Write down a second equation to represent Model A, when the speed is $10\,{\text{m}}\,{{\text{s}}^{ - 1}}$ .
EXM.2.SL.TZ0.3a:
Show that $4a + 2b + c = 34$ .
SPM.2.SL.TZ0.5g:
It is found that once a driver realizes the need to stop their vehicle, 1.6 seconds will elapse, on average, before the brakes are engaged. During this reaction time, the vehicle will continue to travel at its original speed.

A truck approaches an intersection with speed $s\,{\text{m}}\,{{\text{s}}^{ - 1}}$ . The driver notices the intersection’s traffic lights are red and they must stop the vehicle within a distance of $330\,{\text{m}}$ .

Using model B and taking reaction time into account, calculate the maximum possible speed of the truck if it is to stop before the intersection.
21N.2.AHL.TZ0.2a.ii:
minimum value of $h$ .
21N.2.AHL.TZ0.2b.i:
Find the time, in seconds, it takes for the blade $[BC]$ to make one complete rotation under these conditions.
21N.2.SL.TZ0.3a.i:
maximum value of $h$ .
21N.2.SL.TZ0.3a.ii:
minimum value of $h$ .
21N.2.SL.TZ0.3b.i:
Find the time, in seconds, it takes for the blade $[BC]$ to make one complete rotation under these conditions.
21N.2.SL.TZ0.3b.ii:
Calculate the angle, in degrees, that the blade $[BC]$ turns through in one second.
21N.2.SL.TZ0.3c.ii:
Find the period of the function.
21N.2.SL.TZ0.3d:
Sketch the function $h (t)$ for $0 \leq t \leq 5$ , clearly labelling the coordinates of the maximum and minimum points.
21N.2.SL.TZ0.3f.i:
At any given instant, find the probability that point $C$ is visible from Tim’s window.
21N.2.AHL.TZ0.2a.i:
maximum value of $h$ .
21N.2.AHL.TZ0.2b.ii:
Calculate the angle, in degrees, that the blade $[BC]$ turns through in one second.
21N.2.AHL.TZ0.2c.i:
Write down the amplitude of the function.
21N.2.AHL.TZ0.2c.ii:
Find the period of the function.
21N.2.AHL.TZ0.2e.i:
Find the height of $C$ above the ground when $t = 2$ .
21N.2.AHL.TZ0.2e.ii:
Find the time, in seconds, that point $C$ is above a height of $100 m$ , during each complete rotation.
21N.2.AHL.TZ0.2f:
The wind speed increases and the blades rotate faster, but still at a constant rate.

Given that point $C$ is now higher than $110 m$ for $1$ second during each complete rotation, find the time for one complete rotation.

Hide related questions

Topic 2—Functions

Question

Markscheme

Examiners report

Syllabus sections

View options