IB · Mathematics: Applications & Interpretation SL & HL · Topic

Functions: IB Mathematics: Applications & Interpretation questions and answers

10 free IB Mathematics: Applications & Interpretation Functions questions written in the style of IB Mathematics: Applications & Interpretation papers, each with its answer and mark scheme. 3 of them are A* challenge questions, built to catch the mistakes strong students still make.

This topic covers Linear models, Modelling: quadratic, exponential & sinusoidal, Log-log & semi-log models, Logistic models. Work through the questions below, then generate a full Functions paper with a fresh set every time.

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Functions sub-topics

Linear modelsModelling: quadratic, exponential & sinusoidalLog-log & semi-log modelsLogistic models

Functions practice questions

Question 1Structured · 14 marksA* challenge

A biologist models the number of fish in a lake t years after they are introduced by N(t) = 10001 + 39e−0.4t.

(a)

Find the initial number of fish.

[1]
(b)

Find the number of fish after 3 years.

[2]
(c)

Find the time at which the population is growing fastest, and the population at that time.

[4]
(d)

An alternative model is P(t) = 25emt, which agrees with N at t = 0 and t = 3. Find m.

[3]
(e)

Compare the predictions of the two models after 10 years, and explain which model is more realistic.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)25
25 A1
1
(b)78.5
N(3) (M1)
78.5 A1
2
(c)t = 9.16, N = 500
inflexion point / maximum of N′(t) (M1)
N = half the carrying capacity (A1)
t = 9.16 A1
N = 500 A1
4
(d)m = 0.381
25e3m = 78.5 (M1)
taking logs (M1)
m = 0.381 A1
3
(e)N = 583, P = 1131
N(10) = 583 A1
P(10) = 1131 A1
the exponential model grows without limit R1
the logistic model is more realistic as the lake has limited resources R1
4
Question 2Structured · 7 marksA* challenge

The temperature is modelled by f(t) = a cos(b(t − c)) + d, where t is in hours. The maximum value 22 occurs at t = 8, the minimum value is 4, and the period is 12.

(a)

Find a, b, c and d.

[4]
(b)

Find the length of time in one period for which f(t) > 18.4.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)a = 9, b = 0.524, c = 8, d = 13
a = 9 A1
b = 2π12 = 0.524 A1
c = 8 A1
d = 13 A1
4
(b)3.54
f(t) = 18.4 (M1)
t = 6.23, 9.77 A1
3.54 A1
3
Question 3Structured · 7 marksA* challenge

Plan A costs $12 per month plus $3 per GB of data. Plan B costs $27 per month, which includes 8 GB, and $6 for each additional GB.

(a)

Write down the cost of each plan for 6 GB.

[2]
(b)

Write down an expression for the cost of plan B when x > 8.

[1]
(c)

Find the range of values of x for which plan B is cheaper.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)A: $30, B: $27
$30 A1
$27 A1
2
(b)27 + 6(x − 8)
27 + 6(x − 8) A1
1
(c)5.00 < x < 11.0
12 + 3x = 27 (M1)
x = 5.00 A1
12 + 3x = 27 + 6(x − 8) (M1)
5.00 < x < 11.0 A1
4
Question 4Structured · 6 marks

The number of people in a town who have heard a rumour, t hours after it starts, is modelled by N(t) = 8001 + 9e−0.4t.

(a)

Find the number of people who had heard the rumour initially.

[1]
(b)

State the number of people in the town, according to the model, giving a reason.

[2]
(c)

Find the time taken for 75% of the town to have heard the rumour.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)80
80 A1
1
(b)800
as t → ∞, e−0.4t → 0 R1
800 A1
2
(c)8.24 hours
N(t) = 600 (M1)
attempt to solve (M1)
8.24 hours A1
3
Question 5Structured · 6 marks

The cost $C of a taxi journey of d km is modelled by C = md + c. A journey of 8 km costs $21.10 and a journey of 18 km costs $43.10.

(a)

Find the value of m and the value of c.

[3]
(b)

Interpret the value of m in this context.

[1]
(c)

Find the greatest distance that can be travelled for $50.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)m = 2.2, c = 3.5
two equations or gradient from two points (M1)
m = 2.2 A1
c = 3.5 A1
3
(b)cost per km
each additional km costs $2.20 A1
1
(c)21.1 km
2.2d + 3.5 = 50 (M1)
21.1 km A1
2
Question 6Structured · 4 marks

A taxi charges a fixed fee of $5 plus $2 per kilometre.

(a)

Write down a linear model for the cost C of a journey of d km.

[1]
(b)

A second taxi charges $2.5 per kilometre with no fixed fee. Find the distance for which both cost the same.

[2]
(c)

State what the gradient of the first model represents.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)C = 2d + 5
C = 2d + 5 A1
1
(b)10.0 km
equating (M1)
10.0 km A1
2
(c)cost per km
cost per kilometre R1
1
Question 7Structured · 5 marks

A ball is thrown upwards. Its height h metres after t seconds is modelled by h(t) = −4.9t2 + 18t + 1.5.

(a)

Find the maximum height of the ball.

[2]
(b)

Find the time at which the ball hits the ground.

[2]
(c)

State a reason why the model might not be appropriate for large values of t.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)18.0 m
vertex, e.g. GDC or t = −b2a (M1)
18.0 m A1
2
(b)3.75 s
h(t) = 0 (M1)
3.75 s A1
2
(c)h becomes negative
h(t) < 0 after the ball lands, which is not physically possible R1
1
Question 8Structured · 6 marks

The table shows values of two variables x and y, which are thought to be related by y = axb.

x24681012
y8.462443.369.396.8123
Table 8.1
(a)

Show that ln y = b ln x + ln a.

[2]
(b)

By finding the equation of the regression line of ln y on ln x, estimate the values of a and b.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)shown
taking logs of both sides M1
ln y = ln a + b ln x AG A1
2
(b)a = 2.98, b = 1.50
transforming the data to ln x and ln y (M1)
ln y = 1.50ln x + 1.09 (A1)
b = 1.50 A1
a = e1.09 = 2.98 A1
4
Question 9Structured · 4 marks

The number of people who have heard a rumour is modelled by N(t) = 5001 + 9e−0.5t, t in hours.

(a)

Find the initial number of people who have heard the rumour.

[1]
(b)

Write down the long-term number predicted by the model.

[1]
(c)

Find the time when half of the long-term number have heard it.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)50
50 A1
1
(b)500
500 A1
1
(c)4.39 hours
solving N = 250 (M1)
4.39 A1
2
Question 10Structured · 4 marks

Data are modelled by y = axn. A graph of ln y against ln x is a straight line with gradient 1.5 and intercept 1.10.

Find a and n, and use the model to predict y when x = 10.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
a = 3, n = 1.5; y = 94.9
ln y = n ln x + ln a (M1)
n = 1.5 A1
a = 3 A1
94.9 A1
4

Other IB Mathematics: Applications & Interpretation topics

Number & algebraGeometry & trigonometryStatistics & probabilityCalculus

Frequently asked questions

What is in IB Mathematics: Applications & Interpretation Functions?

In the SL & HL syllabus, Functions covers Linear models, Modelling: quadratic, exponential & sinusoidal, Log-log & semi-log models, Logistic models.

Where can I get Functions questions with mark schemes?

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Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in