Free Modelling: quadratic, exponential & sinusoidal questions for IB Mathematics: Applications & Interpretation (SL & HL), part of Functions. Each question has the answer and the marking points an examiner would look for.
Modelling: quadratic, exponential & sinusoidal practice questions
Question 1Structured · 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 = 7.5, the minimum value is 8, 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) > 18.5.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
a = 7, b = 0.524, c = 7.5, d = 15
a = 7 A1 b = 2π12 = 0.524 A1 c = 7.5 A1 d = 15 A1
4
(b)
4.00
f(t) = 18.5 (M1) t = 5.50, 9.50 A1 4.00 A1
3
Question 2Structured · 6 marks
The temperature T °C of a cup of coffee t minutes after it is made is modelled by T(t) = 22 + ae−kt. Initially T = 90 and after 15 minutes T = 54.1.
(a)
Find the value of a and the value of k.
[3]
(b)
Find the time taken for the coffee to cool to 40 °C.
[2]
(c)
Write down the temperature of the room.
[1]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
a = 68, k = 0.0500
a = 68 A1 54.1 = 22 + 68e−15k(M1) k = 0.0500 A1
3
(b)
26.6 minutes
22 + 68e−0.0500t = 40 (M1) 26.6 minutes A1
2
(c)
22 °C
22 °C (the horizontal asymptote) A1
1
Question 3Structured · 6 marks
The temperature T °C, t hours after midnight, varies between a minimum of 12 °C and a maximum of 24 °C, with the maximum at t = 7. It is modelled by T(t) = a cos(b(t − 7)) + d, with a period of 24 hours.
(a)
Find the values of a, b and d.
[4]
(b)
Find the temperature at t = 10.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
a = 6, b = 0.262, d = 18
a = 6 A1 d = 18 A1 b = 2π24(M1) b = 0.262 A1
4
(b)
22.2 °C
T(10) (M1) 22.2 °C A1
2
Question 4Structured · 5 marks
A ball is thrown upwards. Its height h metres after t seconds is modelled by h(t) = −4.9t2 + 20t + 1.2.
(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
Part
Answer and marking points
Marks
(a)
21.6 m
vertex, e.g. GDC or t = −b2a(M1) 21.6 m A1
2
(b)
4.14 s
h(t) = 0 (M1) 4.14 s A1
2
(c)
h becomes negative
h(t) < 0 after the ball lands, which is not physically possible R1
1
Question 5Structured · 7 marksA* challenge
The temperature is modelled by f(t) = a cos(b(t − c)) + d, where t is in hours. The maximum value 18 occurs at t = 7, the minimum value is 8, 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) > 16.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
a = 5, b = 0.524, c = 7, d = 13
a = 5 A1 b = 2π12 = 0.524 A1 c = 7 A1 d = 13 A1
4
(b)
3.54
f(t) = 16 (M1) t = 5.23, 8.77 A1 3.54 A1
3
Question 6Structured · 6 marks
The temperature T °C of a cup of coffee t minutes after it is made is modelled by T(t) = 22 + ae−kt. Initially T = 90 and after 10 minutes T = 47.
(a)
Find the value of a and the value of k.
[3]
(b)
Find the time taken for the coffee to cool to 50 °C.
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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