Free Exponents & logarithmic scales questions for IB Mathematics: Applications & Interpretation (SL & HL), part of Number & algebra. Each question has the answer and the marking points an examiner would look for.
A cup of coffee cools in a room at 20°C. Its temperature, T°C, t minutes after it is made is modelled by T = 20 + Ae−kt. Initially T = 95, and after 5 minutes T = 60.
(a)
Write down the value of A.
[1]
(b)
Find the value of k.
[2]
(c)
Find the time taken for the coffee to cool to 35°C.
[2]
(d)
Find the rate at which the temperature is falling when t = 0.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
75
75 A1
1
(b)
0.126
60 = 20 + 75e−5k(M1) 0.126 A1
2
(c)
12.8 min
setting T = 35 (M1) 12.8 min A1
2
(d)
9.43 °C per min
differentiating (M1) 9.43 °C per min A1
2
Question 2Structured · 5 marks
The pH of a solution is given by pH = −log10[H+], where [H+] is the hydrogen ion concentration in mol dm−3.
(a)
Find [H+] for a solution with pH 3.3. Give your answer in standard form.
[2]
(b)
Find how many times greater [H+] is in a solution of pH 3.3 than in a solution of pH 5.7.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
5.01 × 10−4
10−3.3(M1) 5.01 × 10−4A1
2
(b)
251
10−3.310−5.7(M1) 102.4(A1) 251 A1
3
Question 3Structured · 7 marksA* challenge
A cup of coffee cools in a room at 20°C. Its temperature, T°C, t minutes after it is made is modelled by T = 20 + Ae−kt. Initially T = 90, and after 5 minutes T = 65.
(a)
Write down the value of A.
[1]
(b)
Find the value of k.
[2]
(c)
Find the time taken for the coffee to cool to 30°C.
[2]
(d)
Find the rate at which the temperature is falling when t = 0.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
70
70 A1
1
(b)
0.0884
65 = 20 + 70e−5k(M1) 0.0884 A1
2
(c)
22.0 min
setting T = 30 (M1) 22.0 min A1
2
(d)
6.19 °C per min
differentiating (M1) 6.19 °C per min A1
2
Question 4Structured · 5 marks
The energy E released by an earthquake of magnitude M satisfies log10E = 4.8 + 1.5M.
(a)
Find E for an earthquake of magnitude 6.
[2]
(b)
Find how many times more energy is released by an earthquake of magnitude 8 than one of magnitude 6.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
6.31 × 1013
104.8 + 1.5 × 6(M1) 6.31 × 1013A1
2
(b)
1000
subtracting the logarithms (M1) 101.5 × 2(A1) 1000 A1
3
Question 5Structured · 7 marksA* challenge
A cup of coffee cools in a room at 20°C. Its temperature, T°C, t minutes after it is made is modelled by T = 20 + Ae−kt. Initially T = 90, and after 5 minutes T = 60.
(a)
Write down the value of A.
[1]
(b)
Find the value of k.
[2]
(c)
Find the time taken for the coffee to cool to 40°C.
[2]
(d)
Find the rate at which the temperature is falling when t = 0.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
70
70 A1
1
(b)
0.112
60 = 20 + 70e−5k(M1) 0.112 A1
2
(c)
11.2 min
setting T = 40 (M1) 11.2 min A1
2
(d)
7.83 °C per min
differentiating (M1) 7.83 °C per min A1
2
Question 6Structured · 5 marks
The pH of a solution is given by pH = −log10[H+], where [H+] is the hydrogen ion concentration in mol dm−3.
(a)
Find [H+] for a solution with pH 4.3. Give your answer in standard form.
[2]
(b)
Find how many times greater [H+] is in a solution of pH 4.3 than in a solution of pH 5.6.
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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