Free worked solutions · 2026 Edition

IB Maths AI HL Chapter 24: Exam strategy worked solutions

Worked solutions to the timed questions in Chapter 24 of Photon Guidebook: Mathematics Applications and Interpretation HL for the IB Diploma (2026 Edition). Each solution shows every step with the mark it earns, the way an IB mark scheme does.

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6 timed questions for exam practice.

Question 1

[7 marks]
Priya borrows 24 000 AUD to buy a car. The loan charges 5.4% annual interest, compounded monthly, and she repays it with equal payments at the end of each month for 5 years.
(a) Calculate the amount of each monthly payment.
[3]
(b) Calculate the total interest Priya pays over the 5 years.
[2]
(c) Find the amount Priya still owes immediately after her 24th payment.
[2]
Show solutionHide solution
(a) N=60, I\%=5.4, PV=24\,000, FV=0, \mathrm{P/Y}=\mathrm{C/Y}=12 (M1). A loan received is positive, so PMT<0 (M1). PMT=-457.32, so 457.32 AUD a month (A1).
(b) 60\times457.32-24\,000 (M1) =3439.20 AUD (A1); accept 3439.26 from the unrounded payment.
(c) N=24 with the stored payment, solve for FV (M1). FV=-15\,167.78, so 15 167.78 AUD is owed (A1); accept 15 167.80 from 457.32.

Final answer

(a) 457.32 AUD (b) 3439.20 AUD (c) 15 167.78 AUD

Question 2

[7 marks]
A transport office asks 240 commuters which mode of transport they use most and records their age group.
\begin{array}{l|ccc}&\text{Bus}&\text{Train}&\text{Cycle}\\\hline\text{Under 30}&36&52&32\\\text{Aged 30 or over}&52&54&14\end{array}
A chi-squared test for independence is carried out at the 5% significance level.
(a) State the null hypothesis.
[1]
(b) Write down the number of degrees of freedom.
[1]
(c) Show that the expected number of cyclists under 30 is 23.
[1]
(d) Find the p-value for this test.
[2]
(e) State the conclusion of the test, giving a reason.
[2]
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(a) H_0: the mode of transport used is independent of age group (A1).
(b) (2-1)(3-1)=2 (A1).
(c) \dfrac{120\times46}{240} (M1) =23 (AG).
(d) \chi^2=9.99 (A1), p=0.00677 (A1).
(e) 0.00677<0.05 (R1), so reject H_0: there is evidence that the mode of transport used depends on age group (A1).

Final answer

(a) independence (b) 2 (c) AG (d) p=0.00677 (e) reject H_0

Question 3

[6 marks]
A capsule on an observation wheel is h metres above the ground t minutes after it leaves the lowest point, where h(t)=22-20\cos(12t) and the angle is measured in degrees.
(a) Write down the greatest height of the capsule.
[1]
(b) Find the time taken for one complete rotation.
[1]
(c) Find the length of time, during one rotation, for which the capsule is more than 30 m above the ground.
[4]
Show solutionHide solution
(a) 22+20=42 m (A1).
(b) 360\div12=30 minutes (A1).
(c) 22-20\cos(12t)=30 (M1). t=9.4648\ldots and t=20.5351\ldots (A1 A1). 20.5351\ldots-9.4648\ldots=11.1 minutes (A1).

Final answer

(a) 42 m (b) 30 minutes (c) 11.1 minutes

Question 4

[8 marks]
Calls reach a student help desk at random, at a mean rate of 3.2 calls every 10 minutes.
(a) Find the probability that exactly 5 calls arrive in a 10-minute period.
[2]
(b) Find the probability that at least 10 calls arrive in a 30-minute period.
[3]
A 10-minute period is called busy if more than 5 calls arrive.
(c) Find the probability that more than 3 of the 24 periods in a 4-hour shift are busy.
[3]
Show solutionHide solution
(a) X\sim\mathrm{Po}(3.2) (M1). \mathrm{P}(X=5)=0.114 (A1).
(b) Y\sim\mathrm{Po}(9.6) (M1). \mathrm{P}(Y\ge10)=1-\mathrm{P}(Y\le9) (M1) =0.491 (A1).
(c) p=\mathrm{P}(X>5)=1-\mathrm{P}(X\le5)=0.105408\ldots (A1). B\sim\mathrm{B}(24,\,0.105408\ldots) (M1). \mathrm{P}(B>3)=1-\mathrm{P}(B\le3)=0.243 (A1).

Final answer

(a) 0.114 (b) 0.491 (c) 0.243

Question 5

[8 marks]
The inside of a glass bowl is formed by rotating the curve y=0.1x^2, for 0\le y\le8, through 360^\circ about the y-axis; all lengths are in centimetres.
(a) Write down an integral for the volume of the bowl.
[2]
(b) Hence find the volume of the bowl.
[2]
Water is poured into the empty bowl until it holds 200\text{ cm}^3.
(c) Determine the depth of the water.
[4]
Show solutionHide solution
(a) x^2=10y (M1). V=\pi\int_0^8 10y\,\mathrm{d}y (A1).
(b) 320\pi=1005.3\ldots (A1). V=1010\text{ cm}^3 (A1); accept 1005\text{ cm}^3.
(c) \pi\int_0^h 10y\,\mathrm{d}y=200 (M1). 5\pi h^2=200 (A1). h^2=\dfrac{40}{\pi} (A1). h=3.57 cm (A1).

Final answer

(a) \pi\int_0^8 10y\,\mathrm{d}y (b) 1010\text{ cm}^3 (c) 3.57 cm

Question 6

[9 marks]
A cup of coffee cools in a room. Its temperature, T\,{}^\circ\text{C}, t minutes after it is poured satisfies \dfrac{\mathrm{d}T}{\mathrm{d}t}=-0.08(T-22), with T=90 when t=0.
(a) Taking steps of 1 minute, use Euler's method to estimate T when t=3.
[3]
(b) By separating the variables, show that T=22+68e^{-0.08t}.
[4]
(c) Find the percentage error in your estimate from part (a).
[2]
Show solutionHide solution
(a) T_{n+1}=T_n+1\times(-0.08)(T_n-22) (M1). T_1=84.56, T_2=79.5552 (A1). T_3=74.9508\ldots, so 75.0\,{}^\circ\text{C} (A1).
(b) \int\dfrac{\mathrm{d}T}{T-22}=\int-0.08\,\mathrm{d}t (M1). \ln|T-22|=-0.08t+c (A1). T-22=A\mathrm{e}^{-0.08t} (M1). t=0, T=90 gives A=68 (A1). Hence T=22+68\mathrm{e}^{-0.08t} (AG).
(c) Exact T(3)=75.4906\ldots (A1). \dfrac{|74.9508-75.4906|}{75.4906}\times100=0.715\% (A1).

Final answer

(a) 75.0\,{}^\circ\text{C} (b) AG (c) 0.715%

From the book

Photon Guidebook: Mathematics Applications and Interpretation HL for the IB Diploma

Chapter 24 teaches Exam strategy from the basics to full exam questions: how the IB asks it, every question type with a marked example, and these practice questions in three steps.

About the book

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