Free worked solutions · 2026 Edition

IB Maths AI SL Chapter 15: Exam strategy (SL) worked solutions

Worked solutions to the timed questions in Chapter 15 of Photon Guidebook: Mathematics Applications and Interpretation SL 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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5 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, P/Y=C/Y=12 (M1 for the values, M1 for the signs with FV=0). The payment is −457.32, so Priya pays 457.32 AUD a month (A1). (b) 60\times457.32-24\,000=3439.20 AUD (M1 A1). (c) Set N=24 with the stored payment and solve for FV (M1): FV=-15\,167.78, so she still owes 15 167.78 AUD (A1).

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. Under 30: bus 36, train 52, cycle 32. Aged 30 or over: bus 52, train 54, cycle 14. 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) Expected value =\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

(b) 2 (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). With the GDC in degrees, t=9.4648\ldots and t=20.5351\ldots (A1 A1). The time above 30 m is 20.5351\ldots-9.4648\ldots=11.1 minutes (A1).

Final answer

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

Question 4

[7 marks]
A workshop makes and sells x hundred lamps a month, where 0\le x\le10. Its monthly profit, in thousands of euros (EUR), is modelled by P(x)=-2x^3+15x^2+36x-40. (a) Find P'(x).
[2]
(b) Find the number of lamps that gives the greatest monthly profit, justifying that it is a maximum.
[3]
(c) Find the greatest monthly profit.
[2]
Show solutionHide solution
(a) P'(x)=-6x^2+30x+36 (A1 A1). (b) -6x^2+30x+36=0 (M1), so x=6 since x=-1 is outside the domain (A1). P'(5)=36>0 and P'(7)=-48<0, so P changes from increasing to decreasing and x=6 is a maximum (R1). The workshop should make 600 lamps a month. (c) P(6)=284 (M1), so the greatest profit is 284 000 EUR (A1).

Final answer

(a) -6x^2+30x+36 (b) 600 lamps (c) 284 000 EUR

Question 5

[8 marks]
The mass of a bag of rice filled by a machine is normally distributed with mean 1005 g and standard deviation 6 g. (a) Find the probability that a bag has a mass of less than 1000 g.
[2]
(b) 95% of bags have a mass greater than m grams. Find m.
[2]
(c) A carton holds 12 bags, filled independently. Find the probability that at least 3 bags in a carton have a mass of less than 1000 g.
[4]
Show solutionHide solution
(a) M\sim N(1005,6^2) (M1), P(M<1000)=0.202 (A1). (b) P(M<m)=0.05 (M1), m=995 g (A1). (c) Y\sim B(12,0.202328\ldots) with the stored probability (M1 A1). P(Y\ge3)=1-P(Y\le2) (M1) =0.450 (A1).

Final answer

(a) 0.202 (b) 995 g (c) 0.450

From the book

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

Chapter 15 teaches Exam strategy (SL) 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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