IB · Mathematics: Applications & Interpretation SL & HL · P2S

IB Mathematics: Applications & Interpretation Paper 2 · SL (long questions) · exams to Nov 2028 practice papers

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Sample Paper 2 · SL (long questions) · exams to Nov 2028 questions

Question 1Structured · 13 marks

The volume of drink in bottles has mean 250 ml and standard deviation 15 ml. The distribution of the volumes is not known.

(a)

A sample of 36 bottles is taken. State the approximate distribution of the sample mean X̄, giving a reason.

[3]

(b)

Find the probability that the sample mean is less than 249 ml.

[2]

(c)

A different sample of 36 bottles has mean 245 ml. Find a 95% confidence interval for the population mean, assuming σ = 15.

[3]

(d)

Hence comment on the claim that the mean volume is 250 ml.

[2]

(e)

Find the least sample size needed for the width of a 95% confidence interval to be at most 4 ml.

[3]

Show answers and mark scheme

Part	Answer and marking points	Marks
(a)	X̄ ~ N(250, 6.25) central limit theorem, n large R1 mean 250 A1 variance 15²36 = 6.25 A1	3
(b)	0.345 normal CDF with standard error (M1) 0.345 A1	2
(c)	240.1 < μ < 249.9 z-interval (M1) 245 ± 1.96 × 15√36 (A1) 240.1 < μ < 249.9 A1	3
(d)	not supported 250 is not in the interval R1 the claim is not supported at the 5% level A1	2
(e)	217 2 × 1.96 × 15√n ≤ 4 (M1) solving for n (M1) 217 A1	3

Question 2Structured · 14 marks

An engineer records the time y seconds for a component to fail under a load of x tonnes.

x	2	4	6	8	10	12
y	8.91	24.4	42.4	66.9	88	109

Table 2.1

(a)

Find the equation of the least squares regression line of y on x.

[2]

(b)

Find the sum of the squares of the residuals, SS_res, for this linear model.

[2]

(c)

The engineer believes a model of the form y = ax^b is more appropriate. Using a log-log transformation, find a and b.

[5]

(d)

Find SS_res for the model in (c), and hence state which model fits the data better.

[3]

(e)

Use the power model to estimate y when x = 7.

[2]

Show answers and mark scheme

Part	Answer and marking points	Marks
(a)	y = 10.2x − 15.0 GDC (M1) y = 10.2x − 15.0 A1	2
(b)	32.1 Σ(y − ŷ)² (M1) 32.1 A1	2
(c)	a = 3.41, b = 1.41 ln y = ln a + b ln x (M1) regression of ln y on ln x (M1) ln y = 1.41ln x + 1.23 A1 b = 1.41 A1 a = 3.41 A1	5
(d)	27.9 predicted values from y = ax^b (M1) 27.9 A1 the power model has the smaller SS_res, so fits better R1	3
(e)	53.0 3.41 × 7^1.41 (M1) 53.0 A1	2

Question 3Structured · 14 marks

Each year 20% of the customers of company X move to company Y, and 20% of the customers of company Y move to company X. Initially X has 1200 customers and Y has 1200.

(a)

Write down a transition matrix T for the movement of customers.

[2]

(b)

Find the eigenvalues and corresponding eigenvectors of T.

[4]

(c)

Hence write down matrices P and D such that T = PDP⁻¹.

[2]

(d)

Find an expression for the number of customers of X after n years.

[5]

(e)

Write down the number of customers X can expect in the long term.

[1]

Show answers and mark scheme

Part	Answer and marking points	Marks
(a)	0.8 0.20.2 0.8 columns sum to 1 (M1) 0.8 0.20.2 0.8 A1	2
(b)	λ = 1, λ = 0.6 det(T − λI) = 0 (M1) λ = 1, λ = 0.6 A1 eigenvector for 1: 0.20.2 (or a multiple) A1 eigenvector for 0.6: 1−1 A1	4
(c)	P, D P = 0.2 10.2 −1 A1 D = 1 00 0.6 A1	2
(d)	1200 use of Tⁿ = PDⁿP⁻¹ M1 correct P⁻¹ A1 x_n = 1200 A2 form A + B(λ)ⁿ A1	5
(e)	1200 1200 A1	1

Question 4Structured · 12 marks

A sweet company makes star-shaped and moon-shaped sweets. 80% are stars and the rest are moons. 10% of stars and 30% of moons are yellow.

(a)

A sweet is chosen at random.
(i) Find the probability that it is yellow.
(ii) Given that it is yellow, find the probability that it is a star.

[4]

The company claims the colours in each bag follow the distribution below. A bag of 100 sweets is tested with a χ² goodness of fit test at the 5% level.

Colour	Brown	Red	Green	Orange	Yellow	Purple
Claimed %	15	25	20	20	10	10
Observed	14	31	22	15	4	14

Table 4.1

(b)

Write down the null hypothesis.

[1]

(c)

Find the expected frequencies.

[2]

(d)

Write down the number of degrees of freedom.

[1]

(e)

Find the p-value.

[2]

(f)

State the conclusion of the test, giving a reason.

[2]

Show answers and mark scheme

Part	Answer and marking points	Marks
(a)	0.140; 0.571 0.8 × 0.1 + 0.2 × 0.3 (M1) 0.140 A1 0.08000.140 (M1) 0.571 A1	4
(b)	colours follow the claimed distribution the colours are distributed as the company claims A1	1
(c)	15.0, 25.0, 20.0, 20.0, 10.0, 10.0 percentage × 100 (M1) 15.0, 25.0, 20.0, 20.0, 10.0, 10.0 A1	2
(d)	5 6 − 1 = 5 A1	1
(e)	0.148 χ² = 8.16 (A1) 0.148 A1	2
(f)	do not reject H₀ 0.148 > 0.05 R1 do not reject H₀: the data are consistent with the claim A1	2

Question 5Structured · 14 marks

An aircraft’s position is (x, y, z) km, where x and y are displacements east and north of an airport and z is the height. At 13:00 it is at (40, 15, 5) and moves with constant velocity −200−75−15 km h⁻¹. Let t be the time in hours after 13:00.

(a)

Write down a vector equation for the position r of the aircraft.

[2]

(b)

Verify that the aircraft would pass directly over the airport, and find its height at that time.

[4]

(c)

Find the time when the aircraft is 4 km above the ground, and its distance from the airport then.

[4]

(d)

The aircraft now adjusts its descent so that it lands at (0, 0, 0), keeping the same horizontal velocity −200−75. Find the new vertical component of velocity.

[4]

Show answers and mark scheme

Part	Answer and marking points	Marks
(a)	r = 40155 + t−200−75−15 position + t × velocity (M1) r = 40155 + t−200−75−15 A1	2
(b)	t = 0.200; 2.00 km x = 0 ⇒ t = 0.200 A1 y = 0 at the same t R1 substituting into z (M1) 2.00 km A1	4
(c)	t = 0.0667 h; 28.8 km 5 − 15t = 4 (M1) t = 0.0667 A1 position and distance (M1) 28.8 km A1	4
(d)	−30.0 km h⁻¹ time to reach the airport horizontally (M1) 0.133 h (A1) −4 ÷ time (M1) −30.0 km h⁻¹ A1	4

Question 6Structured · 13 marks

The table shows the average temperature x °C and the number of ice creams sold y at a kiosk on 8 days.

x	15	20	21	25	29	34	33	32
y	33	35	37	67	57	86	87	65

Table 6.1

(a)

Find Pearson’s product-moment correlation coefficient, r, and describe the correlation.

[3]

(b)

Find the equation of the regression line of y on x.

[2]

(c)

Use the regression line to estimate the number of ice creams sold when the temperature is 26 °C, and comment on the reliability of your estimate.

[3]

(d)

Interpret the gradient of the regression line in this context.

[1]

(e)

Find Spearman’s rank correlation coefficient r_s, and explain why it may be a more suitable measure if the relationship is not linear.

[4]

Show answers and mark scheme

Part	Answer and marking points	Marks
(a)	r = 0.912 GDC (M1) r = 0.912 A1 strong positive linear correlation A1	3
(b)	y = 2.86x − 16.3 GDC (M1) y = 2.86x − 16.3 A1	2
(c)	58.0 2.86 × 26 − 16.3 (M1) 58.0 A1 reliable: interpolation and strong correlation R1	3
(d)	increase per degree about 2.86 more ice creams are sold for each 1 °C increase R1	1
(e)	0.905 ranking the data (M1) r_s = 0.905 A2 Spearman measures a monotonic (not necessarily linear) relationship R1	4

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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