IB · Mathematics: Applications & Interpretation SL & HL · P1H

IB Mathematics: Applications & Interpretation Paper 1 · HL (short questions) · exams to Nov 2028 practice papers

HL · 110 marks · 2h · calculator. Each generated paper follows the SL & HL Paper 1 · HL (short questions) · exams to Nov 2028 format, with the same number and style of questions as the real exam, and comes with an IB-style mark scheme.

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Sample Paper 1 · HL (short questions) · exams to Nov 2028 questions

Question 1Structured · 7 marksA* challenge

Two boats leave harbour H at the same time. Boat P sails 18 km on a bearing of 050°. Boat Q sails 10 km on a bearing of 135°.

(a)

Write down the size of angle PHQ.

[1]
(b)

Find the distance PQ.

[3]
(c)

Find the bearing of Q from P.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)85°
85° A1
1
(b)19.8 km
cosine rule (M1)
PQ2 = 182 + 102 − 2(18)(10)cos 85° (A1)
19.8 km A1
3
(c)200°
angle HPQ = 30.2° (sine rule) (M1)(A1)
bearing 200° A1
3
Question 2Structured · 7 marksA* challenge

A pond is 25 m long. Its width, w metres, is measured every 5 m along its length.

Distance (m)0510152025
Width (m)0181214100
Table 2.1
(a)

Use the trapezoidal rule to estimate the surface area of the pond.

[3]
(b)

The pond has a constant depth of 1.2 m. One litre of treatment is needed for every 40 m3 of water. Find the number of litres needed, and explain whether this is likely to be enough.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)270 m2
h = 5 (A1)
trapezoidal rule (M1)
270 m2 A1
3
(b)9 litres
volume 324 m3 (A1)
÷ 40 (M1)
9 litres (round up) A1
the edges are curved outwards, so the rule may underestimate; buy a little extra R1
4
Question 3Structured · 6 marks

A storage tank is a cylinder of radius 6 m and height 12 m with a hemisphere on top.

(a)

Find the volume of the tank.

[3]
(b)

Paint costs $4.50 per m2. Find the cost of painting the curved surface and the hemisphere.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)1810 m3
cylinder (M1)
hemisphere (M1)
1810 A1
3
(b)$3053.63
2πrh (M1)
2πr2 (M1)
$3053.63 A1
3
Question 4Structured · 6 marks

Let R be the matrix representing an anticlockwise rotation of 270° about the origin, and E the matrix representing an enlargement, scale factor 3, centre the origin.

(a)

Write down R and E.

[2]
(b)

Find the single matrix representing the rotation followed by the enlargement.

[2]
(c)

A triangle of area 6 units2 is transformed by this matrix. Find the area of its image.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)R = 0 1−1 0, E = 3 00 3
R = 0 1−1 0 A1
E = 3 00 3 A1
2
(b)0 3−3 0
ER (order matters) (M1)
0 3−3 0 A1
2
(c)54
|det| = 9 (M1)
54 A1
2
Question 5Structured · 5 marks

The population of a town is 25000 at the start of 2025 and decreases by 5% each year.

(a)

Find the population at the start of 2033.

[2]
(b)

Find the first year in which the population at the start of the year is less than 15000.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)16586
25000(0.95)8 (M1)
16586 A1
2
(b)2035
25000(0.95)n < 15000 (M1)
n > 9.96 (A1)
2035 A1
3
Question 6Structured · 6 marks

Two voltages are given by V1 = 3sin(2t) and V2 = 6cos(2t). By writing V1 and V2 as the imaginary parts of 3e2ti and 6ie2ti, or otherwise,

express V1 + V2 in the form R sin(2t + θ), where R > 0 and 0 < θ < π2.

[6]
Show answers and mark scheme
PartAnswer and marking pointsMarks
6.71sin(2t + 1.11)
V1 + V2 = Im((3 + 6i)e2ti) M1A1
|3 + 6i| = 6.71 A1
arg(3 + 6i) = 1.11 (M1)A1
6.71sin(2t + 1.11) A1
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