International Baccalaureate
IB Mathematics AI Higher Level questions organised by topic, paper, and session.
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Unlock from $6/mo1. [Maximum mark: 4] George goes fishing. From experience he knows that the mean number of fish he catches per hour is 1.1. It is assumed that the number of fish he catches can be modelled by a Poisson distribution. On a day in which George spends 8 hours fishing, find the probability that he will catch more than 9 fish.
1. [Maximum mark: 7] Two schools are represented by points A(2 , 20) and B (14 , 24) on the graph below. A road, represented by the line R with equation − x + y = 4 , passes near the schools. An architect is asked to determine the location of a new bus stop on the road such that it is the same distance from the two schools. y x 4 8 12 16 4 8 12 20 16 B A 0 20 24 28 24 0 R (a) Find the equation of the perpendicular bisector of [AB] . Give your equation in the form y = mx + c . [5] (b) Determine the coordinates of the point on R where the bus stop should be located. [2] (This question continues on the following page) 2221 – 7211 (Question 1 continued)
1. [Maximum mark: 6] Dilara is designing a kite ABCD on a set of coordinate axes in which one unit represents 10 cm. The coordinates of A, B and C are (2 , 0), (0 , 4) and (4 , 6) respectively. Point D lies on the x-axis. [AC] is perpendicular to [BD] . This information is shown in the following diagram. diagram not to scale x y A B C D 0 (a) Find the gradient of the line through A and C. [2] (b) Write down the gradient of the line through B and D. [1] (c) Find the equation of the line through B and D. Give your answer in the form ax + by + d = 0 , where a , b and d are integers. [2] (d) Write down the x-coordinate of point D. [1] (This question continues on the following page) 8821 – 7201 (Question 1 continued)
2. [Maximum mark: 6] The Voronoi diagram below shows three identical cellular phone towers, T1, T2 and T3. A fourth cellular phone tower, T4 is located in the shaded region. The dashed lines in the diagram below represent the edges in the Voronoi diagram. Horizontal scale: 1 unit represents 1 km. Vertical scale: 1 unit represents 1 km. y x −2 −15 A B T1 T2 T3 1 10 Tim stands inside the shaded region. (a) Explain why Tim will receive the strongest signal from tower T4. [1] Tower T2 has coordinates (−9 , 5) and the edge connecting vertices A and B has equation y = 3 . (b) Write down the coordinates of tower T4. [2] Tower T1 has coordinates (−13 , 3). (c) Find the gradient of the edge of the Voronoi diagram between towers T1 and T2. [3] (This question continues on the following page) 2221 – 7206 (Question 2 continued)
2. [Maximum mark: 7] A function is defined by f (x) = 2 − 12 ____ x + 5 for −7 ≤ x ≤ 7 , x ≠ −5 . (a) Find the range of f . [3] (b) Find an expression for the inverse function f −1(x) . The domain is not required. [3] (c) Write down the range of f −1(x) . [1]
2. [Maximum mark: 5] Inspectors are investigating the carbon dioxide emissions of a power plant. Let R be the rate, in tonnes per hour, at which carbon dioxide is being emitted and t be the time in hours since the inspection began. When R is plotted against t , the total amount of carbon dioxide produced is represented by the area between the graph and the horizontal t-axis. The rate, R , is measured over the course of two hours. The results are shown in the following table. t 0 0.4 0.8 1.2 1.6 2 R 30 50 60 40 20 50 (a) Use the trapezoidal rule with an interval width of 0.4 to estimate the total amount of carbon dioxide emitted during these two hours. [3] The real amount of carbon dioxide emitted during these two hours was 72 tonnes. (b) Find the percentage error of the estimate found in part (a). [2]
3. [Maximum mark: 8] Charlie and Daniella each began a fitness programme. On day one, they both ran 500 m. On each subsequent day, Charlie ran 100 m more than the previous day whereas Daniella increased her distance by 2 % of the distance ran on the previous day. (a) Calculate how far (i) Charlie ran on day 20 of his fitness programme. (ii) Daniella ran on day 20 of her fitness programme. [5] On day n of the fitness programmes Daniella runs more than Charlie for the first time. (b) Find the value of n . [3]
3. [Maximum mark: 6] At Springfield University, the weights, in kg, of 10 chinchilla rabbits and 10 sable rabbits were recorded. The aim was to find out whether chinchilla rabbits are generally heavier than sable rabbits. The results obtained are summarized in the following table. Weight of chinchilla rabbits, kg 4.9 4.2 4.1 4.4 4.3 4.6 4.0 4.7 4.5 4.4 Weight of sable rabbits, kg 4.2 4.1 4.1 4.2 4.5 4.4 4.5 3.9 4.2 4.0 A t-test is to be performed at the 5 % significance level. (a) Write down the null and alternative hypotheses. [2] (b) Find the p-value for this test. [2] (c) Write down the conclusion to the test. Give a reason for your answer. [2]
3. [Maximum mark: 7] Let the function h(x) represent the height in centimetres of a cylindrical tin can with diameter x cm. h(x) = 640 ___ x 2 + 0.5 for 4 ≤ x ≤ 14 . (a) Find the range of h . [3] The function h −1 is the inverse function of h . (b) (i) Find h −1(10) . (ii) In the context of the question, interpret your answer to part (b)(i). (iii) Write down the range of h −1 . [4]
4. [Maximum mark: 6] Professor Wei observed that students have difficulty remembering the information presented in his lectures. He modelled the percentage of information retained, R , by the function R(t) = 100 e −pt , t ≥ 0 , where t is the number of days after the lecture. He found that 1 day after a lecture, students had forgotten 50 % of the information presented. (a) Find the value of p . [2] (b) Use this model to find the percentage of information retained by his students 36 hours after Professor Wei’s lecture. [2] Based on his model, Professor Wei believes that his students will always retain some information from his lecture. (c) State a mathematical reason why Professor Wei might believe this. [1] (d) Write down one possible limitation of the domain of the model. [1]
4. [Maximum mark: 6] The diagram below shows a helicopter hovering at point H, 380 m vertically above a lake. Point A is the point on the surface of the lake, directly below the helicopter. diagram not to scale 25 380 40 Minta H C B A helicopter Minta is swimming at a constant speed in the direction of point A. Minta observes the helicopter from point C as she looks upward at an angle of 25. After 15 minutes, Minta is at point B and she observes the same helicopter at an angle of 40. (a) Find the distance from A to C. [2] (b) Find the distance from B to C. [3] (c) Find Minta’s speed, in metres per hour. [1]
4. [Maximum mark: 6] There are four stations used by the fire wardens in a national forest. On the following Voronoi diagram, the coordinates of the stations are A(6 , 2), B(14 , 2), C(18 , 6) and D(10.8 , 11.6) where distances are measured in kilometres. The dotted lines represent the boundaries of the regions patrolled by the fire warden at each station. The boundaries meet at P(10 , 6) and Q(13 , 7). 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 B C A D x y 14 16 18 P Q To reduce the areas of the regions that the fire wardens patrol, a new station is to be built within the quadrilateral ABCD. The new station will be located so that it is as far as possible from the nearest existing station. (a) Show that the new station should be built at P. [3] The Voronoi diagram is to be updated to include the region around the new station at P. The edges defined by the perpendicular bisectors of [AP] and [BP] have been added to the following diagram. (b) (i) Write down the equation of the perpendicular bisector of [PC]. (ii) Hence draw the missing boundaries of the region around P on the following diagram. [3] (This question continues on the following page) 8821 – 7201 (Question 4 continued) 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 B C A D x y 14 16 18 P
5. [Maximum mark: 6] A garden has a triangular sunshade suspended from three points A(2 , 0 , 2), B(8 , 0 , 2) and C(5 , 4 , 3), relative to an origin in the corner of the garden. All distances are measured in metres. (a) (i) Find ⟶ CA . (ii) Find ⟶ CB . [2] (b) Find ⟶ CA × ⟶ CB . [2] (c) Hence find the area of the triangle ABC. [2]
5. [Maximum mark: 5] Roger buys a new laptop for himself at a cost of £495. At the same time, he buys his daughter Chloe a higher specification laptop at a cost of £2200. It is anticipated that Roger’s laptop will depreciate at a rate of 10 % per year, whereas Chloe’s laptop will depreciate at a rate of 15 % per year. (a) Estimate the value of Roger’s laptop after 5 years. [2] Roger and Chloe’s laptops will have the same value k years after they were purchased. (b) Find the value of k . [2] (c) Comment on the validity of your answer to part (b). [1]
5. [Maximum mark: 9] In this question, give all answers correct to 2 decimal places. Raul and Rosy want to buy a new house and they need a loan of 170 000 Australian dollars (AUD) from a bank. The loan is for 30 years and the annual interest rate for the loan is 3.8 %, compounded monthly. They will pay the loan in fixed monthly instalments at the end of each month. (a) Find the amount they will pay the bank each month. [3] (b) (i) Find the amount Raul and Rosy will still owe the bank at the end of the first 10 years. (ii) Using your answers to parts (a) and (b)(i), calculate how much interest they will have paid in total during the first 10 years. [6]
6. [Maximum mark: 5] A triangular field ABC is such that AB = 56 m and BC = 82 m , each measured correct to the nearest metre, and the angle at B is equal to 105, measured correct to the nearest 5. diagram not to scale C A B 82 56 105 Calculate the maximum possible area of the field.
6. [Maximum mark: 7] Consider the function f (x) = ax 2 + bx + c . The graph of y = f (x) is shown in the diagram. The vertex of the graph has coordinates (0.5 , −12.5) . The graph intersects the x-axis at two points, (−2 , 0) and ( p , 0) . diagram not to scale y x (0.5 , –12.5) (–2 , 0) (p , 0) (a) Find the value of p . [1] (b) Find the value of (i) a . (ii) b . (iii) c . [5] (c) Write down the equation of the axis of symmetry of the graph. [1] (This question continues on the following page) 2221 – 7211 (Question 6 continued)
6. [Maximum mark: 5] An infinite geometric sequence, with terms u n , is such that u 1 = 2 and ∑ k=1 ∞ u k = 10 . (a) Find the common ratio, r , for the sequence. [2] (b) Find the least value of n such that u n < 1 _ 2 . [3]
7. [Maximum mark: 8] Ellis designs a gift box. The top of the gift box is in the shape of a right-angled triangle GIK. A rectangular section HIJL is inscribed inside this triangle. The lengths of GH, JK, HL, and LJ are p cm, q cm, 8 cm and 6 cm respectively. diagram not to scale H G L K J I q p 8 6 The area of the top of the gift box is A cm2. (a) (i) Find A in terms of p and q . (ii) Show that A = 192 ___ q + 3q + 48 . [4] (b) Find dA ___ dq . [2] Ellis wishes to find the value of q that will minimize the area of the top of the gift box. (c) (i) Write down an equation Ellis could solve to find this value of q . (ii) Hence, or otherwise, find this value of q . [2] (This question continues on the following page) 2221 – 7206 (Question 7 continued)
7. [Maximum mark: 6] A meteorologist models the height of a hot air balloon launched from the ground. The model assumes the balloon travels vertically upwards and travels 450 metres in the first minute. Due to the decrease in temperature as the balloon rises, the balloon will continually slow down. The model suggests that each minute the balloon will travel only 82 % of the distance travelled in the previous minute. (a) Find how high the balloon will travel in the first 10 minutes after it is launched. [3] (b) The balloon is required to reach a height of at least 2520 metres. Determine whether it will reach this height. [2] (c) Suggest a limitation of the given model. [1]
Revision method
Practise 409 authentic past-paper questions from 48 archived papers (2021-2025). These are past-paper questions, not AI-generated exam-style substitutes. Start with a topic, then switch to mixed or timed practice.