International Baccalaureate
IB Mathematics AI Standard Level questions organised by topic, paper, and session.
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Unlock from $6/mo1. [Maximum mark: 4] Katya approximates p, correct to four decimal places, by using the following expression. 3 + 1 ______ 6 + 13 __ 16 (a) Calculate Katya’s approximation of p, correct to four decimal places. [2] (b) Calculate the percentage error in using Katya’s four decimal place approximation of p, compared to the exact value of p in your calculator. [2]
1. [Maximum mark: 5] The amount, in milligrams, of a medicinal drug in the body t hours after it was injected is given by D(t) = 23(0.85) t , t ≥ 0 . Before this injection, the amount of the drug in the body was zero. (a) Write down (i) the initial dose of the drug. (ii) the percentage of the drug that leaves the body each hour. [3] (b) Calculate the amount of the drug remaining in the body 10 hours after the injection. [2]
1. [Maximum mark: 6] Eduardo believes that there is a linear relationship between the age of a male runner and the time it takes them to run 5000 metres. To test this, he recorded the age, x years, and the time, t minutes, for eight males in a single 5000 m race. His results are presented in the following table and scatter diagram. x, years 18 24 28 36 40 46 52 62 t, minutes 29.4 29.2 31.1 33.6 32.2 33.1 35.2 40.4 0 10 20 30 40 50 60 70 25 30 35 40 45 Age Time x t (a) For this data, find the value of the Pearson’s product-moment correlation coefficient, r . [2] Eduardo looked in a sports science text book. He found that the following information about r was appropriate for athletic performance. Value of | r | Description of the correlation 0 ≤ | r | < 0.4 weak 0.4 ≤ | r | < 0.8 moderate 0.8 ≤ | r | ≤ 1 strong (b) Comment on your answer to part (a), using the information that Eduardo found. [1] (c) Write down the equation of the regression line of t on x , in the form t = ax + b . [1] (This question continues on the following page) 8821 – 7204 (Question 1 continued) A 57-year-old male also ran in the 5000 m race. (d) Use the equation of the regression line to estimate the time he took to complete the 5000 m race. [2]
2. [Maximum mark: 4] Deb used a thermometer to record the maximum daily temperature over ten consecutive days. Her results, in degrees Celsius (C), are shown below. 14 , 15 , 14 , 11 , 10 , 9 , 14 , 15 , 16 , 13 For this data set, find the value of (a) the mode. [1] (b) the mean. [2] (c) the standard deviation. [1]
2. [Maximum mark: 5] An inclined railway travels along a straight track on a steep hill, as shown in the diagram. diagram not to scale x y z B(20, 5, 250) A(140, 15, 0) The locations of the stations on the railway can be described by coordinates in reference to x , y , and z-axes, where the x and y axes are in the horizontal plane and the z-axis is vertical. The ground level station A has coordinates (140 , 15 , 0) and station B, located near the top of the hill, has coordinates (20 , 5 , 250). All coordinates are given in metres. (a) Find the distance between stations A and B. [2] Station M is to be built halfway between stations A and B. (b) Find the coordinates of station M. [2] (c) Write down the height of station M, in metres, above the ground. [1] (This question continues on the following page) 2221 – 7214 (Question 2 continued) Please do not write on this page. Answers written on this page will not be marked. 2221 – 7214
2. [Maximum mark: 5] The Bermuda Triangle is a region of the Atlantic Ocean with Miami (M), Bermuda (B), and San Juan (S) as vertices, as shown on the diagram. diagram not to scale M S B 1670 km 1550 km 1660 km θ The distances between M, B and S are given in the following table, correct to three significant figures. Distance between Miami and Bermuda 1670 km Distance between Bermuda and San Juan 1550 km Distance between San Juan and Miami 1660 km (a) Calculate the value of θ , the measure of angle MŜB . [3] (b) Find the area of the Bermuda Triangle. [2] (This question continues on the following page) 8821 – 7204 (Question 2 continued) Please do not write on this page. Answers written on this page will not be marked. 8821 – 7204
3. [Maximum mark: 6] A piece of candy is made in the shape of a solid hemisphere. The radius of the hemisphere is 6 mm. 6 (a) Calculate the total surface area of one piece of candy. [4] The total surface of the candy is coated in chocolate. It is known that 1 gram of the chocolate covers an area of 240 mm2. (b) Calculate the weight of chocolate required to coat one piece of candy. [2]
3. [Maximum mark: 7] A storage container consists of a box of length 90 cm, width 42 cm and height 34 cm, and a lid in the shape of a half-cylinder, as shown in the diagram. The lid fits the top of the box exactly. The total exterior surface of the storage container is to be painted. Find the area to be painted. diagram not to scale 42 90 34
3. [Maximum mark: 4] Natasha carries out an experiment on the growth of mould. She believes that the growth can be modelled by an exponential function P(t ) = Ae kt , where P is the area covered by mould in mm2, t is the time in days since the start of the experiment and A and k are constants. The area covered by mould is 112 mm2 at the start of the experiment and 360 mm2 after 5 days. (a) Write down the value of A . [1] (b) Find the value of k . [3]
4. [Maximum mark: 7] The price of gas at Leon’s gas station is $1.50 per litre. If a customer buys a minimum of 10 litres, a discount of $5 is applied. This can be modelled by the following function, L , which gives the total cost when buying a minimum of 10 litres at Leon’s gas station. L(x) = 1.50x - 5 , x ≥ 10 where x is the number of litres of gas that a customer buys. (a) Find the total cost of buying 40 litres of gas at Leon’s gas station. [2] (b) Find L -1(70) . [2] The price of gas at Erica’s gas station is $1.30 per litre. A customer must buy a minimum of 10 litres of gas. The total cost at Erica’s gas station is cheaper than Leon’s gas station when x > k . (c) Find the minimum value of k . [3]
4. [Maximum mark: 7] 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) Write down the size of the angle of depression from H to C. [1] (b) Find the distance from A to C. [2] (c) Find the distance from B to C. [3] (d) Find Minta’s speed, in metres per hour. [1]
4. [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 – 7204 (Question 4 continued)
5. [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 – 7209 (Question 5 continued)
5. [Maximum mark: 5] The number of sick days taken by each employee in a company during a year was recorded. The data was organized in a box and whisker diagram as shown below: 0 5 10 15 20 Number of sick days (a) For this data, write down (i) the minimum number of sick days taken during the year. (ii) the lower quartile. (iii) the median. [3] Paul claims that this box and whisker diagram can be used to infer that the percentage of employees who took fewer than six sick days is smaller than the percentage of employees who took more than eleven sick days. (b) State whether Paul is correct. Justify your answer. [2]
5. [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]
6. [Maximum mark: 5] Arriane has geese on her farm. She claims the mean weight of eggs from her black geese is less than the mean weight of eggs from her white geese. She recorded the weights of eggs, in grams, from a random selection of geese. The data is shown in the table. Weights of eggs from black geese 136 134 142 141 128 126 Weights of eggs from white geese 135 138 141 140 136 134 In order to test her claim, Arriane performs a t-test at a 10 % level of significance. It is assumed that the weights of eggs are normally distributed and the samples have equal variances. (a) State, in words, the null hypothesis. [1] (b) Calculate the p-value for this test. [2] (c) State whether the result of the test supports Arriane’s claim. Justify your reasoning. [2]
6. [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 – 7214 (Question 6 continued)
6. [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]
7. [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]
7. [Maximum mark: 5] 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 the value of f −1(0) . [2]
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