IB Questionbank Test



1.4 1a. [1 mark] A solid glass paperweight consists of a hemisphere of diameter 6 cm on top of a cuboid with a square base of length 6?cm, as shown in the diagram.The height of the cuboid, x?cm, is equal to the height of the hemisphere.Write down the value of x. 1b. [3 marks] Calculate the volume of the paperweight. 1c. [2 marks] 1?cm3 of glass has a mass of 2.56?grams.Calculate the mass, in grams, of the paperweight. 2a. [2 marks] A solid right circular cone has a base radius of 21 cm and a slant height of 35 cm.A smaller right circular cone has a height of 12 cm and a slant height of 15 cm, and is removed from the top of the larger cone, as shown in the diagram.Calculate the radius of the base of the cone which has been removed. 2b. [2 marks] Calculate the curved surface area of the cone which has been removed. 2c. [2 marks] Calculate the curved surface area of the remaining solid. 3a. [2 marks] Julio is making a wooden pencil case in the shape of a large pencil. The pencil case consists of a cylinder attached to a cone, as shown.The cylinder has a radius of r cm and a height of 12 cm.The cone has a base radius of r cm and a height of 10 cm.Find an expression for the slant height of the cone in terms of r. 3b. [4 marks] The total external surface area of the pencil case rounded to 3 significant figures is 570 cm2.Using your graphic display calculator, calculate the value of r. 4a. [3 marks] A park in the form of a triangle, ABC, is shown in the following diagram. AB is 79?km and BC is 62?km. Angle AC is 52°.Calculate the length of side AC in km. 4b. [3 marks] Calculate the area of the park. 5a. [5 marks] The Tower of Pisa is well known worldwide for how it leans.Giovanni visits the Tower and wants to investigate how much it is leaning. He draws a diagram showing a non-right triangle, ABC.On Giovanni’s diagram the length of AB is 56 m, the length of BC is 37 m, and angle ACB is 60°. AX is the perpendicular height from A to BC.Use Giovanni’s diagram to?show that angle ABC, the angle at which the Tower is leaning relative to thehorizontal, is 85° to the nearest degree. 5b. [2 marks] Use Giovanni's diagram to calculate the length of AX. 5c. [2 marks] Use Giovanni's diagram to find the length of BX, the horizontal displacement of the Tower. 5d. [2 marks] Giovanni’s tourist guidebook says that the actual horizontal displacement of the Tower, BX, is 3.9 metres.Find the percentage error on Giovanni’s diagram. 5e. [3 marks] Giovanni adds a point D to his diagram, such that BD = 45 m, and another triangle is formed.Find the angle of elevation of A from D. 6a. [3 marks] The speed of light is kilometres per second. The average distance from the Sun to the Earth is 149.6 million km.Calculate the time, in minutes, it takes for light from the Sun to reach the Earth. 6b. [3 marks] A light-year is the distance light travels in one year and is equal to million km. Polaris is a bright star, visible from the Northern Hemisphere. The distance from the Earth to Polaris is 323 light-years.Find the distance from the Earth to Polaris in millions of km. Give your answer in the form with and . 7a. [2 marks] A type of candy is packaged in a right circular cone that has volume and vertical height 8 cm.Find the radius, , of the circular base of the cone. 7b. [2 marks] Find the slant height, , of the cone. 7c. [2 marks] Find the curved surface area of the cone. 8a. [2 marks] The following table shows the average body weight, , and the average weight of the brain, , of seven species of mammal. Both measured in kilograms (kg).Find the range of the average body weights for these seven species of mammal. 8b. [2 marks] For the data from these seven species calculate , the Pearson’s product–moment correlation coefficient; 8c. [2 marks] For the data from these seven species describe the correlation between the average body weight and the average weight of the brain. 8d. [2 marks] Write down the equation of the regression line on , in the form . 8e. [2 marks] The average body weight of grey wolves is 36 kg.Use your regression line to estimate the average weight of the brain of grey wolves. 8f. [2 marks] In fact, the average weight of the brain of grey wolves is 0.120 kg.Find the percentage error in your estimate in part (d). 8g. [2 marks] The average body weight of mice is 0.023 kg.State whether it is valid to use the regression line to estimate the average weight of the brain of mice. Give a reason for your answer. 9a. [3 marks] A pan, in which to cook a pizza, is in the shape of a cylinder. The pan has a diameter of 35 cm and a height of 0.5 cm.Calculate the volume of this pan. 9b. [4 marks] A chef had enough pizza dough to exactly fill the pan. The dough was in the shape of a sphere.Find the radius of the sphere in cm, correct to one decimal place. 9c. [2 marks] The pizza was cooked in a hot oven. Once taken out of the oven, the pizza was placed in a dining room.The temperature, , of the pizza, in degrees Celsius, °C, can be modelled bywhere is a constant and is the time, in minutes, since the pizza was taken out of the oven.When the pizza was taken out of the oven its temperature was 230 °C.Find the value of . 9d. [2 marks] Find the temperature that the pizza will be 5 minutes after it is taken out of the oven. 9e. [3 marks] The pizza can be eaten once its temperature drops to 45 °C.Calculate, to the nearest second, the time since the pizza was taken out of the oven until it can be eaten. 9f. [1 mark] In the context of this model, state what the value of 19 represents. 10a. [2 marks] A water container is made in the shape of a cylinder with internal height cm and internal base radius cm.The water container has no top. The inner surfaces of the container are to be coated with a water-resistant material.Write down a formula for , the surface area to be coated. 10b. [1 mark] The volume of the water container is .Express this volume in?. 10c. [1 mark] Write down, in terms of and , an equation for the volume of this water container. 10d. [2 marks] Show that . 10e. [3 marks] The water container is designed so that the area to be coated is minimized.Find . 10f. [3 marks] Using your answer to part (e), find the value of which minimizes . 10g. [2 marks] Find the value of this minimum area. 10h. [3 marks] One can of water-resistant material coats a surface area of .Find the least number of cans of water-resistant material that will coat the area in part (g). 11a. [2 marks] Temi’s sailing boat has a sail in the shape of a right-angled triangle, ,angle and angle .Calculate , the height of Temi’s sail. 11b. [2 marks] William also has a sailing boat with a sail in the shape of a right-angled triangle, .. The area of William’s sail is .Calculate , the height of William’s sail. 12a. [4 marks] Assume that the Earth is a sphere with a radius, , of .i) ? ??Calculate the surface area of the Earth in .ii) ? ?Write down your answer to part (a)(i) in the form , where and . 12b. [2 marks] The surface area of the Earth that is covered by water is approximately .Calculate the percentage of the surface area of the Earth that is covered by water. 13a. [1 mark] A snack container has a cylindrical shape. The diameter of the base?is . The height of the container is . This is shown in the following diagram.Write down the radius, in , of the base of the container. 13b. [2 marks] Calculate the area of the base of the container. 13c. [3 marks] Dan is going to paint the curved surface and the base of the snack container.Calculate the area to be painted. 14a. [1 mark] A ladder is standing on horizontal ground and leaning against a vertical wall. The length of?the ladder is metres. The distance between the bottom of the ladder and the base of the?wall is metres.Use the above information to sketch a labelled diagram showing the ground, the ladder?and the wall. 14b. [2 marks] Calculate the distance between the top of the ladder and the base of the wall. 14c. [3 marks] Calculate the obtuse angle made by the ladder with the ground. 15a. [3 marks] When Bermuda , Puerto Rico , and Miami are joined on a map using straight lines,?a triangle is formed. This triangle is known as the Bermuda triangle.According to the map, the distance is , the distance is and?angle is .Calculate the distance from Bermuda to Puerto Rico, .? 15b. [3 marks] Calculate the area of the Bermuda triangle. 16a. [2 marks] Each day a supermarket records the midday temperature and how many cold drinks are sold?on that day. The following table shows the supermarket’s data for the last 6 days. This data?is also shown on a scatter diagram.Write downi) ? ??the mean temperature, ;ii) ? ?the mean number of cold drinks sold, .? 16b. [2 marks] Draw the line of best fit on the scatter diagram. 16c. [2 marks] Use the line of best fit to estimate the number of cold drinks that are sold on a day?when the midday temperature is . 17a. [2 marks] For an ecological study, Ernesto measured the average concentration of the fine dust,?, in the air at different distances from a power plant. His data are represented on?the following scatter diagram. The concentration of ?is measured in micrograms per?cubic metre and the distance is measured in kilometres.His data are also listed in the following table.Use the scatter diagram to find the value of and of in the table. 17b. [4 marks] Calculatei) ? ? ??, the mean distance from the power plant;ii) ? ? ?, the mean concentration of ;iii) ? ? , the Pearson’s product–moment correlation coefficient. 17c. [2 marks] Write down the equation of the regression line on . 17d. [4 marks] Ernesto’s school is located from the power plant. He uses the equation of the?regression line to estimate the concentration of in the air at his school.i) ? ? Calculate the value of Ernesto’s estimate.ii) ? ?State whether Ernesto’s estimate is reliable. Justify your answer. 18a. [1 mark] A distress flare is fired into the air from a ship at sea. The height, , in metres, of the flare?above sea level is modelled by the quadratic functionwhere is the time, in seconds, and at the moment the flare was fired.Write down the height from which the flare was fired. 18b. [2 marks] Find the height of the flare seconds after it was fired. 18c. [2 marks] The flare fell into the sea seconds after it was fired.Find the value of ?. 18d. [2 marks] Find? 18e. [3 marks] i) ? ? Show that the flare reached its maximum height seconds after being fired.ii) ? ?Calculate the maximum height reached by the flare. 18f. [3 marks] The nearest coastguard can see the flare when its height is more than metres above?sea level.Determine the total length of time the flare can be seen by the coastguard. 19a. [3 marks] The Great Pyramid of Giza in Egypt is a right pyramid with a square base. The pyramid is made of solid stone. The sides of the base are long. The diagram below represents?this pyramid, labelled . is the vertex of the pyramid. ?is the centre of the base, . is the midpoint of .?Angle .Show that the length of is metres, correct to three significant figures. 19b. [2 marks] Calculate the height of the pyramid, . 19c. [2 marks] Find the volume of the pyramid. 19d. [2 marks] Write down your answer to part (c) in the form ?where and . 19e. [4 marks] Ahmad is a tour guide at the Great Pyramid of Giza. He claims that the amount of stone?used to build the pyramid could build a wall metres high and ?metre wide stretching from?Paris to Amsterdam, which are apart.Determine whether Ahmad’s claim is correct. Give a reason. 19f. [6 marks] Ahmad and his friends like to sit in the pyramid’s shadow, , to cool down.At mid-afternoon, ?and angle?i) ? ??Calculate the length of ?at mid-afternoon.ii) ? ?Calculate the area of the shadow, , at mid-afternoon. 20a. [4 marks] A boat race takes place around a triangular course, , with , ?and angle . The race starts and finishes at point .Calculate the total length of the course. 20b. [3 marks] It is estimated that the fastest boat in the race can travel at an average speed of .Calculate an estimate of the winning time of the race. Give your answer to the nearest minute. 20c. [3 marks] It is estimated that the fastest boat in the race can travel at an average speed of .Find the size of angle . 20d. [3 marks] To comply with safety regulations, the area inside the triangular course must be kept clear of other boats, and the shortest distance from ?to ?must be greater than ?metres.Calculate the area that must be kept clear of boats. 20e. [3 marks] To comply with safety regulations, the area inside the triangular course must be kept clear of other boats, and the shortest distance from ?to ?must be greater than ?metres.Determine, giving a reason, whether the course complies with the safety regulations. 20f. [2 marks] The race is filmed from a helicopter, , which is flying vertically above point .The angle of elevation of ?from ?is .Calculate the vertical height, , of the helicopter above . 20g. [3 marks] The race is filmed from a helicopter, , which is flying vertically above point .The angle of elevation of ?from ?is .Calculate the maximum possible distance from the helicopter to a boat on the course. 21a. [6 marks] The following diagram shows a perfume bottle made up of a cylinder and a cone.The radius of both the cylinder and the base of the cone is 3 cm.The height of the cylinder is 4.5 cm.The slant height of the cone is 4 cm.(i) ? ? Show that the vertical height of the cone is ?cm correct to three significant figures.(ii) ? ? Calculate the volume of the perfume bottle. 21b. [2 marks] The bottle contains of perfume. The bottle is not full and all of the perfume is in the cylinder part.Find the height of the perfume in the bottle. 21c. [4 marks] Temi makes some crafts with perfume bottles, like the one above, once they are empty. Temi wants to know the surface area of one perfume bottle.Find the total surface area of the perfume bottle. 21d. [4 marks] Temi covers the perfume bottles with a paint that costs 3 South African rand (ZAR) per millilitre. One millilitre of this paint covers an area of .Calculate the cost, in ZAR, of painting the perfume bottle. Give your answer correct to two decimal places. 21e. [2 marks] Temi sells her perfume bottles in a craft fair for 325 ZAR each. Dominique from France buys one and wants to know how much she has spent, in euros (EUR). The exchange rate is 1 EUR = 13.03 ZAR.Find the price, in EUR, that Dominique paid for the perfume bottle. Give your answer?correct to two decimal places. 22a. [2 marks] A cross-country running course consists of a beach section and a forest section. Competitors run from to , then from to and from back to .The running course from to is along the beach, while the course from , through and back to , is through the forest.The course is shown on the following diagram.Angle is .It takes Sarah minutes and seconds to run from to at a speed of .Using ‘distance = speed time’, show that the distance from??to??is metres correct to 3 significant?figures. 22b. [1 mark] The distance from??to??is metres. Running this part of the course takes Sarah minutes and seconds.Calculate the speed, in , that Sarah runs from??to?. 22c. [3 marks] The distance from??to??is metres. Running this part of the course takes Sarah minutes and seconds.Calculate the distance, in metres, from??to?. 22d. [2 marks] The distance from to??is metres. Running this part of the course takes Sarah minutes and seconds.Calculate the total distance, in metres, of the cross-country running course. 22e. [3 marks] The distance from to??is metres. Running this part of the course takes Sarah minutes and seconds.Find the size of angle . 22f. [3 marks] The distance from to??is metres. Running this part of the course takes Sarah minutes and seconds.Calculate the area of the cross-country course bounded by the lines , and . 23a. [2 marks] Ross is a star that is 82 414 080 000 000 km away from Earth. A spacecraft, launched from Earth, travels at 48 000 kmh–1 towards Ross.Calculate the exact time, in hours, for the spacecraft to reach the star Ross. 23b. [2 marks] Give your answer to part (a) in years. (Assume 1 year = 365 days) 23c. [2 marks] Give your answer to part (b) in the form a×10k, where 1 ≤ a < 10 and . 24a. [1 mark] José stands 1.38 kilometres from a vertical cliff.Express this distance in metres. 24b. [3 marks] José estimates the angle between the horizontal and the top of the cliff as 28.3° and uses it to find the height of the cliff.Find the height of the cliff according to José’s calculation. Express your answer in metres, to the nearest whole metre. 24c. [2 marks] José estimates the angle between the horizontal and the top of the cliff as 28.3° and uses it to find the height of the cliff.The actual height of the cliff is 718 metres. Calculate the percentage error made by José when calculating the height of the cliff. 25a. [2 marks] A race track is made up of a rectangular shape by with semi-circles at each end as shown in the diagram.Michael drives around the track once at an average speed of .Calculate the distance that Michael travels. 25b. [4 marks] Calculate how long Michael takes in seconds.Printed for International School of Europe ? International Baccalaureate Organization 2019 International Baccalaureate? - Baccalauréat International? - Bachillerato Internacional? ................
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