Rates of Change GCSE MATHEMATICS. These questions have been taken or modified from previous AQA GCSE Mathematics Papers.

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GCSE MATHEMATICS Rates of Change These questions have been taken or modified from previous AQA GCSE Mathematics Papers. Instructions Use black ink or black ball-point pen. Draw diagrams in pencil.

Information The marks for questions are shown in brackets. The quality of your written communication is specifically assessed in questions that are indicated with an asterisk (*).

1 GCSE MATHEMATICS Rates of Change These questions have been taken or modified from previous AQA GCSE Mathematics Papers. Instructions Use black ink or black ball-point pen. Draw diagrams in pencil. Answer all questions. You must answer the questions in the spaces provided. If your calculator does not have a π button, take the value of π to be 3.14 unless another value is given in the question. Information The marks for questions are shown in brackets. The quality of your written communication is specifically assessed in questions that are indicated with an asterisk (*). Advice Read each question carefully before you start to answer it. In all calculations, show clearly how you work out your answer. Use the number of marks for the question as a guide to the amount of time you need to spend. Look at previous parts of the question, e.g. a), b), c) i) as there may be information there you need to answer later parts. Check your answer is realistic and appropriate. For calculator decimal numbers always write your full calculator display in the working out area and then, if you need to, round the answer on the answer line. This booklet was curated and modified using AQA examination papers between , for thecalculatorguide.com, where you can find many more booklets on further topics. All questions used are reproduced for educational purposes only.

2 1 Four containers are of equal height. These diagrams show the cross section of each container. A B C D Water flows into each container at a constant rate until the container is full. These sketch graphs show how the depth of the water changes with time, for each container. Graph 1 Graph 2 Depth Depth Graph 3 Graph 4 Depth Depth

3 1 (a) Complete this table to match each container to a graph. [2 marks] Container A Graph... Container B Graph... Container C Graph... Container D Graph... 1 (b) Which graph shows that the depth of water increases at a constant rate until the container is full? [1 mark] Answer...

4 2 On the opposite page are five sketch graphs that show depth of water against time. Water is poured into each container at a steady rate. For each part match the container to its graph opposite. 2 (a) Circle your answer. [1 mark] A B C D E 2 (b) Circle your answer. [1 mark] A B C D E 2 (c) Circle your answer. [1 mark] A B C D E

5 A B Depth Depth C D Depth Depth E Depth

6 3 Four empty containers are shown. A B C D Each container is filled with water at a constant rate. Write the letter of each container in the box next to its graph. Leave the two remaining boxes blank. [4 marks] Graph 1 Graph 2 Height Height Graph 3 Graph 4 Height Height Graph 5 Graph 6 Height Height

7 4 The diagram shows a water tank in the shape of a cuboid. 6 cm 10 cm 25 cm The height of the water in the tank is 6 cm Water leaks from the bottom of the tank at the rate of 30 cm 3 per minute. How many minutes will it take the tank to empty? [3 marks] Answer minutes

8 *5 This water tank is a cuboid. 3 m 1.5 m 8 m You are given that 1 cubic metre holds 1000 litres. The tank is five-sixths full of water. Water is leaking from the tank at a rate of 20 litres per minute. How long will it take the tank to empty? Give your answer in hours. [5 marks] Answer... hours

9 6 A tank has a volume of cm 3 6 (a) What is the volume of the tank in litres? Circle your answer. [1 mark] 10.8 litres 108 litres 1080 litres litres 6 (b) Water is poured into the tank at a constant rate. It takes 4 minutes 30 seconds to fill the tank. Work out the rate at which the water is poured in. Give your answer in litres per minute. [2 marks] Answer litres/minute

10 7 A tank is in the shape of a cylinder of radius 15 cm and height 50 cm 50 cm 15 cm 7 (a) Work out the volume of the tank. [3 marks] Answer... cm 3 7 (b) The volume of another tank is cm 3 The tank is empty. The tank is filled at the rate of 0.22 litres a second. How many minutes will it take to fill the tank? [4 marks] Answer... minutes

11 8 The diagram shows an empty container. Each part is a cylinder. 16 cm 12 cm Water is added to the container at a steady rate. The container is full after T seconds. The sketch graph shows the height, in cm, of the water as the container fills. B Height (cm) A 8 (a) State the values of A and B. 0 0 T 10 (seconds) Answer A =..., B =... (2 marks) 8 (b) The water is added at 250 millilitres per second. When full, the container holds 3.25 litres. After how many seconds is the height of the water 20 cm? You must show your working. Answer... seconds (3 marks)

12 9 The diagram shows an empty container of height 17 cm The container consists of a cylinder on a frustum of a cone. 17 cm Water is added to the container at a constant rate for 7 seconds. The sketch graph shows the depth of the water as the container fills. The graph is a curve for the first 5 seconds and a straight line for the next 2 seconds. 17 Not drawn accurately Depth (cm) (seconds)

13 9 (a) Circle the height of the cylinder. [1 mark] 5 cm 8.5 cm 12 cm 17 cm 9 (b) Work out the rate of increase of the depth of water between 5 seconds and 7 seconds. State the units of your answer. [3 marks] Answer...

14 10 Here is a sketch graph showing the height of a candle as it burns. h is the height, in millimetres, of the candle. t is the time, in minutes, after the candle starts burning. h t 10 (a) Work out the rate at which the height of the candle decreases. Give your answer in millimetres per minute. [2 marks] Answer mm/min 10 (b) The relationship between h and t can be written as h = a bt Work out the values of a and b. [2 marks] Answer a = b =

15 10 (c) When the candle is 80 mm high, a new candle is used. Work out the amount of time that the candle burns before a new candle is used. Give your answer in hours and minutes. [4 marks] Answer minutes hours

16 11 A ball is kicked horizontally from the top of a cliff. The top of the cliff is 50 metres above sea level. Top of cliff Path of ball Not drawn accurately 50 m y m Sea level The height of the ball is modelled by the equation y = t 2 y is the height of the ball, in metres, above sea level. t is the time, in seconds, after the ball is kicked. 11 (a) Complete this table of values for y = t 2 Values of y are given to 1 decimal place. [2 marks] t y

17 11 (b) Draw the graph of y = t 2 for values of t from 0 to 3.5 [2 marks] y t (c) Use your graph to estimate the time the ball takes to reach sea level. [1 mark] Answer... seconds

18 12 Sachin throws a ball. The graph shows the height of the ball above the ground, in metres, after he throws it. 5 Height above the ground (metres) (seconds) 4 12 (a) How high above the ground is the ball when Sachin throws it? Answer... m (1 mark) 12 (b) After how many seconds does the ball hit the ground? Answer... s (1 mark) 12 (c) For how many seconds is the ball more than 3 metres above the ground? Answer... s (2 marks)

19 13 The graph shows the temperature, T ( C) of bread, m (minutes) after it is placed in a freezer. T ( C) m (minutes) 13 (a) How many minutes does it take for the temperature to reach 0 C? [1 mark] Answer... min 13 (b) Estimate the rate at which the temperature is decreasing when m = 3 You must show your working. [3 marks] Answer... C per minute

20 14 Asif throws a cricket ball to Ben. The ball is in the air for 5 seconds. The graph shows the height of the ball above the ground Height above ground (metres) (seconds) 4 5

21 14 (a) Give a reason why the graph shows that Ben catches the ball. (1 mark) 14 (b) After how many seconds is the ball at its greatest height? Answer Answer... seconds (1 mark) 14 (c) What is the greatest height of the ball? Answer Answer... metres (1 mark)

22 15 Leroy goes to a gym to exercise. The graph shows his heart rate, H (beats per minute) during 12 minutes of exercise Heart rate, H (beats per minute) , T (minutes)

23 15 (a) What was his heart rate when he started to exercise? Answer... beats per min (1 mark) 15 (b) How many minutes of exercise did it take for him to reach his highest heart rate? Answer... min (1 mark) 15 (c) By drawing a tangent, work out the rate of increase of H when T = 4 You must show your working. Answer... beats per min 2 (3 marks)

24 16 A dish contains some bacteria. An antibiotic is added to the dish. The antibiotic reduces the number of bacteria in the dish. N is the number of bacteria t hours after the antibiotic is added. The relationship between N and t is modelled by N = a t where a is a positive constant. A sketch graph of N = a t is shown. N 0 0 t 16 (a) Show that there are bacteria in the dish when the antibiotic is added. [1 mark]

25 16 (b) There are 6144 bacteria in the dish after 3 hours. Work out the value of a. [2 marks] Answer (c) Show that approximately one-sixth of the bacteria are left in the dish after 8 hours. [1 mark]

26 17 A farmer wants to make a triangular enclosure of area 60 m 2. This graph shows the relationship between the base, b (metres), and the perpendicular height, h (metres), of the triangle Perpendicular height, h (metres) Base, b (metres) 40

27 17 (a) Explain how the graph shows that the area of the triangle is 60 m 2. (2 marks) 17 (b) Complete the graph for values of b up to 40. (2 marks) 17 (c) The farmer decides to make the base twice as long as the perpendicular height. 17 (c) (i) Plot these points on the graph opposite and join them with a straight line. b h (1 mark) 17 (c) (ii) Use your line to write down approximate values for the base and perpendicular height that the farmer will use. Base... m (3 marks) Perpendicular height... m (2 marks)

28 18 The diagram shows an empty cone of radius 1.5 metres and height 4 metres. Sand is poured into the cone at a rate of 0.2 m 3 per minute. Work out the number of minutes it takes to fill the cone. [3 marks] Answer... minutes

Trigonometry Problems

GCSE MATHEMATICS Trigonometry Problems These questions have been taken or modified from previous AQA GCSE Mathematics Papers. Instructions Use black ink or black ball-point pen. Draw diagrams in pencil.