CARDINAL NEWMAN CATHOLIC SCHOOL Mathematics Calculator Paper 3 PRACTICE Yr11 Nov Pre Public Exam Name : Subject Teacher : Examination Instructions Tutor Group: 11 Tutor: For Examiners Use Use black ink or black ball-point pen. Draw diagrams in pencil. Answer all questions. You must answer the questions in the spaces provided. Do not write outside the answer space around each page or on blank pages. Do all rough work in this book. Cross through any work you do not want to be marked. Examination Information The marks for questions are shown in brackets. The maximum mark for this paper is 80. You have 80 minutes to complete the paper. You may ask for more answer paper, graph paper and tracing paper. These must be tagged securely to this answer book. In all calculations, show clearly how you work out your answer. Question Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 22 23 24 25 26 27 Total Marks Percentage Grade 1-9: Target Range (Please Delete) Mark Awarded 3-4 5-6 7-9
Self-Assessment and Reflection Paper 3 Calculator Mathswatch Clip Corbett Maths Clip Q1.Adding Vectors 174 353a Q2.Cube Numbers 181 212 Q3. Changing the Subject 136 8 Q4. Bearings 124 26 Q5. Relative Frequency 125 248 Q6. Solving Inequalities 139 178 Q7. Error Intervals 155 377 Q8. Congruent 12 66 Q9. Conditional Probability 204 247 Q10. Angles in parallel lines 120 25 Q11.Sharing in each ratio 106 271 Q12. Index Law 82 174 Q13. Means 62 53 Q14.Proprtionality 199 254 Q15.Depth Graphs - 171 Q16. Percentage Decrease 108 238 Q17. Speed, distance, time 142 299 Q18. Exponential Functions 194 345 Q19. Circle Theorems 183 64 Q20. Error Intervals 155 377 Q21. Quadratic Inequalities 212 378 Q22. Quadratic Sequences 213 388a 388b and 388c Q23. Turning Points 160 - Q24. Acceleration Graph 216 - Q25. Application of Pythagoras 217 261 Q26. Area of compound shapes 119 41 Q27. Proof 193 365 Areas of Strength Areas for development and feedback Page 2
Q1. Here are two column vectors. f = g = Work out 3f 2g............ Answer... (Total 2 marks) Q2.(a) Write down the value of 10 3... Answer... (1) (b) Work out the value of 0.4 0.2... Answer... (1) (Total 2 marks) Q3.Rearrange the formula 3c = to make d the subject............. Page 3
...... Answer... (Total 4 marks) Q4. Here is a map of France. Scale: 1 cm represents 80 km (a) What is the three-figure bearing of Lyon from Bordeaux? Circle your answer. 005 085 095 175 (1) Page 4
(b) Work out the actual straight-line distance from Paris to Marseille. Answer...km (2) (Total 3 marks) Q5. (a) The arrow on this spinner is equally likely to land on each section. The arrow is spun 72 times. How many times do you expect the arrow to land on 4?... Answer... (2) (b) An arrow on a different spinner is spun 250 times. Some of the results are shown below. Number shown 1 2 3 4 5 Frequency 25 53 62 The relative frequency of landing on a 4 is the same as the relative frequency of landing on a 5 Work out the relative frequency of landing on a 4... Page 5
... Answer... (3) (Total 5 marks) Q6.(a) Solve 5x 2 < 6...... Answer... (2) (b) List the whole number values of n that satisfy 1.5 < n 6...... Answer... (2) (Total 4 marks) Q7.(a) Garage A sold 4960 vehicles. The garage takes a sample of customers, stratified by type of vehicle sold. Some information about the sample is shown. Car People carrier Van Total Number sold 2520 4960 Number in sample 126 44 Complete the table. (3) Page 6
(b) Garage B sold 3790 vehicles, to 3 significant figures. Write down the minimum and maximum possible number sold by Garage B. Minimum... Maximum... (2) (Total 5 marks) Q8. (a) These triangles are congruent. Not drawn accurately State the condition they satisfy. Answer... (1) (b) These triangles are congruent. Not drawn accurately State the condition they satisfy. Answer... Page 7
(1) (Total 2 marks) Q9. A bag contains n beads. One bead is black and the rest are white. Two beads are taken from the bag at random. (a) Show that the probability that both beads are white is (2) (b) The probability that both beads are white is greater than 0.9 Work out the least possible value of n. Answer... (3) (Total 5 marks) Q10. Page 8
(a) Which angles are vertically opposite? Circle your answer a and b a and c b and c b and d c and d (1) (b) Which angles are alternate? Circle your answer a and b a and c b and c b and d c and d (1) (c) Which angles are corresponding? Circle your answer a and b a and c b and c b and d c and d (1) (Total 3 marks) Q11. The table shows the ratio of teachers to children needed for two activities. teachers : children Climbing 1 : 4 Page 9
Walking 1 : 9 (a) There are 7 teachers to take children climbing. What is the greatest number of children that can go climbing?... Answer... (1) (b) 49 children want to go walking. What is the smallest number of teachers needed?... Answer... (1) (Total 2 marks) Q12. Write as a single power of 9 Answer... (Total 2 marks) Q13. Adam and six other men ran a race. The times, in seconds, of the six other men are shown. 9.75 9.79 9.80 9.88 9.94 9.98 The mean time for all seven men was 9.83 seconds. Did Adam win the race? Page 10
You must show your working......................... (Total 3 marks) Q14.M is directly proportional to r 3 When r = 5, M = 200 (a) Work out the value of M when r = 8.................. Answer... (4) (b) Work out the value of r when M = 3125.................. Page 11
Answer... (3) (Total 7 marks) Q15. In an experiment, different masses are hung on a spring. The length of the spring is measured for each mass. Mass (g) 10 20 30 40 Length (cm) 20.8 21.6 22.4 23.2 (a) Draw a graph to show the length of the spring for masses from 10 g to 40 g Page 12
(2) (b) Estimate the length of the spring with no mass hung on it. Answer... cm (1) (c) How much longer is the spring with a 35 g mass than with a 15 g mass?......... Answer... cm (2) (Total 5 marks) Page 13
Q16. (a) Which calculation works out the total amount after decreasing 50 by 8%? Circle the correct answer. 50 0.08 50 0.92 (1) (b) Adrian is going on holiday. He has two bags. The mass of one bag is 9 kg This is 45% of the total mass of the two bags. What is the mass of his other bag?............... Answer... kg (3) (Total 4 marks) Q17. The speed-time graph for a car s journey is shown. Page 14
(a) Estimate the acceleration at 6 seconds. You must show your working. Answer...m/s 2 (3) (b) Estimate the average speed of the car for the journey. You must show your working. Page 15
Answer...m/s (4) (c) Evaluate your answer to part (b). Tick a box. underestimate exact overestimate Comment... (1) (Total 8 marks) Q18. A painting has a value of 2000 The value increases at a rate of 8% per year. The value, V pounds, of the painting after x years is V = 2000 1.08 x (a) Complete the table of values. Values of V are given to the nearest 100 x 0 5 10 15 20 Page 16
V 2000 2900 4300 6300 (1) (b) Draw the graph of V = 2000 1.08 x for x values from 0 to 20 (2) (c) Use the graph to estimate the number of years it takes for the painting to have a value of Page 17
5000 You must show your working. Answer... years (2) (Total 5 marks) Q19. Points A, B, C and D are on the circumference of the circle. CDE is a straight line. Not drawn accurately (a) Work out the size of angle BCD. Give a reason for your answer. Answer... degrees Reason... (2) (b) Work out the size of angle ABC.... Answer... degrees (1) (Total 3 marks) Q20.Luke has a rectangular garden. The length is 40 m Page 18
The width is 25 m Both measurements are given to the nearest metre. Mira also has a garden. The area is 970 m 2 to the nearest 10 m 2 Mira thinks her garden has a bigger area. Is she correct? Tick a box. You must show your working. Correct Incorrect Cannot tell............... (Total 3 marks) Q21.The diagram shows a ball being thrown. It is thrown from a height h metres above level ground. It lands 7.5 metres from where it was thrown. Not drawn accurately The path of the ball can be modelled by the equation y = (2x + 1)(2x 15) Page 19
The sketch shows the graph of the equation. (a) Work out the value of h. You must show your working.......... Answer... (2) (b) Show that the maximum height reached by the ball is metres. Use the symmetry of the graph to help you. You must show your working................ Page 20
............... (2) (Total 4 marks) Q22. Work out the next term of this quadratic sequence. 4 12 24 40... Answer... (Total 2 marks) Q23. The graph y = a + bx x 2 is shown. Page 21
(a) Circle the coordinates of the turning point of the curve. ( 2, 0) (0, 12) (2, 16) (6, 0) (1) (b) Circle the value of a. 2 12 16 6 (1) (c) Circle the two roots of a + bx x 2 = 0 2 and 6 2 and 6 2 and 6 2 and 6 Page 22
(1) (Total 3 marks) Q24. Here is the velocity-time graph of a car for 50 seconds. (a) Work out the average acceleration during the 50 seconds. Give the units of your answer.......... Answer... (2) (b) Estimate the time during the 50 seconds when the instantaneous acceleration = the average acceleration You must show your working on the graph.... Answer... (2) (Total 4 marks) Page 23
Q25. A runner starts at A. She follows this route. A B C D A Three of the distances are shown on the diagram. Not drawn accurately Work out the distance, in kilometres, from D to A......................... Answer... km (Total 5 marks) Q26. Work out the area of this pentagon. Not drawn accurately Page 24
.................. Answer... cm 2 (Total 3 marks) Q27. ABCD is a parallelogram. CE = CF Prove that y = x Page 25
................................. (Total 5 marks) Page 26
. or or SC1 Answer or [2] M2.(a) 1000 (b) 0.08 Additional Guidance Accept use of comma eg 0,08 Accept or 0.080 or 0.0800 [2] M3.3cd = 4(c d) or 3c = 3cd = 4c 4d Page 27
3cd + 4d = 4c or d(3c + 4) = 4c Alternative method 3c = 3c = 3c + 4 = or [4] M4. (a) 085 (b) [8, 8.4] May be implied by correct answer Page 28
[640, 672] ft their [8, 8.4] 50 ft [3] M5. (a) or 72 6 or 12 or 72 6 2 24 Additional Guidance 24 out of 72 A0 2 out of 6 or 1 out of 3 M0 (b) 250 25 53 62 or 110 their 110 2 or 55 or 1 0.56 or 0.44 dep ignore fw Additional Guidance Page 29
55 in table A0 Do not allow misreads for 250 [5] M6.(a) 5x < 6 + 2 or 5x < 8 or 1.6 seen x < Additional Guidance Sight of 1.6 or score (b) 2, 3, 4, 5, 6 for one extra or one missing eg 2, 3, 4, 5 1, 2, 3, 4, 5, 6 2, 3, 4, 5, 6, 7 2, 3, 5, 6 B2 [4] M7.(a) 2520 126 or 20 or Page 30
126 2520 or 0.05 44 their 20 or 44 their 0.05 or 4960 their 20 or 4960 their 0.05 or 880 or 248 M2 44 126 2520 or 4960 2520 126 dep 2520 880 1560 4960 126 44 78 248 (b) (minimum) 3785 (maximum) 3794 SC1 correct answers interchanged [5] M8. (a) SAS or Side, Angle, Side or two sides and the included angle Additional Guidance 2 sides and included angle Page 31
2 sides and angle B0 (b) RHS or Right angle, Hypotenuse, Side e.g. RSH [2] M9. (a) or with cancelling shown (b) > 0.9 or n 2 > 0.9n 0.1n > 2 or n > 20 dep 21 SC1 n = 20 [5] 0.(a) a and b (b) b and c (c) a and c [3] Page 32
1. (a) 28 (b) 6 [2] 2. (9⁵ 9⁷ =) 9¹² or 9 ( ¹ ) 9⁷ or 9⁵ 9³ or 5 + 7 or 5 4 or 7 4 9⁸ SC1 9³¹ [2] 3. 9.83 7 or 68.81 their 68.81 9.75 9.79 9.80 9.88 9.94 9.98 or 9.67 dep 9.67 and Yes [3] 4. (a) M r 3 or M r 3 = k or M = r 3 k Accept any letter for k 200 = k 5 3 or (k =) or k = 1.6 dep Page 33
8 3 8 3 their 1.6 or 8 3 their k 819.2 or 819 (b) 3125 = r 3 their Accept 3125 = r 3 their 1.6 Accept or dep 12.5 [7] 5. (a) (10, 20.8), (20, 21.6), (30, 22.4) and (40, 23.2) plotted Straight line through their points ft line of best fit following plotting error ft (b) [19.9, 20.1] Page 34
(c) Alternative method 1 21.2 or 22.8 1.6 ft their graph ft Alternative method 2 (20.8 + 21.6) 2 or 21.2 or (22.4 + 23.2) 2 or 22.8 1.6 Alternative method 3 23.2 21.6 or 22.4 20.8 or 21.6 20 or (22.4 21.6) 2 or (23.2 22.4) 2 Finds the difference for any two masses 20 kg apart or Doubles the difference for any two masses 10 kg apart 1.6 [5] 6. (a) 50 0.92 Alternative method 1 (b) 9 0.45 or 20 or 9 45 or 0.2 5% = 1 (kg) or 1% = 0.2 (kg) or 10% = 2 (kg) Page 35
their 20 9 or their 0.2 55 55 5 or 9 + 2 11 dep Alternative method 2 e.g. y : 9 = 55 : 45 dep 11 [4] 7. (a) Draws a tangent at t = 6 for their tangent Correct answer for their tangent ft (b) Attempts to work out area below straight lines eg 4 12 or 48 and 6 12 or 36 Attempts to work out estimate of area under the curve eg 14 12 or 84 14 (4 + 12) or 112 Page 36
their total distance 24 Their answer worked out correctly with no errors in area below straight lines Their area must be in the range [168, 196] (c) Correct box ticked with suitable comment ft their answer to part (b) eg their (b) 168 Underestimate ticked and triangle less than area under curve their (b) 196 Overestimate ticked and trapezium more than area under curve ft [8] 8. (a) 9300 (b) Plots the 4 given points Within half a square May also plot their (20, 9300) Joins these points with a smooth curve Within half a square May also join to their (20, 9300) (c) Line from 5000 to their graph or Mark on x-axis corresponding to V = 5000 on their graph or Mark on curve corresponding to V = 5000 on their graph 12 Page 37
SC1 12 with no working on graph seen ft [5] 9. (a) 108 Opposite angle of a cyclic quadrilateral Strand (i) (add up to 180) Must have 108 Q1 (b) 125 Additional Guidance Must see opposite and cyclic ( e.g. quadrilateral in a circle) [3] M20.39.5 or 24.5 or 40.5 or 25.5 or 965 or 975 One correctly evaluated trial using at least one bound or one correctly evaluated trial giving an answer in range 965 to 975 eg 39.5 24.5 = 967(.75) or 39.7 24.5 = 972(.65) or 40.5 25.5 = 1032(.75) Trial values must be in range of bounds Ticks cannot tell and 965 seen and One correctly evaluated trial giving an answer in range 965 to 970 Page 38
or Ticks cannot tell and 975 seen and One correctly evaluated trial giving an answer in range 970 to 975 eg 967.75 eg 972.6 Alternative method 1 One correctly evaluated trial giving an answer below 970 (or their value [965, 975]) One correctly evaluated trial giving an answer below 970 (or their value [965, 975]) and One correctly evaluated trial giving an answer above 970 (or their value [965, 975]) dep Ticks cannot tell and One correctly evaluated trial giving an answer below 970 (or their value [965, 975]) and One correctly evaluated trial giving an answer above 970 (or their value [965, 975]) eg 967.75 and 1032.75 or 967.75 and 1000 or 967.75 Additional Guidance Page 39
Trial values must be within range of bounds, e.g. 39.5 26 = 1027 scores M0 25 40 = 1000 on its own scores zero but see Alt method 2 [3] M21.(a) Substitution of x = 0 into equation 1 15 1 (b) (2 3.5 + 1)(2 3.5 15) for (2 their midpoint + 1)(2 their midpoint 15) or substituting into their expanded expression. for graph intersects at x = 0.5 or midpoint = 3.5 B2 [4] M22. + 8, + 12, + 16 seen or implied or 40 + 20 60 [2] M23. (a) (2, 16) (b) 12 (c) 2 and 6 Page 40
[3] M24. (a) 0.6 or fraction Accept 36 m/s per min m/s² Accept m/s per min only if their acceleration is 36 m/s per min (b) Chord from (0, 0) to (50, 30) and attempt at tangent to curve that is parallel to chord [11, 14] Must see working on the graph [4] M25. 2.5² 2² or 2.5² = BD² + 2² e.g. working in metres 2.25 or 1.5 3.4² + their 2.25 or 3.4² + their 1.5² or 13.81 Page 41
[3.7, 3.72] Allow as further work [3.7, 3.72] + 2 + 2.5 + 3.4 = [11.6, 11.62] [5] M26. Alternative method 1 10 12 or 120 10 12 or 120 150 Alternative method 2 10 18 or 180 10 18 or 180 Page 42
150 Alternative method 3 150 [3] M27. Angle BCD = 2x Opposite angles of parallelogram are equal Angle FCE = 360 90 90 2x or Angle FCE = 180 2x Angles at a point sum to 360 Angle CFE = y or Angle FCE = 180 2y 180 2x + y + y = 180 eg 2y + FCE = 180 Isosceles triangle Angles in a triangle sum to 180 Page 43
2y = 2x y = x All reasons must be stated [5] Page 44
E2.Part (a) was usually correct with 30 being a common incorrect answer. The correct answer to part (b) was rarely seen with incorrect answers of 0.8 or 0.6 common. E3.This question was a good discriminator of the more able students. Algebraic misconceptions were common. Many successfully obtained 3cd = 4c 4d but some students, instead of collecting terms with d on the left hand side, subtracted 3c to get d = c 4d. Some reached 3cd +4d = 4c but did not factorise with common incorrect next steps being: dividing by 3c and getting d + 4d = or subtracting 3c to get d + 4d = c. E5. Part (a) of this question was well answered. In part (b) the majority of students gave 55 as their final answer. E6.This question was not well answered. In part (a) very few students rearranged the inequality correctly. In part (b) common errors were missing 6 from the list, adding 1 to the list and including decimals in the answer. E7.Many students were able to deal with the problem-solving nature of part (a). Weaker students ignored the need for a stratified sample to match the proportions of the population or rounded prematurely. Part (b) was poorly answered with few students able to give both an accurate minimum and maximum. Many gave an accurate value for the minimum but a common error was to treat the data as continuous and give the maximum value as 3795. Page 45
E8. Many students had little knowledge of the conditions for congruency giving answers such as rotation for part (a) and reflection for part (b). E10.All parts of this question were well answered with part (b) the most successful. In part (a) the most popular incorrect answer was 'a and c. In parts (b) and (c), the most popular incorrect answers were 'a and b' in part (b) and 'c and d' in part (c). E12. Some students began by working out 9 5 and 9 7; others multiplied the nines to get 81. For some students the question became an exercise in successive multiplications by 9 in order to work out 9⁵, 9 7 and 9⁴. Those who obtained 9¹² / 9⁴ often simplified this to 9³ or 9¹⁶. The correct answer was rarely seen. E14. Many students appeared unprepared for this question. The most successful used a formal algebraic approach. Those using a ratio method often gave unclear solutions. Common errors in part (a) were to ignore the cube, to give 200 = 125k leading to k = 75, changing r 3 to r 2. In part (b) 3125 40 and working out the cube root of 3125 were quite common. E16. Responses to this question were very mixed. The second most popular answer in part (a) was 50 0.08. In part (b) many students gave the mass of both bags as their final answer. Others divided 45 by 9 to give 5 and could not progress. E18. Page 46
In part (a) most students evaluated V but some did not round to the nearest 100. Plotting the points was usually completed accurately in part (b). Some joined the points with lines and some curves were out of tolerance. In part (c) most showed their method on the graph but using the horizontal scale incorrectly was quite a common error. E19. In part (a), approximately half of the students worked out the angle to be 108 and approximately half of those gave a completely correct reason. 72 was the common incorrect answer. Less than half of students correctly answered part (b) E20.Although there were many ways to approach this question, most students were unsuccessful and made no significant progress. Many students compared 25 40 = 1000 with 970. E21.This challenging question was not well answered. There were many non-attempts with over half the students not attempting part (b). Part (a) was more successful than part (b). The main errors in part (a) were to attempt to expand the brackets or to substitute x = 7.5. In part (b), the instruction to use the symmetry of the graph helped students to progress. Often the midpoint was usually taken as 3.75. E25. Nearly all successful attempts used two applications of Pythagoras theorem. Although trigonometric methods were acceptable, most who used this approach made errors. Pythagoras theorem was sometimes incorrectly applied with terms being transposed. Some worked out BD or BD² correctly but made errors when using these in triangle ABD. Page 47
E26. This question was quite well answered by approximately half of the students. Common errors were to attempt to work out the area of the rectangular part only or to use the base multiplied by the height for the area of the triangle leading to a final answer of 180 cm 2. Page 48