CARDINAL NEWMAN CATHOLIC SCHOOL Mathematics Non Calculator Paper 1 PRACTICE Yr11 Nov Pre Public Exam

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1 CARDINAL NEWMAN CATHOLIC SCHOOL Mathematics Non Calculator Paper 1 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 Total Marks Percentage Grade 1-9: Target Range (Please Delete) Mark Awarded

2 Self-Assessment and Reflection Paper 1 Non-Calculator Mathswatch Clip Corbett Maths Clip Q1. Index Law Q2. Congruent Triangles Q3. Geometric Progression Q4. Ratio and fractions a Q5. Product of Primes Q6. Means from a Table Q7. Fractions of an amount Q8. Speed Q9. Density Q10. Simultaneous Equations Q11. Using Percentages Q12. Area of a Circle Q13. Standard Form Q14. Solving Inequalities Q15. Recurring Fractions Q16. Probability Mutually Exclusive outcomes Q17. Gradient of a straight line Q18. Best Buys Q19. Cumulative Frequency Graphs Q20. Equation of A circle Q21. Reflection & Rotation 48 & Q22. Similarity Q23. Velocity time graphs Q24. Fractional Indices Q25. Two way Tables Q26. Expanding Brackets Q27. Tangents to a Circle f Q28. Cones and Frustums 171 & Q29. Exact Trigonometric Ratios Areas of Strength Areas for development and feedback Page 2

3 Q1. Write as a single power of 9 Answer... (Total 2 marks) Q2.ABCD and AEFG are identical squares. CD = EF = 10 cm Angle BAG = 45 Not drawn accurately Prove that triangles AGD and ABE are congruent Page 3

4 (Total 4 marks) Q3.Here are the first five terms of a linear sequence Work out the nth term. Answer... (Total 2 marks) Q4.Work out the value of x when x 20 : x simplifies to 1 : 4 Answer... (Total 4 marks) Q5. Five integers have a mode of 6 a median of 8 a mean of 10 Page 4

5 What is the greatest possible range of the five integers? You must show your working. Answer... (Total 3 marks) Q6.Write 56 as a product of prime factors. Answer... (Total 2 marks) Q7. There are 20 students. 12 are boys. What fraction are boys? Circle your answer. (Total 1 mark) Q8.(a) Three electric cars are tested by driving them around a track until the battery runs out. The table shows some information about their performance. Car Total time travelled Average speed (km/h) Total distance travelled Page 5

6 (hours) (km) A 4 35 B C Complete the table. (3) (b) Two cars are driven around a 10 kilometre track. Both cars leave from the start line at the same time. Car X travels at exactly 40 km/h Car Y travels at exactly 30 km/h How many minutes will it be before they pass the start line together again? Answer... minutes (2) (Total 5 marks) Q9.A scooter is travelling at a constant speed of 75 kilometres per hour. (a) The scooter travels at this speed for 20 minutes. How many kilometres has the scooter travelled in this time? Answer... km (2) Page 6

7 (b) The speed limit is 50 miles per hour. Is the scooter travelling faster or slower than the speed limit? Faster Slower You must show your working. (2) (Total 4 marks) Q10. 2x + 3y = 15.5 x + y = 6 Work out the values of x and y. x =... y =... (Total 3 marks) Q11. The cash price for a boiler is 2000 Customers can pay the cash price or pay monthly. Page 7

8 Cash Price 2000 Pay Monthly 60 monthly payments of 40 Work out the percentage increase from the cash price when paying monthly Answer... % (Total 4 marks) Q12.Which has the greater area? Not drawn accurately Use π = 3.1 You must show your working. Page 8

9 Answer... (Total 3 marks) Q13. Which of these is not a square number? Circle your answer (Total 1 mark) Q14. (a) w and x are whole numbers. w > 40 x < 30 Work out the smallest possible value of w x Answer... (2) (b) y and z are whole numbers. y < 60 z 50 Work out the largest possible value of y + z Page 9

10 Answer... (2) (Total 4 marks) Q15. Write as a recurring decimal. Answer... (Total 2 marks) Q16. A prime number between 300 and 450 is chosen at random. The table shows the probability that the number lies in different ranges. Prime number, n Probability 300 n < n < n < 390 x 390 n < n < (a) Work out the value of x. Page 10

11 Answer... (2) (b) Work out the probability that the prime number is greater than 390 Answer... (1) (c) There are four prime numbers between 300 and 330 How many prime numbers are there between 300 and 450? Answer... (2) (Total 5 marks) Q17.Here is a straight-line graph. Page 11

12 (a) Use the graph to work out the value of x when x = 8... Answer... (1) (b) Work out the gradient of the line Answer... (3) (Total 4 marks) Page 12

13 Q18.Kamil looks at two cars. The cars are part of this offer. The normal price of Car A is 1550 The normal price of Car B is 1950 After all reductions, which car is cheaper, Car A or Car B? You must show your working. Answer... (Total 5 marks) Page 13

14 Q19.The times that 100 customers spent queuing in a post office were recorded. The cumulative frequency diagram shows the results. (a) How many customers queued for more than 15 minutes? Answer... (1) (b) Work out the median queuing time. Answer... minutes (1) (c) A new serving window was opened in the post office. The times that 100 customers spent queuing were then recorded. Page 14

15 The box plot shows the results. Work out the inter-quartile range of these times. Answer... minutes (2) (d) Compare the queuing times before and after the new serving window was opened. Give two comparisons. Comparison 1... Comparison 2... (2) (Total 6 marks) Q20.A circle radius 3 units, centre (7, 5) is shown. Not drawn accurately Page 15

16 Work out the coordinates of any point that lies on the circumference of the circle. You must show your working, which may be on the diagram. Answer (...,... ) (Total 2 marks) Q21.Reflect the triangle in the line y = 2 (Total 2 marks) Page 16

17 Q22.Triangles ABC and PQR are similar. Not drawn accurately Work out the length QR. Answer... cm (Total 2 marks) Q23.Paul travels from Rye to Eston at an average speed of 90 km/h He travels for T hours. Mary makes the same journey at an average speed of 70 km/h She travels for 1 hour longer than Paul. Work out the value of T Answer... hours (Total 4 marks) Page 17

18 Q24. Steph is solving a problem. Cube A has a surface area of 150 cm² Cube B has sides half the length of cube A What is the volume of cube B? To solve this problem, Steph decides to halve the surface area calculate the square root of the answer then divide by 6 then cube this answer to work out the volume. Evaluate Steph s method. (Total 2 marks) Q25. In the table, a, b and c represent numbers. The total for each row is given. Work out the numbers for the column totals. Row totals a a a 12 b b a 24 2a 2c b 30 Column totals Page 18

19 (Total 4 marks) Q26. Expand and simplify (2x + 5)(2x 5)(3x + 7) Answer... (Total 3 mark) Q27. The diagram shows the circle x² + y² + 10 P lies on the circle and has x-coordinate 1 The tangent at P intersects the x-axis at Q. Not drawn accurately Page 19

20 Work out the coordinates of Q. Answer (...,...) (Total 5 marks) Q28.The cone below has radius 3 cm and slant height l cm. Page 20

21 The total surface area, including the base, is 24π cm 2. Work out the length l. Answer... cm (Total 3 marks) Q29. Show that 12 cos 30 2 tan 60 can be written in the form where k is an integer. Page 21

22 (Total 3 marks) Page 22

23 . (9⁵ 9⁷ =) 9¹² or 9 ( ¹ ) 9⁷ or 9⁵ 9³ or or 5 4 or 7 4 9⁸ SC1 9³¹ [2] M2.AD = AE (10 (cm) or sides of a square) or sides marked as 10 on diagram Must give a reason or mark sides as 10 on diagram B1 AB = AG (10 (cm) or sides of a square) or sides marked as 10 on diagram Must give a reason or sides as 10 on diagram B1 Angle DAG = angle EAB (135 or ) Must state 135 or or 135 shown for both angles on diagram B1 Congruent due to SAS (could be expressed in words eg two sides and angle between them the same) or congruent due to ASA or AAS or SAA with 22.5 shown or stated (after 135 seen) as one of the other angles. (could be in words eg two angles and the side between them, or two angles and a side) Q0 for congruent without SAS, AAS etc or the appropriate reason for their proof stated in words (strand (ii)) Q1 [4] Page 23

24 M3.6n + 3 or 3(2n + 1) B1 for 6n Accept 6 n or n 6 but not n6 B1 for n6 + 3 Accept any letter B2 [2] M4.Alternative method 1 or 4(x 20) = x x 80 = x oe 4x x = oe correct expansion of their brackets or division scores M2 or 3x = 360 collecting their four terms scores M3 x = 120 SC3 380 Alternative method 2 x (x 20) (= 3 parts) Page 24

25 300 (= 3 parts) and 100 (= 1 part) scores M2 x 20 = 100 or x = 400 scores M3 x = 120 SC3 380 Alternative method 3 x 20 + x = 5(x 20) 2x = 5x 100 scores M2 3x = 360 scores M3 x = 120 SC3 380 Additional Guidance x 20 = 4(x + 280) M0 x 20 = 4x = 3x [4] M5. 15 from B2 5 integers with at least two criteria mode 6 or median 8 and total 50 do not award B2 for mode and median only Page 25

26 B1 5 integers with any one of these criteria mode 6 median 8 total 50 B3 [3] M6.28 ( ) 2 or 8 ( ) 7 or 14 ( ) 2 ( ) 2 or 2 ( ) 4 ( ) 7 or 2, 2, 2, 7 allow on prime factor tree or repeated division ignore incorrect products if at least one correct product seen or Additional Guidance Ignore any 1 for but not [2] M7. B1 [1] M8.(a) or 4.50 or 4 h 30m 50 B1 each Do not accept 4.30 B3 (b) Indication that car X passes start at 15, 30, 45, 60 mins or Indication that car Y passes start at 20, 40, 60 mins or 15 for X and 20 for Y NB time in hours can score ie ¼, ½, ¾ etc Page 26

27 ¼ for X and ⅓ for Y 60 Answer of 1 hour is, A0 Additional Guidance 60 from wrong work is zero marks but 60 from no work or no incorrect work is full marks [5] M9.(a) or 1.25 km per minute 25 (b) Any correct conversion between miles and km seen, eg 5 miles = 8 km or 1 mile = 1.6 km or 1km = 5 / 8 mile 75 Slower as limit is 80 km Slower as < 50 [4] 0. (2x + 3y = 15.5) (2x + 3y = 15.5) 2x + 2y = 12 3x + 3y = 18 Equates coefficients Page 27

28 y = 3.5 or x = 2.5 oe x = 2.5 and y = 3.5 [3] 1. Alternative method or 2400 oe their or 400 or 2000 their 2400 dep 20(%) oe dep Alternative method or 2400 oe their or 400 or 2000 their % = or 1% = and correctly finds multiplier using build up or division to find percentage equivalent to total their 400 oe Correct build up to find percentage equivalent to total their (their ) or their (2000 their 2400) implies M3 20(%) dep Page 28

29 Alternative method or 2400 dep their or their 1.2(0) 1(.00) or 100 their 120 or 1(.00) their 1.2(0) or 0.2 oe 20(%) dep Additional Guidance 20% on answer line and no working (= 2400) from 5 years scores minimum = 1800 and 200 scores minimum A0 A0 [4] or 27.5 or or 27.9 or 9π Allow Accept as meaning 27.5 cm and 27.9 or Do not accept 9π at this stage as comparison of values cannot be made without evaluation to a number. Correct conclusion based on both their areas using correct methods with at least one correct area Strand (iii) Ignore any incorrect subtraction of Page 29

30 Q1 Additional Guidance Indication of which is bigger shape can be done by the name, the value or the calculation = = 27.9 Circle Both methods, one value incorrect, correct conclusion using name of shape, A0, Q = = = Both methods, one value incorrect, correct conclusion using calculation, A0, Q = = 18.6 Rectangle Both methods, correct conclusion but Q0 as both values incorrect., A0, Q = = = One method correct, Q0 as one method wrong, therefore one value wrong = = = 27.9 Circle bigger by 0.3 Fully correct, ignore wrong subtraction., A0, Q0,, Q1 [3] B1 [1] 4. (a) 41 or 29 used Page 30

31 12 (b) 59 or 50 used 109 [4] 5. Divides 8 by 11, showing at least 0.7 Strand (i) Correct notation Accept Q1 [2] 6. (a) or 0.8(0) 0.2 oe (b) 0.4(0) B1 (c) Alternative method or 1 number 0.04 Page 31

32 oe 25 oe Alternative method 2 4 or 6 or 4 or 5 oe Attempt to work out how many prime numbers in the range 361 n < 390 or 421 n < 450 or 331 n < [5] 7.(a) [2.3, 2.5] Ignore x = B1 (b) Alternative method 1 A triangle drawn on graph or a y and corresponding x length clearly shown ir stated. their y length their x length Allow lengths to be ±½ small square ie ± 0.2 vertically or ± 0.1 horizontally dep 5 Page 32

33 Only award if y length x length = 5 and does not round to 5 Accept y = 5x 4 Alternative method 2 Substitutes a coordinate value into y = mx + c, eg (2, 6) Shows a correct equation, eg 6 = 2m 4 dep 5 Accept y = 5x 4 Additional Guidance = 5 dep A0 8 = m = 2.4m dep A0 Page 33

34 M0dep A0 [4] or 1240 oe or 1300 oe their their 1300 or 0.95 their 1300 or 1235 their 1300 can be their 1240 if greater than and 1235 (Car) B as final values Strand (iii) ft for correct decision based on their values, with at least M2 scored and one correct final value SC and 1200 or (0) and (0) SC or (0) or (0) Q1ft Additional Guidance Car A = 1240 and Car B = 1300 with correct decision of Car A M0A0Q1ft [5] 9.(a) 20 Page 34 B1

35 (b) 9 B1 (c) 11 and 3 seen Could be written on diagram 8 (d) Comment on average and the implication, eg waiting times decreased after new window as median lower ft their medians if valid conclusion reached B1 Comment on range or inter-quartile range and the implication, eg Spread of waiting times decreased after new window as range decreased or Not much effect on waiting times as IQR about the same ft their values if a valid conclusion reached B1 [6] M20.A point that lies on the circumference, eg (4, 5), (10, 5), (7, 2), (7, 8) Additional Guidance B1 (4, y) or (10, y) or (x, 2) or (x, 8) B1 for 4 or 10 clearly shown as min or max horizontal value B1 for 2 or 8 clearly shown as min or max vertical value Circle measurement is 2.6 cm so if subtracted or added then rounded can lead to correct answer, but allow as 2.6 rounds to 3, so mark answer line, ignore any other working B2 [2] M21. Page 35

36 B2 correct B1 reflection in x = 2 or y = 2 drawn with no other lines drawn B2 [2] M22.12 : 16 or 15 : 12 or or 0.75 or or 1.33 or or 1.25 or or oe From accurate working, eg 19.5 rounded to 20 is A0 Additional Guidance = 1.3, = 19.5, A = Page 36

37 , A = 19.5 M0, A0 [2] M23.(P:) (D =) 90T or (M:) (D =) 70(T + 1) oe 90T = 70(T + 1) Condone missing bracket, ie 90T = 70T + 1 but no further marks unless bracket recovered, dep 90T 70T = 70 oe NB is M3 dep 3.5 oe 3.30 is M3, A0 Alternative Method 1 Chooses a value for distance travelled and correctly works out time taken at 90kph and time taken at 70kph Lists distance travelled for Paul and Mary (for at least 2 hours) Eg 90, 180, 270, 360, 70, 140, 210, 280, 350,. Subtracts their values or repeats above with a different value Trying a new value implies that the difference between previously calculated times was not 1. dep Page 37

38 Chooses a different value for distance travels and correctly works out time taken at 90kph and time taken at 70kph, but the difference in times must be closer to 1 hour than the previous choice. oe dep 3.5 oe 3.30 is M3, A0 SC2 315 km Alternative Method 2 (P:) (D =) 90(t 1) or (M:) (D =) 70t NB this scheme is for working out the time that Mary takes. It can be recovered for full marks but if it ends at 4.5 then 2 marks maximum. oe 20t = 90, and t = 4.5 NB = 4.5 is M2 dep Their oe dep 3.5 oe 3.30 is M3, A0 [4] M24. Full evaluation referencing that the steps are right but the order is wrong, giving the correct order oe B1 for a partial explanation eg references incorrect order Page 38

39 without being specific B2 [2] M25. Alternative method 1 (a = ) 12 3 or 4 2b + their a = 24 or 2b + 4 = 24 or b = 10 2 their a + their b + 2c = 30 or c = 30 or 2c = 12 or c = 6 or sum of middle column is 30 their a 22, 26 and 18 SC2 first and third column totals correct SC1 totals of 3a + b, a + b + 2c, 2a + b Alternative method 2 (a = ) 12 3 or 4 2b + their a = 24 or 2b + 4 = 24 or b = 10 ( ) their totals for first and third columns or 66 their 22 their 18 22, 26 and 18 SC2 first and third column totals correct Page 39

40 SC1 totals of 3a + b, a + b + 2c, 2a + b [4] M26. 4x 2 10x + 10x 25 or 4x 2 25 or 6x x 15x 35 or 6x 2 x 35 or 6x x + 15x + 35 or 6x x + 35 Allow one error 4x 2 10x + 10x 25 or 4x 2 25 or 6x x 15x 35 or 6x 2 x 35 or 6x x + 15x + 35 or 6x x + 35 Fully correct 12x x 2 75x 175 [3] M27. Alternative method 1 P (1, 3) or y = 3 or grad OP = 3 B1 Page 40

41 and substitutes (1, their 3) or oe dep Substitutes y = 0 in their equation dep (10, 0) Alternative method 2 P (1, 3) or y = 3 or grad OP = 3 B1 dep their 3 their 3 or 9 dep Page 41

42 N is on the x-axis PN is perpendicular to the x-axis (10, 0) [5] M28.πrl + πr 2 = 24π 15π 3l + 9 = 24 oe e.g. 3πl = 15π 5 SC1 for 8 from πrl = 24π Must see working SC1 for 6 from πrl + 2πr = 24π Must see working NB if height calculated after 5 seen ignore [3] M29. Page 42

43 ft value seen in the form where a and b are integers > 1 B1ft [3] Page 43

44 E1. 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. E2.Many students had only limited success with this question. Having values for lengths and angles meant that the required reasons had to be given, either as a statement or shown clearly on the diagram. Very few students stated the conditions for congruency. E3.This question was quite well answered. Common errors were n + 6, 6n 3 or 3n + 6. E4.This question was not well answered. A successful approach was to start from 4(x 20) = x However, a common approach was to incorrectly start from x 20 = 4(x + 280). Some students used ratio notation to achieve 100 : 400 and then solved x 20 = 100 or x = 400; and others simply stated = 100 or = 400. E8.Part (a) discriminated well between differing levels of ability. Part (b), however, proved to be extremely challenging for all but the most able students with a high proportion of nonattempts. E9.Part (a) was well answered. The common error was confusing the relationship between time, distance and speed. Page 44

45 E11. This question was very well answered. Some students did not understand how to find the percentage increase after correctly working out the 400 increase and some left their answer as 1.20 or 120%. E12.This question was well answered. Poor arithmetic in working out or π 9 were common errors. Another common error was using the wrong formula for the area of a circle. E15. This question proved to be very challenging. A minority of students attempted a division, and most of those who did divided 11 by 8. Some students changed the fraction to, with most of these going on to give the correct answer. E17.Part (a) was quite well answered, with misreading the scale a common error. Part (b) was not well answered. Those students who knew to find an equivalent y and x value often misread the scale. There were some algebraic methods seen but these were rarely successful. E18.There were many fully correct answers and most students had some success with this question. Students were generally able to calculate percentage reductions but many were unable to work out off 1950, incorrectly using 30% or 33%. Arithmetic errors and poor setting out of work caused problems. E19.The first three parts of this question were well answered, but part (d) was poorly answered. When comparing distributions, students should make a comparison based on an average and a comparison based on a measure of spread. These comparisons should also explain the effect of the change in average or spread. Page 45

46 E20.This question was not well answered. Many students tried to put scales on the axes. An answer of (7, 1) was common. E21.As a starter question responses were generally poor. Fewer than half the students drew the correct reflection. The expected error of reflecting in line x = 2 was seen, but another very common error seen was to reflect in the y-axis. E22.This question was not well answered. The most common answer was 19 from Students who did identify a scale factor were often let down by poor arithmetic, for example, 1.3 being used for E23.This question was quite well answered. Many students gave the correct answer, although the working, whilst not contradictory, was not always convincing. Main errors were not using the correct combination of speed, time and distance. There were few trial and improvement attempts. E25. A good number of students correctly worked out the values of a, b and c and completed the table. Some could find a correctly by dividing 12 by 3, but then calculated b as Others who correctly found a and b calculated c as 30 a b. A few students gave algebraic expressions as the column totals. Page 46

47 E28.This question was not well answered. Despite the question stating that the base was included, this was often ignored. However, the majority who made an attempt to equate something to 24π used a variety of combinations of the base, curved surface area and volume. Page 47