CARDINAL NEWMAN CATHOLIC SCHOOL Mathematics PRACTICE Calculator Paper 2 HIGHER TIER Year 10 End of Year Exam Name : Subject Teacher : Examination Instructions Tutor Group: 10 Tutor: For Examiners Use Use black ink or ball point pen. Draw diagrams in pencil Fill in the boxes at the top of this page. You must not use a calculator for this paper Check that you have the correct question paper. Answer ALL the questions. Write your answers in the spaces provided in this question paper. Show all stages in any calculations, show clearly how you work out your answer to award marks for problem solving and reasoning. Examination Information This examination will last 80 mins. Mark allocations are shown in brackets. You may ask for tracing paper or graph paper at any time. This paper is out of 67 marks Question Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Marks Grade: Mark Awarded
Self-Assessment and Reflection Question Number Maths Watch Clip Number Working Towards Expectations (Grade 3-4) Working At Expectations (Grade 5-6) Working Above Expectations (Grade 7-9) I can express values as the product of primes I can use multipliers to increase values and reverse I can convert into standard form percentage change I can expand brackets I can expand two brackets I can apply y=mx+c I can select a method to find a median value I can apply medians in a table I can interpolate from histograms I can organise frequency into tree diagrams I can apply relative frequency I can interpolate from cumulative frequency diagrams I can equate fractions, decimals and percentages I can apply density formulae I can apply direct proportionality I can apply the properties of a I can find the volume of a I can use iteration processes quadrilateral cylinder I can draw plans and I can calculate angles using I can find angles in every elevations I can apply set notation in probability Topic (Maths Watch CD Title) 1 142 Density 2 85 Fractions Decimals and Percentage equivalence 3 96 Equations of a straight line 4 43 & 10 Properties of a Quadrilateral 5 51 Plans and Elevations 6 130 Medians within a table 7 57 Frequency Trees 8 125 Relative frequency 9 78 Product of Prime Factors 10 134 Expanding two binomials 11 168 Trigonometry in right angled triangles 12 119 Volume of a cylinder 13 93 Expanding brackets 14 110 Reversing Percentages 15 83 Standard Form 16 97 & 96 Y=mx+c 17 205 Histograms 18 186 Cumulative Frequency Diagrams 19 199 Proportionality 20 180 Iteration 21 123 Angles in Quadrilaterals and Triangles 22 127 &185 Venn Diagrams Trigonometry I can draw and complete Venn Diagrams Indicative grade as a result of the level of current mastery U 3 4 5 6 7 8 9 polygon I can find probability from Venn Diagrams Page 2
Q1.Three items were bought at a car boot sale. Cardinal Newman VA RC High School Item A Item B Item C Mass = 9.5 grams Volume = 2 cm 3 Mass = 57 grams Volume = 3 cm 3 Mass = 76 grams Volume = 4 cm 3 The density of gold is approximately 19 grams per cm 3. Which item or items cannot be gold? You must show your working. Answer... (Total 4 marks) Q2. Circle the decimal that is closest in value to 0.6 0.66 0.667 0.67 (Total 1 mark) Q3. Page 3
(a) Circle the equation of line A. y = 2 x = 2 x + y = 2 y = x + 2 (1) (b) On the grid draw the line y = x (1) (c) Write down the coordinates of the point where the line y = x crosses line A. Answer (...,... ) (1) (Total 3 marks) Q4. Tick ( ) or cross ( ) the properties of the quadrilaterals shown. The square has been done as an example. Property Diagonals cross at right angles One pair of equal opposite angles All sides equal Exactly one line of symmetry Rotational symmetry of order 2 Square X X X Page 4
Rhombus Kite (Total 4 marks) Q5.This solid shape is made from identical cubes. (a) On the grids draw the side elevations L and R. Page 5
(2) (b) How many cubes must be added to the shape to make this cuboid? Answer... (2) (Total 4 marks) Q6.The table shows information about the pay per hour of 40 people. Pay per hour, x ( ) Frequency 5 < x 15 14 15 < x 25 12 25 < x 35 11 35 < x 45 2 45 < x 55 1 Total = 40 (a) Which group contains the median pay per hour? Circle your answer. 5 < x 15 15 < x 25 25 < x 35 35 < x 45 45 < x 55 (1) (b) Work out an estimate of the mean pay per hour. Page 6
............ Answer... (4) (Total 5 marks) Q7.50 cars arrive at a car park. The table shows the number of people in each car. Number of people Number of cars 1 9 2 12 3 18 4 7 5 4 (a) One of the cars is chosen at random. Work out the probability that there are more than 3 people in the car. Answer... (2) (b) Work out the total number of people in the 50 cars. Answer... (2) Page 7
(Total 4 marks) Q8. (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.. Answer... (3) (Total 5 marks) Page 8
Q9.Write 56 as a product of prime factors. Cardinal Newman VA RC High School Answer... (Total 2 marks) Q10. Expand and simplify (2x 3y)(4x 5y) Answer... (Total 3 marks) Q11. The diagram shows a design for a zipwire. The zipwire will run between the top of two vertical posts, AB and CD. Work out the distance AD. Page 9
Answer... m (Total 4 marks) Q12.(a) The diagram shows a cylinder. The radius of the base is 6 cm The height is 15 cm Work out the volume.......... Answer... cm 3 (3) Page 10
(b) 1000 cm 3 = 1 litre A tank contains 45 000 cm 3 of water. The tank leaks at 0.75 litres/minute. How long does the tank take to empty?.................. Answer... (4) (Total 7 marks) Q13.(a) Expand and simplify (2x + 1)(x 2) Answer... (3) (b) Factorise fully 3x 2 48y 2 Answer... (3) (Total 6 marks) Page 11
Q14.I increase a number by 24% The answer is 6014. Cardinal Newman VA RC High School What number did I start with? Answer... (Total 3 marks) Q15. a 10 4 + a 10² = 24 240 Work out a 10 4 a 10² where a is a number. Give your answer in standard form. Answer... (Total 2 marks) Q16.(a) The graph shows the line y = ax + b Work out the values of a and b. Page 12
Answer a =... b =... (2) (b) Work out the value of y when x = 80............ Answer... (2) (Total 4 marks) Page 13
Q17. The histogram shows the ages, in years, of members of a chess club. Cardinal Newman VA RC High School There are 22 members with ages in the range 40 age < 65 Work out the number of members with ages in the range 25 age < 40 Answer... (Total 4 marks) Q18. Here are the examination marks for 60 pupils. mark, m (%) Frequency 0 m < 20 8 20 m < 40 9 40 m < 60 21 Page 14
60 m < 60 10 80 m < 100 12 Molly drew this cumulative frequency graph to show the data. Make two criticisms of Molly s graph. Criticism 1... Criticism 2... Page 15
(Total 2 marks) Q19. y is directly proportional to x 36 a y 2 5 Work out the value of a. Answer... (Total 4 marks) Q20. An approximate solution to an equation is found using this iterative process. (a) Work out the values of x₂ and x₃ Page 16
x₂ =... x₃ =... (2) (b) Work out the solution to 6 decimal places. x =... (1) (Total 3 marks) Q21. Here is a metal badge in the shape of a kite. Not drawn accurately Page 17
(a) Set up and solve an equation to work out the value of x................... x =... (3) (b) The badge is cut from a rectangular sheet of metal as shown. Not drawn accurately Page 18
Cathy says, The area of the badge is exactly half the area of the rectangle. Give reasons why she is correct. You may use the diagram to help you................ (2) (Total 5 marks) Q22. Here is a Venn diagram. It shows information about the number of students who have a laptop or a TV. Set L represents students with a laptop. Set T represents students with a TV. Page 19
There are 50 students altogether. A student is chosen at random. (a) Work out the probability that the student has a laptop. Answer... (1) (b) Work out the probability that the student has a laptop and a TV. Answer... (1) (c) Complete the sentence to make it true. The probability that the student...... is (1) (Total 3 marks) Page 20
.9.5 2 (= 4.75) or 19 2 (= 38) or 9.5 19 (= 0.5) 57 3 (= 19) or 19 3 (= 57) or 57 19 (= 3) 76 4 (= 19) or 19 4 (= 76) or 76 19 (= 4) A with full verification eg A and 4.75 (19 and 19) Checking density or A and 38 (57 and 76) Checking masses or A and 0.5 (3 and 4) Checking volumes [4] M2. 0.667 B1 [1] M3.(a) x = 2 B1 Page 21
(b) Correct straight line drawn at least 3 diagonal squares long B1 (c) 2, 2 ft their intersection with line A only if B0 in part (b) B1ft [3] M4. B1 for 4 correct, 1 wrong B0 for 2 or more wrong B2 B1 for 4 correct, 1 wrong B0 for 2 or more wrong B2 [4] M5.(a) Drawings can be anywhere on the grids B1 for shapes reversed or B1 for one correct B2 Page 22
(b) 6 2 + 3 Cardinal Newman VA RC High School or 4 + 7 + 4 or 2 + 2 + 2 + 2 + 7 or 28 or 13 15 SC1 for 17 [4] M6.(a) 15 < x 25 B1 (b) Mid values seen 10, 20, 30, 40 and 50 or 10.005, 20.005, 30.005, 40.005, 50.005 or 10.01, 20.01, 30.01, 40.01, 50.01 B1 10 14 (+) 20 12 (+) 30 11 (+) 40 2 (+) 50 ( 1) or 140 (+) 240 (+) 330 (+) 80 (+) 50 or 840 Accept use of mid values 10.005, 20.005 etc or 10.01, 20.01 etc Allow one error eg one mid value incorrect or one calculation incorrect their 840 40 21 or 21.01 Accept 21.005 SC2 for 16 or 16.005 or 16.01 or 21.5(0) or 21.505 or 21.51 Page 23 dep
or 26 or 26.005 or 26.01 or 791.25 Cardinal Newman VA RC High School Additional Guidance 21 and then states answer is in 15 < x 25 class is fw and can be ignored 140 + 240 + 330 + 80 + 50 40 = 21 (recovered) 4 marks 4 marks = 791.25 140 + 240 + 330 + 80 + 50 40 = 791.25 Answer 791.25 implies at least B1 840 840 5 = 168 140, 240, 330, 80, 50 168 with no working Note: Two or more midpoints incorrect B1A0 B1 B1 B1M0 B1 M0 B0M0 [5] M7.(a) or 0.22 oe B1 for numerator 11 or denominator 50 or 11 out of 50 or 11 in 50 Ignore fw B2 (b) 1 9 (+) 2 12 (+) 3 18 (+) 4 7 (+) 5 4 or 9 (+) 24 (+) 54 (+) 28 (+) 20 oe Allow one error May be in table Page 24
135 Cardinal Newman VA RC High School [4] M8. (a) or 72 6 or 12 or 72 6 2 oe 24 oe 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 oe Page 25
Additional Guidance 55 in table A0 Do not allow misreads for 250 [5] M9.28 ( ) 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 2 2 2 7 or 2 3 7 Additional Guidance Ignore any 1 for but not [2] 0.8x 2 12xy 10xy + 15y 2 Allow one term error 8x 2 12xy 10xy + 15y 2 8x 2 22xy + 15y 2 ft their four terms if awarded Do not ignore fw for final mark ft [3] Page 26
1. 6.5 2.3 or 4.2 and 5 or 85 seen Cardinal Newman VA RC High School oe oe dep [48, 48.2] [4] 2.(a) π 6 2 or 3.14 6 2 or [113, 113.2] May be embedded oe π 6 2 15 or 3.14 6 2 15 or [113, 113.2] 15 oe dep [1695, 1698] or 1700 or 540π Ignore fw after 540π Page 27
Additional Guidance π 6 2 = π 12 15 π 6 2 15 = π 12 15 π 6 2 30 2 π 6 2 15 π 6 2 = π 12 π6 2 π 12 π 12 15 M0 M0 M0 M0 M0 (b) Alternative method 1 45 000 1000 or 45 45 0.75 or 45 1.33... or their 45 0.75 oe eg 45 3 4 60 60 minutes or 60 min(s) or 1 hour or 1h(r) Strand (i) Correct notation Q1 Alternative method 2 0.75 1000 or 750 45 000 750 or 45 000 their 750 oe Page 28
60 60 minutes or 60 min(s) or 1 hour or 1h(r) Strand (i) Correct notation Q1 Additional Guidance For the Q mark 60 minutes or 1 hour must not come from incorrect working Ignore fw after 60 minutes or 1 hour Digit 6 implies M0 eg 60 000, 6000, 600, 6 or 0.6 750 45 000 = 0.016 (units would be minutes 1 ) 750 45 000 = 0.016 and 0.016 60 = 1 hour (method is incorrect) Do not accept 60 m for the Q mark M0 M0A0Q0 M0A0Q0 Q0 [7] 3.(a) 2x 2 + x 4x 2 4 terms, allow one error but must have a term in x 2 2x 2 + x 4x 2 2x 2 3x 2 oe ft their 4 terms if awarded SC1 answer of 2x 2 5x 2 or 2x 2 + 3x 2 or 2x 2 3x + 2 without working worth at least fit (b) 3(x 2 16y 2 ) Page 29
(3)(x + ay)(x + by) where ab = 16 Cardinal Newman VA RC High School 3(x 4y)(x + 4y) oe Alternative method (3x + ay)(x + by) where ab = 48 (3x + 12y)(x 4y) or (3x 12y)(x + 4y) 3(x 4y)(x + 4y) oe [6] 4.1.24 or 124% or seen B1 6014 1.24 oe 6014 124 100 4850 [3] Page 30
5. 2.376 10⁴ B1 (a =) 2.4 or 24 000 and 240 or 23 760 or value calculated that is correctly converted to standard form B2 [2] 6.(a) a = 6 Allow 6x B1 b = 100 SC1 if values reversed. y = 6x + 100 seen in script with no contradictory answers for a and b given allow B2 B1 (b) Substitution of 80 into their formula y = their 6 80 + their 100. Their 6 must have a value, ie not 0. 580 ft their formula ft Alternative Method 400 + (280 100) Or use of values from graph 580 Page 31
[4] 7. Alternative method 1 25 11 or 275 their 275 22 or 12.5 dep 15 30 their 12.5 36 Alternative method 2 25 11 or 275 15 30 their 275 or [1.6, 1.64] dep their [1.6, 1.64] 22 36 Alternative method 3 11 squares or Page 32
275 squares Cardinal Newman VA RC High School 22 11 or 2 or 22 275 or 0.08 dep their 2 18 or their 0.08 450 36 Alternative method 4 oe fraction dep 36 Alternative method 5 25 h = 22 or or 0.88 Page 33
oe Cardinal Newman VA RC High School 0.88 11 or 0.08 oe eg frequency density axis labelled with correct scale dep their 0.08 30 15 36 [4 marks] 8. Cumulative frequency 46 should be 48 oe B1 Points should be plotted at end of class intervals oe B1 [2] 9. Alternative method 1 dep Page 34
oe 225 Alternative method 2 dep oe 225 Alternative method 3 dep on dep Page 35
225 [4 marks] M20. (a) or 0.5 B1 or 0.390625 B1ft (b) 0.381966 B1ft [3] M21. (a) Any correct equation e.g.1 2x + x + 96 + 96 = 360 e.g.2 2x + x + 96 + 96 = 360 e.g.3 x + x + 96 = 180 B1 Correct rearrangement of their equation to the form ax = b or Page 36
Follow through their equation of form px + q = r a, b, p, q and r all non-zero 56 ft their ax = b if gained ft (b) Fully correct explanation e.g.1 Labels large rectangle a and b or labels diagonals of kite a and b Area rectangle = a b Area kite = product of diagonals e.g.2 Labels each part of top edge with w and the side parts with x and y Area rectangle = 2w(x + y) = 2wx + 2wy Area kite = wx + wx + wy + wy = wx + wy e.g.3 Draws both diagonals of kite and indicates there are 4 pairs of equal areas e.g.4 Draws at least one diagonal of the kite and states that the area of a triangle is half the area of a rectangle e.g.5 Uses compatible numbers and correctly works out areas of kite and rectangle For example Labels each part of top edge with 4 and the side parts with 3 and 6 Rectangle area = 8 9 = 72 Kite area = 0.5 8 3 + 0.5 8 6 = 12 + 24 = 36 B1 Partially correct statement or correct step towards correct explanation e.g.1 Labels large rectangle a and b or labels diagonals of kite a and b Area rectangle = a b e.g.2 Labels each part of top edge with w and the Page 37
side parts with x and y Cardinal Newman VA RC High School Area rectangle = 2w(x + y) e.g.3 Draws both diagonals of kite e.g.4 Uses compatible numbers and works out areas of kite and rectangle with correct method but makes arithmetic error(s) For example Labels each part of top edge with 4 and the side parts with 3 and 6 Rectangle area = 8 9 = 82 Kite area = 0.5 8 3 + 0.5 8 6 = 12 + 24 = 36 B2 [5] M22. (a) oe B1 (b) oe SC1 Incorrect but consistent denominator, greater than 29, used in (a) and (b) with correct numerators B1 (c) Only has a TV oe B1 [3] Page 38
E1.A variety of methods were used and generally the question was very well answered. A few students did not check all three items. E3.In part (a) most students correctly circled x = 2, but it was fairly common for students to circle y = 2 or x + y = 2. Part (b) was not well answered by a good proportion of students. Common incorrect lines drawn were y = 0 or y = 2 or y = x. In part (c) many students correctly followed through from their incorrect line in part (b) and hence scored 1 mark. E4. This question was not well answered. Full marks were rare. The main errors were in the first two columns. E5.Part (a) was very well answered. The most common errors were to attempt to draw in three dimensions or to double the size of the view. In part (b) many students made no progress. Of those who did use volumes 28 was a very common final answer. E6.Both parts of this question were quite well answered. In part (a) the middle class was chosen by approximately one-third of the students. In part (b) a significant minority of students did not know the method to use. Many knew how to start the question with the common error being to correctly obtain 840 but then significant numbers divided by 5. E7.Part (a) of this question was not well answered, with many reading more than 3 as 3 or Page 39
more. Performance on part (b) was much better. The most common incorrect answer being 15, obtained from adding up the first column. E8. Part (a) of this question was well answered. In part (b) the majority of students gave 55 as their final answer. E10.This question was also a good discriminator. The most common errors were incorrect indices for 8x 2 or 15y 2 or making 15y 2 negative. Some students added the coefficients resulting in 6x 7xy 7xy 8y or 6x 8y. E12.Both parts of this question were well attempted with a significant number working out part (a) in two distinct steps. The most common error in part (a) was to use the circumference formula. Part (b) was very well answered at this tier. A minority of students obtained 45 litres but then multiplied by 0.75. E13.The expansion in part (a) was quite well answered although there were errors when collecting terms. Part (b) was poorly answered with very few fully correct answers. Most students only took out the factor 3. E14.For those students who recognised this as a reverse percentage problem it was a relatively straightforward process to give a fully correct solution. However, many students worked out 24% of 6014 and then subtracted it. Page 40
E16.In part (a), many students did not know the method to find the gradient. Of those who did there were often arithmetic errors. Part (b) was slightly better answered as students used their values from part (a). A common error was to read the value for x = 40 from the graph and double it. E21. Part (a) was not well answered. Setting up an equation was rarely seen. However, a significant number of students gained some credit for obtaining the correct answer without the use of an equation. Centres are reminded that setting up and solving an equation is part of the specification and that students need to experience problems that lend themselves to using a simple equation. Some candidates presented an equation, usually 2x = 112, after finding the solution. This did not meet the requirements for a fully correct solution. Students found part (b) challenging and a large number either gained no credit or made no attempt. Those who gained some credit did so by drawing the diagonals of the kite, and then attempting to explain the equality of the area of each triangle in the kite and each equivalent triangle left in the rectangle. E22. A common error in part (a) was to ignore the 23 in the intersection and give an answer of 6 / 50. There was a much better response to part (b), although some interpreted the question as meaning the union of the sets instead of the intersection. A few gave the numbers in the sets instead of the probabilities asked for. In part (c) the most common incorrect response was to complete the sentence with has a TV. Page 41