PAKURANGA COLLEGE. 12MAT Mathematics and Statistics Practice Exams, Apply calculus methods in solving problems (5 credits)

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1 Name: PAKURANGA COLLEGE Teacher: 2 12MAT Mathematics and Statistics Practice Exams, 2014 AS AS AS Apply algebraic methods in solving problems (4 Credits) Apply calculus methods in solving problems (5 credits) Apply probability methods in solving problems (4 credits) Time allowed: 3 hours You should answer ALL parts of ALL the questions in this booklet You should show ALL your working for ALL questions. The questions in this booklet are NOT in order of difficulty. If you need more space for any answer, use the page provided at the back and clearly number the question. YOU MUST HAND THIS WORKBOOK TO THE SUPERVISOR AT THE END OF THE EXAMINATION. Achievement Achievement with Merit Achievement with Excellence Score Grade Apply algebraic methods in solving problems. Apply algebraic methods, using relational thinking, in solving problems. Apply algebraic methods, using extended abstract thinking, in solving problems. Apply calculus methods in solving problems. Apply calculus methods, using relational thinking, in solving problems. Apply calculus methods, using extended abstract thinking, in solving problems. Apply probability methods in solving problems. Apply probability methods, using relational thinking, in solving problems. Apply probability methods, using extended abstract thinking, in solving problems.

2 AS Apply algebraic methods in solving problems Page 3 AS Apply calculus methods in solving problems Page 12 AS Apply probability methods in solving problems Page End of year L2 MAT exam

3 Algebra You are advised to spend 60 minutes answering the questions in this booklet. QUESTION ONE (a) (i) Factorise 5x 2 6x 8 (ii) Solve 5x 2 6x 8 = 0 (b) Write as a single fraction in its simplest form: 5 m 1 3m m 4 (c) Solve 4x 2 16x 84 2 x 9 = End of year L2 MAT exam

4 (d) (i) Find the value of c which satisfies this equation: 2 x x x 35x 28c = x 5 7(2x 3) (ii) What does the value of c found in (i) above tell us about the nature of the roots of 14x 2 35x +28c = 0? End of year L2 MAT exam

5 (e) An elliptical running track has internal lengths of 3x and 2x as in the diagram. The width of the track is 5 metres. If the area inside the track is the same as the area of the track itself, find the inside dimensions of the track. 2x 3x Note: The area of an ellipse = πab where a and b are the internal lengths End of year L2 MAT exam

6 QUESTION TWO (a) Solve for x : (i) log x 125 = 3 (ii) x = log 2 64 (b) The Department of Conservation begin a control programme on the number of stoats on Karera Island. The number of stoats can be modelled by S = 300 x (0.95) t where S is the number of stoats and t is the time in months since the controls began. (i) After how long will the stoat population first drop below 200? End of year L2 MAT exam

7 (ii) With less stoats the kiwi population on the island grows faster. It can be modelled by K = 100 x (1.015) t where K is the number of kiwis and t is the number of months since the stoat control programme was started. After how many months would the numbers in the kiwi and stoat populations be the same if the programme continued? (iii) After three years the stoat control programme stopped. The rate of growth of the stoat population increased to 4% per month and the kiwi population began to decrease at a rate of 2% per month. How long after the programme was stopped will the numbers in the populations of kiwi and stoats be equal again? End of year L2 MAT exam

8 (c) The function is undefined for values of x less than 0. Find two solutions for m in the equation ( ) and explain why the negative solution is acceptable End of year L2 MAT exam

9 QUESTION THREE (a) Simplify (i) m (ii) 2 / m 5 3 (b) Solve x 9 x 1 = End of year L2 MAT exam

10 (c) Jonny is throwing a ball over the fence to his friend Ngaire. Both children are 1.1 metres tall. Jonny is 2.5 metres away from the fence and Ngaire is 1.5 metres away on the other side. The ball reaches a peak on Jonny s side of the fence at 4 metres, just beneath a tree. Fence (i) Form an equation to represent the path of the ball. (ii) Assuming that the ball just skims over the top of the fence, what is the height of the fence? End of year L2 MAT exam

11 (d) Find the possible value(s) for k if the quadratic 3x 2 2kx + 4k = 0 has two real roots End of year L2 MAT exam

12 Calculus You are advised to spend 60 minutes answering the questions in this booklet. QUESTION ONE: 5 2 (a) Find the gradient at the point where x = 1 on the curve y 0.6x 3x 7x 2. (b) (i) Give the coordinates of the points on the curve y x 3 4x 2 3x 1 where the gradient is 6. (ii) Find the equation of the tangent to the curve y x 3 4x 2 3x 1 at the point (2, -1) End of year L2 MAT exam

13 (b) (iii) The graph below shows the function y f (x). x On the axes below, sketch the graph of the gradient function. x End of year L2 MAT exam

14 (c) The outline of an Easter Egg can be modelled by two functions: the top curve is f ( x) 0.125x x 3x 4x and the bottom curve is g( x) x 2 4x y 4 2 f(x) x -2-4 g(x) (i) The point (1, 1.875) lies on the outline of the Easter Egg. Find the gradient of the outline of the Easter Egg at the point (1, 1.875) and give the other value of x where the outline has the same gradient. (ii) Explain why the outline is not smooth where the two curves meet. Justify your answer using calculus concepts End of year L2 MAT exam

15 QUESTION TWO 3 (a) Is the function f ( x) 5x 7x 6 increasing, decreasing or stationary when x = 1. Use calculus to justify your answer. 3 (b) The derivative of a function is f '( x) 8x 2x 2. Find f(x), if the graph of f(x) passes through the point (1, 0). (c) Give the coordinates of the maximum turning point of the function 3 2 f ( x) x 1.5x 18x 1 and explain how this can be proved to be a maximum by investigating the second derivative End of year L2 MAT exam

16 (d) (i) Melting point of chocolate is about 45degrees Celsius; once it goes over 48 degrees the chocolate burns and becomes lumpy. The temperature T of chocolate when heated is given by the function T = 4 3 t where t is the time in minutes from when the chocolate is placed in the heat. Find the rate that the temperature of the chocolate is changing after 3 minutes. (ii) Explain what your answer means in terms of heating or cooling of the chocolate (iii) If the chocolate is cooled then heated again it is called tempered chocolate. At the Cadberry factory, this temperature is controlled by the function T = -0.01x x 3 5x x + 18 where T is the temperature of the chocolate and x is the time in minutes that the chocolate has been heated. Show that there are two peaks in temperature, first at 3 minutes and the next at around minutes. You must use calculus methods to show that these times give maximum values End of year L2 MAT exam

17 QUESTION THREE (a) (i) Find the equation for the curve, if the gradient function is given by dy dx = 12x2 + 4x and the local minimum point is (0,0). (ii) Give the coordinates of any other point with the same gradient as the point (0,0) and describe how this point differs from (0,0) (b) (i) Find the gradient of the tangent to the curve y = 2x 2 6x + 1 where the curve cuts the y axis End of year L2 MAT exam

18 (ii) Find the minimum value of the curve y = 2x 2 6x + 1 (c) A large helium balloon is released from the top of a building above the ground. The balloon s initial velocity is measured at 10ms -1 and rises at a constant acceleration of 2ms -2 for the first minute after being released. If the balloon is 620m above the ground after 20seconds, how high is the top of the building? End of year L2 MAT exam

19 Probability You are advised to spend 60 minutes answering the questions in this booklet. At Hoopsdunk High School basketball is a major sport with both boys and girls teams. Question 1 (a ) ( i) The height of male basketball players at Hoopdunk High School is assumed to be normally distributed with a mean of 185cm and a standard deviation of 8cm. Calculate the probability that a player selected at random is between 185cm and 195cm tall. (ii) Calculate the percentage of male basketball players at the school with heights between 175cm and 195cm. (b) The height of female basketball players at the school is normally distributed with a mean of 175cm and a standard deviation of 5cm. There are 215 female basketball players at the school. Calculate the expected number of female basketball players with heights less than 169cm or more than 181cm End of year L2 MAT exam

20 Frequency (c) (i) Full size basketballs are used for the Premier Boys team. They are manufactured to have a mean circumference of 765mm when correctly inflated. The manufacturer will reject the smallest 5% (less than 755mm) and the largest 5% (more than 775mm). Calculate the standard deviation of the balls assuming the circumference of basketballs is normally distributed. (ii) At the end of the season all basketballs from all teams, boys and girls, are weighed and graphed as shown in the histogram Weight of basketball (in grams) Discuss the assumption that the weights of basketballs are normally distributed. In your answer you should refer to key features such as the centre, shape and spread in relation to the context End of year L2 MAT exam

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22 Question 2 Carl and Tua, the Premier Boys team s best shooters have a shoot off against each other. This involves each player taking shots from the same distance. The highest score wins the game. The probability of Carl winning the first game is 0.4. If Carl wins a game then the probability of him winning the following game is 0.7. If Tua wins a game then the probability of him winning the following game is 0.8. The boys decide to play three games against each other. 1 st game Carl wins Tua wins (a) (i) Calculate the probability that Tua will win all three games. (ii) Calculate the probability that Carl will win at least two of the three games End of year L2 MAT exam

23 (iii) If the boys play 40 matches of three games each over a season, find the expected number of matches where Tua wins exactly two games (Carl wins one game and Tua wins two games). (iv) If Tua wins the first game, calculate the probability that Carl will win at least one of the second and third games End of year L2 MAT exam

24 (b) A free throw is part of a game in basketball where a player takes two shots from a set distance from the hoop and can score either zero, one or two shots. Suppose, from each free throw, the probability of getting zero shots is a and the probability of getting one shot is b. Show that the probability of getting either zero shots or four shots from two consecutive free throws is 2a 2 + b 2 2a - 2b + 2ab + 1 (Assume that the second turn at a free throw is not affected by the first) End of year L2 MAT exam

25 Frequency Question 3 Hoopsdunk High School is thinking of charging people to watch the Premier Girls team play as part of a fundraiser to improve the gym facilities. Over the last four years (80 games) the school has recorded the number of spectators that have come to watch each of the Premier Girls games. The results are shown in the graph. Spectators at Hoopsdunk High School for Premier Girls team basketball games Number of spectators (a) Describe a feature of the graph that shows the data collected over the four years does NOT follow a normal distribution End of year L2 MAT exam

26 (b) The school will pay for gym improvements earlier than planned if the chance of more than 150 people attending three games in a row is likely to happen. Calculate the probability of this happening and explain whether the school is likely to pay early for improvements. Assume that the attendance of one game is not affected by the attendance of another. (c) The coach of the Premier Girls team is interested to know if there is a link between the number of spectators who turn up to watch the team play and the team s results. The data from the previous four years is summarised in the table. Lose Win 100 or less spectators 101 or more spectators What is the probability that a game chosen at random (i) had 100 or less spectators? (ii) was a game that the Premier Girls team won? (iii) was a game the team lost if it was known that 100 or less spectators attended? End of year L2 MAT exam

27 (d) The school conducts a survey of students who are known to have attended at least one Premier Girls game over the last four years, to find their likelihood of attending the next game. The results are summarised in the table. Likely to attend NOT likely to attend Team won previous game Team lost previous game Using the idea of relative risk, use calculations to explain how much more likely a student is to attend the next game based on whether the team lost the previous game End of year L2 MAT exam

28 Extra paper for continuing your answers, if required. Clearly number the question. Question number End of year L2 MAT exam

29 Assessment Schedule: Mathematics and Statistics 91261: Algebra Question Evidence Achievement (u) Merit (r) Excellence (t) Apply algebraic methods in solving problems. Apply algebraic methods, using relational thinking, in solving problems. Apply algebraic methods, using extended abstract thinking, in solving problems. TWO (a)(i) x 3 = 125 x = 5 Correct (a)(ii) 2 x = 64 x = 6 Correct (b)(i) (b)(ii) ln(0.95) t = ln(⅔) ln( 2 t = 3 ) = 7.9 ln (0.95) t = 100(1.015) t ln = ln t = 16.6 t Correct answer. Accept 8 months. Correct answer. Accept 17 months (b)(iii) S = 300(0.95) 36 = 47 K = 100(1.015) 36 = (1.04) t = 171(0.98) t ln(3.638) t = ln t = 21.8 See my notes Finding and equating the two equations t = 21.8 or 22 months. Logical working shown clearly. (c) log 3 (m 2-14m + 49) = 4 m 2-14m + 49 = 3 4 = 81 m 2-14m-32 = 0 m = 16, -2 m = -2 is an acceptable solution as when substituted (m -7) 2 will be positive. Correct expansion and equating to 81 Solutions found Solutions found and explanation given. N0 No response; no relevant evidence M5 1 of r N1 attempt at one question M6 2 of r N2 1 of u E7 1 of t A3 2 of u E8 2 of t A4 3 of u End of year L2 MAT exam

30 Assessment Schedule: Mathematics and Statistics 91261: Algebra ONE (a)(i) (5x + 4)(x 2) Correct (a)(ii) x = -0.8, 2 Correct (b) 5( m 4) 3m( m 1) ( m 1)( m 4) 2 2m 3m 20 ( m 1)( m 4) One denominator Simplified correctly (c) 4x 2-16x - 84 = 5x 2 45 x x + 39 = 0 (x + 3)(x + 13) = 0 x = -13 x -3 as cannot divide by 0 Correct factorisation x = -13 only solution (d)(i) ( x 5)( x 4) = 2 7(2x 5x 4c) ( x 5) 7(2x 3) 2x 2 5x + 4c = (2x +3)(x 4) 2x 2 5x + 4c = 2x 2 5x 12 c = -3 Factorisation and elimination of common factors c = -3 found (d)(ii) If c = -3 14x 2 35x 84 = 0 2x 2 5x 12 = 0 a = 2, b = -5, c = -12 b 2 4ac = 121 > 0 Roots are real and distinct Discriminant found Nature of roots described (e) Area track = π(2x + 5)(3x + 5) -π.2x.3x = 25πx + 25π Quadratic equation formed Internal lengths found Inside area = 6πx 2 Equating: 25πx + 25π = 6πx 2 6x 2 25x 25 = 0 x = 5, (eliminate) Internal lengths are 10 and 15 N0 No response; no relevant evidence M5 1 of r N1 attempt at one question M6 2 of r N2 1 of u E7 1 of t A3 2 of u E8 2 of t A4 3 of u End of year L2 MAT exam

31 Assessment Schedule: Mathematics and Statistics 91261: Algebra, THREE m (a) 16 Correct (b) (0.4m 2 ) 2 Correct = 0.16m 4 (c) x 1 3 = 27 2x 3 Expressed as powers of 3 x = -2 found 3 (x+1-2x) = 27 1 x = 3 x = -2 (d)(i) Using midpoint as y-axis Equation found y = x (d)(ii) Fence is at x = y = - (0.5) = 3.82 metres Height found from correct equation (e) a = 3, b = -2k, c = 4k b 2 4ac > 0 4k k > 0 Discriminant found and set > 0 One value found Correct range of values found 4k(k 12) > 0 k < 0 and k > 12 N0 No response; no relevant evidence M5 1 of r N1 attempt at one question M6 2 of r N2 1 of u E7 1 of t A3 2 of u E8 2 of t A4 3 of u Grade point scoring (this can be varied for school requirements) 1-6 = Not Achieved 7 13 = Achieved = Merit = Excellence End of year L2 MAT exam

32 Assessment Schedule: Mathematics and Statistics 91262: Calculus, 2014 Achievement Merit Excellence Apply calculus methods in solving problems involves: selecting and using methods demonstrating knowledge of calculus concepts and terms communicating using appropriate representations. Apply calculus methods using relational thinking, in solving problems must involve one or more of: selecting and carrying out a logical sequence of steps connecting different concepts and representations demonstrating understanding of concepts forming and using a model and relating findings to a context, or communicating thinking using appropriate mathematical statements. Apply calculus methods using extended abstract thinking, in solving problems must involve one or more of: devising a strategy to investigate a situation demonstrating understanding of abstract concepts developing a chain of logical reasoning, or proof forming a generalisation and using correct mathematical statements, or communicating mathematical insight End of year L2 MAT exam

33 Assessment Schedule: Mathematics and Statistics 91262: Calculus, 2014 Evidence Statement One Expected Coverage Achievement(u) Merit(r) Excellence(t) NØ = No response; no relevant evidence. N1 = a valid attempt at ONE question. N2 = ONE question demonstrating limited knowledge of calculus techniques.(1u) A3 = TWO of u. A4 = THREE of u. M5 = ONE of r. M6 = TWO of r. E7 = ONE of t, with minor errors ignored. E8 = 2 of t (a) dy dx = 3x4 + 6x 7 when x = 1, gradient = 2 Derivative found with one error and consistently used in finding the gradient at the point where x = 1. derivative and gradient at x = 1 found (b) (i) Provides graph of the gradient function Graph of the gradient function sketched with x intercepts near x= 1 and x=-1, parabolic shape (b) (ii) dy dx = 3x2 8x + 3 = 6 x = 3 or 1 3 Derivative found, equated to 6, partially solved Coordinates of both points found points (3,1) (-0.33,-0.48) dy dx = 3x2 8x + 3 derivative found evaluated with x=2 equation of the tangent (b) (iii) at x = 2, gradient = -1 tangent equation y = x + 1 (c) (i) f (x) = 0.5x 3 + 3x 2 6x + 4 at x = 1 on f(x), gradient = 0.5 g (x) = 2x 4 = 0.5, so finds the derivative for both functions finds the gradient at x=1 for f(x) finds the value of x where the gradients are the same x = 2.25 the other x value is 2.25 (c) (ii) The gradients of the curves need to be the same at the points of intersection for the outline to be smooth. At x=0 the gradient of f(x) is 4 and the gradient of g(x) is -4. So the outline is not smooth discusses the slope of the two curves needing to be equal justifies the claim that the outline is not smooth by providing evidence the slopes are different where the curves meet End of year L2 MAT exam

34 Assessment Schedule: Mathematics and Statistics 91262: Calculus, 2014 Two Expected Coverage Achievement Merit Excellence NØ = No response; no relevant evidence. N1 = a valid attempt at ONE question. N2 = ONE question demonstrating limited knowledge of calculus techniques.(1u) A3 = TWO of u. A4 = THREE of u. M5 = ONE of r. M6 = TWO of r. E7 = ONE of t, with minor errors ignored. E8 = 2 of t. (a) f (x) = 15x 2 7 when x = 1, gradient = 8 This means that f(x) is derivative found and evaluated at x=1 derivative found and states that f(x) is increasing increasing f(x) = 2x 4 x 2 2x + c function integrated and c evaluated (b) 0 = 2(1) 4 (1) 2 2(1) + c c = 1 so f(x) = 2x 4 x 2 2x + 1 (c) f (x) = 3x 2 3x 18 = 0 x = 3 or 2 f (-2)=-15 f (3)=15 so (-2,21) is a local maximum. derivative found and both x values found Finds the second derivative gives the coordinates for the maximum TP and explains the second derivative being positive means that (-2,21) is a maximum. (d) (i) (d) (ii) T ' = 8 3 t At t=3, T = 8 C per min. The chocolate is heating up derives T evaluates T when t=3 (units not required) and explains it is increasing in temperature (heating) T ' = -0.04x x 2 10x shows T '= 0,when x=3 and17.1 derives T, makes T =0 shows T =0 for both x values justifies both points are maximums by calculus methods (d) (iii) T '' = -0.12x x 10 at x=3 and 17.1 shows T''<0 or otherwise using Calculus End of year L2 MAT exam

35 Assessment Schedule: Mathematics and Statistics 91262: Calculus, 2014 Three Expected Coverage Achievement Merit Excellence NØ = No response; no relevant evidence. N1 = a valid attempt at ONE question. N2 = ONE question demonstrating limited knowledge of calculus techniques.(1u) A3 = TWO of u. A4 = THREE of u. M5 = ONE of r. M6 = TWO of r. E7 = ONE of t, with minor errors ignored. E8 = 2 of t.minor error. E8 = Excellence correct. (a) (i) y = 4x 3 + 2x 2 + c at (0,0) c = 0 so y = 4x 3 + 2x 2 anti-differentiates to give an expression for y, includes some evidence of considering the constant of integration, and that c=0. (a) (ii) 0 = 12x 2 + 4x x = 0 or , 2 27 finds both coordinates of the other point where the gradient is 0 (b) (i) y ' = 4x 6 at x=0, gradient = -6 derivative and gradient found (b) (ii) 0 = 4x 6 x = 1.5 equates derivative to 0 and solves for x finds minimum value minimum value y= -3.5 (c) v = 2t + c at t=0, v=10 so c=10 v = 2t + 10 S = t t + c when t=20, s= 620, so c=20 S = t t + 20 when t=0, s=20 Finds the velocity equation, finds the velocity equations and integrates a second time to find an expression for distance, with minor errors integrates twice and interprets the distance at t=0 as the height of the building t= with minor errors or omissions 2t completely correct the height of the building is 20m End of year L2 exam

36 Assessment Schedule: Mathematics and Statistics 91267: Probability, 2014 Evidence Statement Expected Coverage Question ONE (a) i Answers will vary depending on whether the candidate uses the tables or a graphing calculator. P(185< X < 195) = Achievement (u) Merit (r) Excellence (t) Probability found. (a) ii P(175 < X < 190) = = 78.9% (b) P (X < 169 or X > 181) = x 2 = 0.23 Probability found but must be expressed as a percentage. Rounded to 79% is acceptable. Probability of found. Correct number of girls found 0.23 x 215 = 49 or 50 girls (c) z-score = so so z-score found Equation set up Equation solved to at least 1 d.p. so σ = 6.08 mm (d) Comments should be in context and for t students should have linked the previous context to realise there are two separate normal distributions (slightly overlapping), one for girls basketball team and one for boys basketball team OR other valid reason discussed and validated in context. Evidence to support this, in context, could include - Each distribution is symmetrical - Mean and median of each distribution are centrally located to that distribution - Unimodal - Spread from the middle of each distribution fits the shape of a Normal distribution - Idea of overlapping distributions does not mean that basketballs are not normally distributed, it just means that the lightest of the boys set are lighter than the heaviest of the girls set.. One appropriate comment whether arguing for or against Normal Distribution. Two relevant comments recognising weight of basketballs are normally distributed = t This must include some numerical evidence and also a suggested reason e.g. boys and girls basketballs some inflated and some left less inflated etc. Some specific numerical evidence needs to be mentioned e.g. one ND (girls) has a mean of around 535grams N0 no relevant comments N1 attempt at 1 question A3 two of u M5 one of r E7 one of t N2 one of u A4 three of u M6 two of r E8 two of t End of year L2 exam

37 Assessment Schedule: Mathematics and Statistics 91267: Probability, 2014 Evidence Statement Question Expected Coverage Achievement (u) Merit (r) Excellence (t) TWO (a) - i P(RRR) = 0.6 x = Correct probability (a) - ii P( C 2games) = CCC + CCR + CRC + RCC = 0.4 x (0.4 x 0.7 x 0.3) + (0.4 x 0.3 x 0.2) + (0.6 x 0.2 x 0.7) = Probability calculated using at least three of the four combinations possible. Correct probability (a) - iii P( Rua 2 games of 3) = RRC + RCR + CRR = (0.6 x 0.8 x 0.2) + (0.6 x 0.2 x 0.3) + (0.4 x 0.3 x 0.8) = x 0.228= 9 or 10 matches At least one probability of the three combinations calculated Correct expected number allow answer of 9.12 as long as workings seen (a) - iv So would need CC + CR + RC Or 1 RR = = 0.36 Correct probability calculated (d) Three branch tree diagram with branches labelled a, b and 1 a b Or a, b and 1 (a + b) P (0 or four points from two free shots) can only happen by two 0 s or two 2 s So a 2 + [(1 a b)(1 a b)] = a 2 + (1 a b a + a 2 + ab b + ab + b 2 ) e.g. Tree diagram set up with branches labelled or evidence at least that the three branches will add up to 1... or other similar evidence. Attempt to set up expression and multiply out but no errors. Logical reasoning and workings lead to correct final expression. Must see at least some evidence of multiplying out and collection of like terms. = 2a 2 + b 2 +2ab 2a 2b +1 as required. N0 no relevant comments N1 attempt at 1 question A3 two of u M5 one of r E7 one of t N2 one of u A4 three of u M6 two of r E8 two of t End of year L2 exam

38 Assessment Schedule: Mathematics and Statistics 91267: Probability, 2014 Evidence Statement Question Expected Coverage Achievement (u) Merit (r) Excellence (t) THREE (a) Not unimodal No tailing off Not bell shaped or other relevant evidence Correct and relevant evidence stated. (b) P(spec > 150) = 25/80 3 games in a row (25/80) 3 = Very unlikely BoT will take the view of paying early for gym improvements as that level of attendance 3 games in a row is only 3.1%. They can hold onto their money!! Probability for more than 150 spectators attending calculated Probability for three games in a row calculated. Correct and reasonable conclusion reached based on their probability calculation from. CON allowed here. (c)- i P( spec 100) =34/80 =17/40 Correct probability or equivalent.. (c)- ii P(win) = 55/80 = 11/16 Correct probability or equivalent. (c)- iii P (lost spec 100) = 22/34 = 11/17 Correct probability or equivalent. (d) P(attend won previous game) = 153/245 P(attend lost previous game) = 32/ /245 32/105 = 2.05 Just over twice as likely to attend the next game if the team had won the previous game. One of either probability calculated Relative risk calculation done correctly Making meaningful conclusion to explain the difference.. N0 no relevant comments N1 attempt at 1 question A3 two of u M5 one of r E7 one of t N2 one of u A4 three of u M6 two of r E8 two of t Judgement Statement Not Achieved Achievement Achievement with Merit Achievement with Excellence Score range End of year L2 exam