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NAME: TEACHER: Mathematics and Statistics 2013 Level 2 Standards: AS 91261 Apply algebraic methods in solving problems Page 2 AS 91262 Apply calculus methods in solving problems Page 13 AS 91267 Apply probability methods in solving problems Page 24 All Assessment schedules Produced by SINCOS Page 35 Page 1 2013 EOY Level 2 23 Writer: David Fortune NAME: TEACHER: Level 2 Mathematics and Statistics 2013 91261 Apply algebraic methods in solving problems Credits: Four Achievement Apply algebraic methods in solving problems. Achievement with Merit Apply algebraic methods, using relational thinking, in solving problems. Achievement with Excellence Apply algebraic methods, using extended abstract thinking, in solving problems. 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(s) provided at the back of this booklet and clearly number the question. Check that you have a copy of the formulae sheet. You are required to show algebraic working in this paper. Guess and check methods do not demonstrate relational thinking. Guess and check methods will limit grades to Achievement. YOU MUST HAND THIS BOOKLET TO YOUR TEACHER AT THE END OF THE EXAMINATION. TOTAL Produced by SINCOS Page 2 2013 EOY Level 2 23 Writer: David Fortune You are advised to spend 60 minutes answering the questions in this booklet. Assessor’s use only QUESTION ONE: (a) Solve: (i) log 9 x 3 _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (ii) x 8 65536 _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (iii) 6x 3 2x 2 _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 3 2013 EOY Level 2 23 Writer: David Fortune (b) 1 2 and x 2 5 2 Write the quadratic equation in standard form ax bx c 0 Assessor’s use only A quadratic equation has solutions x _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (c) Cody has three wooden spherical balls. The radius of the middle sized ball is twice the radius of the small ball. The radius of the largest ball is 5 cm larger than the radius of the smallest ball. The surface area of the largest ball is equal to the sum of the surface areas of the two smaller balls. Find the radius of the middle sized ball. Note: The surface area of a sphere is given by the formula SA 4 r 2 _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 4 2013 EOY Level 2 23 Writer: David Fortune (d) Find the set of values for k for which the equation kx 2 x k 1 0 has real roots, one Assessor’s use only of which is positive and one of which is negative. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 5 2013 EOY Level 2 23 Writer: David Fortune Assessor’s use only QUESTION TWO (a) Simplify 15 x 2 y 2 (3 y ) 2 7 wx 5w _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (b) (i) Simplify by writing as a single fraction 4 8 x 2 2x 1 _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (ii) Solve 4 8 10 x 2 2x 1 _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 6 2013 EOY Level 2 23 Writer: David Fortune (c) Assessor’s use only Hemi buys a new car for $25000. The value of her second hand car, V , after t years, can be modelled by the equation V C (0.75) t where C is the Cost price of the car in dollars. (i) How much has the car lost in value after Hemi has owned the car for 5 years? _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (ii) After how long would Hemi’s car be worth 40 % of the purchase price? _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 7 2013 EOY Level 2 23 Writer: David Fortune (d) Assessor’s use only For what value(s) of q does the straight line y q 2 x intersect with the hyperbola xy 5 in two separate places. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 8 2013 EOY Level 2 23 Writer: David Fortune Assessor’s use only QUESTION THREE (a) (i) Factorise 3x 2 5 x 2 _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (ii) Solve 3x 2 5 x 2 0 _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (b) Simplify 64 x 14 4 x 2 1 4 _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 9 2013 EOY Level 2 23 Writer: David Fortune (c) Assessor’s use only Solve log 4 ( x 4) log 4 ( x 4) 2 _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (d) Two numbers differ by 5. Prove that the difference of their squares is a multiple of 5. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 10 2013 EOY Level 2 23 Writer: David Fortune Assessor’s use only (e) Solve the equation 4 2 x 9 4 x 20 . Hint: Let u 4 x . _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 11 2013 EOY Level 2 23 Writer: David Fortune Assessor’s use only Extra paper for continuing your answers, if required. Clearly number the question(s). Question Number _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 12 2013 EOY Level 2 23 Writer: David Fortune NAME: TEACHER: Level 2 Mathematics and Statistics 2013 91262 Apply calculus methods in solving problems Credits: Four Achievement Apply calculus methods in solving problems. Achievement with Merit Apply calculus methods, using relational thinking, in solving problems. Achievement with Excellence Apply calculus methods, using extended abstract thinking, in solving problems. 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(s) provided at the back of this booklet and clearly number the question. Check that you have a copy of the formulae sheet. YOU MUST HAND THIS BOOKLET TO YOUR TEACHER AT THE END OF THE EXAMINATION. TOTAL Produced by SINCOS Page 13 2013 EOY Level 2 23 Writer: David Fortune You are advised to spend 60 minutes answering the questions in this booklet. Assessor’s use only QUESTION ONE: (a) A function f is given by f ( x) 3x 2 4 x 7 Find the gradient at the point where x 2 . _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (b) The gradient at a point on a curve is given dy 3x 2 4 dx Find the equation of the curve given that the curve passes through the point (1, 2) . _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 14 2013 EOY Level 2 23 Writer: David Fortune (c) The equation of a parabola is y 3x 2 2 x 4 . Assessor’s use only Find the equation of the tangent to the parabola at the point (1, 1) . _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (d) If g ( x) 2 , g (2) 2 , g ( 2) 2 and g (0) 2 then find g (x ) . _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 15 2013 EOY Level 2 23 Writer: David Fortune (e) Assessor’s use only A rectangle is formed with two sides on the x and y axes as shown. One corner of the rectangle is at the origin and the opposite corner is on the curve y 9 2 x 2 . Find the maximum area of the rectangle. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 16 2013 EOY Level 2 23 Writer: David Fortune Assessor’s use only QUESTION TWO: (a) The graph shows the function y f (x) . On the axes below, sketch the graph of the gradient function y f (x) . Produced by SINCOS Page 17 2013 EOY Level 2 23 Writer: David Fortune (b) Assessor’s use only A curve y f (x) passes through the origin and has a gradient function dy 2 x 3x 2 dx Find the coordinates of the point on the curve where x = 3. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (c) The velocity, v, in m s-1, of a train is v(t ) 0.2t 3 0 t 120 where t is the time in seconds after leaving the station. Three seconds after starting it is 9 m from the station. Find the distance from the station after 20 seconds. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 18 2013 EOY Level 2 23 Writer: David Fortune (d) Assessor’s use only The profit function, P (t ) , of the Acme Manufacturing Company is given by P(t ) 0.01t 2 0.75kt 1000 where t is time in months after starting the company and k is a constant. Given the profit is decreasing for t > 45 months find the value of k. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (e) When a person coughs the windpipe (trachea) contracts, allowing air to be expelled at maximum velocity. The velocity v of airflow during a cough can be modelled by the function v kr 2 (R r ) where r is the trachea’s radius, in cm, during a cough and R is the normal radius of the trachea, in cm and k is a positive constant that depends on the length of the trachea. Find the radius for which the airflow is greatest during a cough. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 19 2013 EOY Level 2 23 Writer: David Fortune Assessor’s use only QUESTION THREE: (a) Find the coordinates of the function h( x) 2.5x 2 3.5x 7 where the gradient is 1.5. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (b) The surface area, SA, of a cube changes as the length, x, of the sides change. SA 6x 2 Find the rate of change of the surface area, with respect to the length of the side, when the side is 3 units. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 20 2013 EOY Level 2 23 Writer: David Fortune (c) The graph shows the derivative function h(x) . Assessor’s use only Sketch the graph of h(x) . Produced by SINCOS Page 21 2013 EOY Level 2 23 Writer: David Fortune (d) A rocket starts from rest. Its acceleration, in m s-2, is given by a (t ) Assessor’s use only 1 2 (t 4) . 4 How far does it travel in the tenth second? _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (e) Find the maximum vertical distance between these curves y 4 x 3 15x 2 20 x 7 and y 3x 2 16 x 10 . [NOTE: Only consider 5 x 1.4 , the region between the intersections] ________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 22 2013 EOY Level 2 23 Writer: David Fortune Assessor’s use only Extra paper for continuing your answers, if required. Clearly number the question(s). Question Number _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 23 2013 EOY Level 2 23 Writer: David Fortune NAME: TEACHER: Level 2 Mathematics and Statistics 2013 91267 Apply probability methods in solving problems Credits: Four Achievement Apply probability methods in solving problems. Achievement with Merit Apply probability methods, using relational thinking, in solving problems. Achievement with Excellence Apply probability methods, using extended abstract thinking, in solving problems. 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(s) provided at the back of this booklet and clearly number the question. Check that you have a copy of the formulae sheet. YOU MUST HAND THIS BOOKLET TO YOUR TEACHER AT THE END OF THE EXAMINATION. TOTAL Produced by SINCOS Page 24 2013 EOY Level 2 23 Writer: David Fortune Assessor’s use only You are advised to spend 60 minutes answering the questions in this booklet. QUESTION ONE A manufacturer of digital watches finds that the watches vary in how accurately they keep the time. The time gained per day is modelled by a normal distribution. On average they gain one second per day, with a standard deviation of 2.5 seconds per day. The questions below relate to this manufacturer’s digital watches (a) (i) What is the probability that a watch, selected at random, gains between 1 and 4 seconds per day? _________________________________________________________________________ _________________________________________________________________________ (ii) What percentage of watches gain at least 8 seconds per day? _________________________________________________________________________ _________________________________________________________________________ (iii) What is the probability a watch, selected at random, loses time? _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (iv) In a batch of 1200 watches what is the expected number of watches that are out by at least 1.5 seconds per day? _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 25 2013 EOY Level 2 23 Writer: David Fortune Assessor’s use only (v) Because of the issue of warranties the manufacturer wants to limit the number of watches gaining more than 3 seconds per day to 10 % of the total number produced. The manufacturer can reset the standard deviation of the time lost per day, whilst keeping the same mean. What should the new standard deviation be after resetting? _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 26 2013 EOY Level 2 23 Writer: David Fortune (b) Assessor’s use only A garden centre purchases potting mix from two different suppliers, Growquick and Readyfast. The percentage moisture content in the potting mix from each supplier is normally distributed with mean and standard deviation as shown. Mean Growquick Readyfast 25.1% 23.5% Standard deviation 4.2% 4.9% The garden centre wishes to purchase a large amount of potting mix with moisture content under 30%. Which supplier should they choose? Give statistical reasons. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (c) The time that people take to complete a puzzle is normally distributed with mean of 18 minutes and a standard deviation of 5 minutes. A large group of people were given the puzzle. If 40 people completed the puzzle in under 10 minutes then estimate how many people took longer than 10 minutes. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 27 2013 EOY Level 2 23 Writer: David Fortune Assessor’s use only QUESTION TWO (a) The table is below is from the Ministry of Transport “CRASH FACT SHEET 2012”. BAC 0 30 50 80 Relative risk of fatal crash by blood alcohol level 30+ 20-29 15-19 years years years 1 3 5.3 2.9 8.7 15 5.8 17.5 30.3 16.5 50.2 86.8 BAC is the driver’s Blood Alcohol Concentration, in mg of alcohol per 100mL of blood The calculation of risk is in relation to that of a sober driver aged 30+ years. (i) Explain what the 5.3 in the first row means. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (ii) Estimate the relative risk, for all 3 age groups, for a driver with a BAC of 80 compared to a driver with a BAC of 50. Comment on your findings. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 28 2013 EOY Level 2 23 Writer: David Fortune Assessor’s use only (b) The Newtown district kept statistics on driving licence applicants. They recorded whether the applicants had professional (paid) driving lessons from an instructor, as well as whether they passed or failed. The results are shown in the table below: Passed test Failed test Had paid lessons Didn’t have paid lessons TOTALS TOTAL 53 8 61 47 27 74 100 35 135 (i) What proportion of applicants in Newtown passed the driving test? _________________________________________________________________________ _________________________________________________________________________ (ii) What percentage of those people in Newtown, who paid for driving lessons, passed their driving test? _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (iii) Show that the risk of failing the driving test in Newtown is about 1 in 4. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 29 2013 EOY Level 2 23 Writer: David Fortune (iv) ACE driving school claimed in an advertisement that people that had paid for driving Assessor’s use only lessons were one third more likely to pass than if they had not had paid for lessons. State whether or not you agree, stating full reasons and showing relevant calculations. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (v) A researcher looked at the gender of all applicants in the above group of applicants. There were 56 male applicants of whom 38 passed. A newspaper headline stated that the risk for males failing their driving test in Newtown was approximately the same as the risk of females failing. Comment on the claim. You should provide suitable reasons, backed by calculations from the results table. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 30 2013 EOY Level 2 23 Writer: David Fortune QUESTION THREE (a) Assessor’s use only On average 1 in 20 people have the particular antibody mGF in their blood. The machine that tests people for this antibody does not always give the correct result. Of the people with mGF 94% test positive on the machine (ie test as having mGF), 4% test negative and 2 % are inconclusive. Of the people without the antibody mGF, 92 % test negative, 5% test positive and the rest are inconclusive. (i) What is the probability that a person, selected at random, had the mGF antibody and the test was positive. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (ii) What is the percentage of people with a negative test? _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 31 2013 EOY Level 2 23 Writer: David Fortune (iii) If a person selected at random had a positive test what is the probability that they in Assessor’s use only fact had the mGF antibody? _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (iv) In one town where everybody was tested there were 710 inconclusive results. What is the expected number of people in this town? _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 32 2013 EOY Level 2 23 Writer: David Fortune Assessor’s use only (b) Venture Tours takes tours to popular destinations. The leader of a tour to Kingstown tells the group members that the weather in Kingstown is “very reliable”. He explains “very reliable” by stating that if the weather is good one day there is an seventy five percent chance it will also be good the next day, however if the weather is bad one day there is only 40% chance it will be bad the next day. The tour arrives on Tuesday and the weather is good. (i) What is the probability that the weather will also be good on Friday (3 days later)? _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (ii) The tour has one outdoor experience day planned. What is the probability that at least 1 of the next 3 days is a good day? _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 33 2013 EOY Level 2 23 Writer: David Fortune Assessor’s use only Extra paper for continuing your answers, if required. Clearly number the question(s). Question Number _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 34 2013 EOY Level 2 23 Writer: David Fortune ASSESSMENT SCHEDULE 91261 Apply algebraic methods in solving problems Achievement Achievement with Merit Achievement with Excellence Apply algebraic methods in solving problems involves: Apply algebraic method, using relational thinking, in solving problems involves one or more of: Apply algebraic methods, using extended abstract thinking, in solving problems involves one or more of: • selecting and using methods • demonstrating knowledge of algebraic concepts and terms • communicating using appropriate representations. • selecting and carrying out a logical sequence of steps • connecting different concepts or representations • demonstrating understanding of concepts • forming and using a model; and also relating findings to a context, or communicating thinking using appropriate mathematical statements. • devising a strategy to investigate or solve a problem • identifying relevant concepts in context • developing a chain of logical reasoning, or proof • forming a generalisation; and also using correct mathematical statements, or communicating mathematical insight. Sufficiency for each question: N0: No response, no relevant evidence. N1: Attempt at ONE question with some algebraic knowledge. N2: 1 u A3: 2 u A4: 3 u M5: 1 r M6: 2 r E7: 1 t E8: 2 t Judgement Statement Score range Produced by SINCOS Not Achieved Achievement Achievement with Merit Achievement with Excellence 0-7 8 - 13 14 - 18 19 -24 Schedule Page 35 2013 EOY Level 2 Writer: David Fortune 23 Question ONE Evidence Achievement (u) Apply algebraic methods in solving problems. Merit (r) Apply algebraic methods, using relational thinking, in solving problems. Excellence (t) Apply algebraic methods using extended abstract thinking, in solving problems. 1a(i) x 9 3 729 Correct solution. 1a(ii) x4 Correct solution. 1(iii) 2x 2 6x 3 0 x 2.366 x 0.634 Written as a quadratic equation = 0 and ONE solution found. Both correct solutions. (2 x 1)(5 x 2) 0 Correct equation written in any form. Correct equation found in correct form. Correct quadratic equation found. Quadratic equation solved. Problem solved correctly. Found solutions to quadratic equation. Found 1 4k (k 1) 1 Problem solved. 1b Equation must be 10 x 2 x 2 0 1c [or multiple] Let r be radius of small ball. 4 r 4 (2r ) 4 (r 5) 5 r 2 (r 5) 2 r 2 10r 25 4r 2 10r 25 0 r 4.045 r 1.545 2 2 2 Radius of middle ball = 8.1 cm 1d Using quadratic formula 1 1 4k (k 1) x 2k [Accept with signs.] For one solution to be positive 1 4k (k 1) 1 (numerator positive) k (k 1) 0 0 k 1 Produced by SINCOS Schedule Page 36 2013 EOY Level 2 Writer: David Fortune 23 Question TWO Evidence Achievement (u) Apply algebraic methods in solving problems. 2a Merit (r) Apply algebraic methods, using relational thinking, in solving problems. Excellence (t) Apply algebraic methods using extended abstract thinking, in solving problems. 15 x 2 y 2 (3 y ) 2 15 xy 2 5w 2 7 wx 5w 7w 9y 25 x 21 Correct expression, fully simplified. 2b(i) 4 8 20 x 2 2x 1 ( x 2)( 2 x 1) Correct expression, fully simplified. 2b(ii) 20 10 ( x 2)( 2 x 1) 2 ( x 2)( 2 x 1) 2 x 2 3x 0 x(2 x 3) 0 x 0 x 1.5 Correct quadratic equation set up. Both correct solutions. 2c(i) 25000 25000(0.75) 5 Value after 5 years found. Correct solution. Correct simplified equation (line 2). Problem solved but not in context. OR Set up equation t = Problem solved in context. Correct equation set up to solve. q 2 40 obtained Both correct solutions. [Accept with ] = $19 067.38 [Accept $19 067] 2c(ii) 10000 25000 (0.75) t 0.4 (0.75) t log 0.4 t 3.185 log 0.75 ie has value 40% after 3.185 years 2d x( q 2 x) 5 2 x 2 qx 5 0 q q 2 40 4 For 2 solutions q 2 40 q 6.32 q 6.32 but not correct solution. x Produced by SINCOS Schedule Page 37 2013 EOY Level 2 Writer: David Fortune 23 Question THREE Evidence Achievement (u) Apply algebraic methods in solving problems. 3a(i) 3a(ii) 3b (3x 1)( x 2) Correct factorisation. (3x 1)( x 2) 0 1 x x 2 3 Both correct solutions. 64 x 14 4x 16 x 16 1 16 x 16 1 4 1 2 4 64 x14 2 4x 1 4 1 4 Merit (r) Apply algebraic methods, using relational thinking, in solving problems. Expression simplified to line 2 OR equivalent. Correct expression. fully simplified. Correct expression for x2. Both correct solutions. Excellence (t) Apply algebraic methods using extended abstract thinking, in solving problems. 1 2x 4 3c log 4 ( x 2 16) 2 x 2 16 4 2 x 2 32 x 32 5.66 3d x 2 y 2 ( x y)( x y) 5( x y ) as ( x y ) =5 Almost correct proof with mainly justified steps shown. Correct proof with justified steps shown. One correct solution. Both correct solutions. (numbers differ by 5) Thus x 2 y 2 is a multiple of 5 OR equivalent 3e u 2 9u 20 0 (u 5)(u 4) 0 u 5 or u 4 4x 5 4x 4 log 5 x = 1.161 log 4 Produced by SINCOS Quadratic equation solved. x=1 Schedule Page 38 2013 EOY Level 2 Writer: David Fortune 23 ASSESSMENT SCHEDULE 91262 Apply calculus methods in solving problems Achievement Achievement with Merit Achievement with Excellence Apply calculus methods in solving problems involves: Apply calculus methods, using relational thinking, in solving problems involves one or more of: Apply calculus methods, using extended abstract thinking, involves one or more of: • selecting and using methods • demonstrating knowledge of calculus concepts and terms • communicating using appropriate representations. • selecting and using a logical sequence of steps • connecting different concepts or representations • demonstrating understanding of concepts • forming and using a model; and also relating findings to a context, or communicating thinking using appropriate mathematical statements. • devising in solving problems a strategy to investigate a problem • demonstrating understanding of abstract concepts • developing a chain of logical reasoning, or proof • forming a generalisation; and also using correct mathematical statements, or communicating mathematical insight. Sufficiency for each question: N0: No response, no relevant evidence. N1: Attempt at ONE question demonstrating limited knowledge of calculus techniques. N2: 1 u A3: 2 u A4: 3 u M5: 1 r M6: 2 r E7: 1 t E8: 2 t Judgement Statement Score range Produced by SINCOS Not Achieved Achievement Achievement with Merit Achievement with Excellence 0-7 8 - 14 15 - 20 21 -24 Schedule Page 39 2013 EOY Level 2 Writer: David Fortune 23 Question ONE Evidence Achievement (u) Apply calculus methods in solving problems. 1a 1b f ( x) 6 x 4 Merit (r) Excellence (t) Apply calculus methods, using relational thinking, in solving problems. Apply calculus methods, using extended abstract thinking, in solving problems. Gradient correct. At x = 2 gradient = 8 y x 3 4x c At (1, 2) 2 1 4 c Correct function. y x 3 4x 1 1c 1d 1e dy 6x 2 dx At (1, 1) the gradient = 8 Tangent is y 1 8( x 1) y 8 x 7 g ( x) 2 Thus g ( x) 2 x c But g ( 2) 2 ie 2 4 c and c 2 ie g ( x) 2 x 2 Thus g ( x) x 2 2 x c But g ( 2) 2 ie 2 4 4 c c 2 ie g ( x) x 2 2 x 2 1 Thus g ( x) x 2 x 2 2 x c 3 But g (0) 2 ie c 2 1 g ( x) x 2 x 2 2 x 2 3 Gradient correct. Area = xy x(9 2 x 2 ) A found. 9x 2x3 A 9 6 x 2 Correct tangent found. Correct function for g (x ) found. Correct function for g (x ) found. Values of x OR y found at maximum. Maximum area correctly found. 3 2 y6 = 0 when x 2 Max area = Produced by SINCOS 3 6 7.35 2 Schedule Page 40 2013 EOY Level 2 Writer: David Fortune 23 Question TWO Achievement (u) Evidence Apply calculus methods in solving problems. 2a 2b Merit (r) Apply calculus methods, using relational thinking, in solving problems. Excellence (t) Apply calculus methods, using extended abstract thinking, in solving problems. Sketch is a straight line with negative gradient and x intercept between 1 and 2. Correct coordinates. dy 2 x 3x 2 dx y x 2 x3 c Through origin so c = 0 Equation y x 2 x 3 So coordinates are (3, -18) 2c v(t ) 0.2t 3 s(t ) 0.1t 3t c At t = 3 9 0.9 9 c c = 0.9 s(t ) 0.1t 2 3t 0.9 After 20 sec distance = 99.1 m 2 2d P 0.02t 0.75k Decreasing when 0.02t 0.75k 0 t 75 k ie for t 2 Correct antiderivative. Correct distance. P(x) correctly differentiated and made less than zero.. Correct inequality in terms of t and k. Correct value for k. Also decreasing for t > 45 45 2e 75 k 2 k 1.2 v kr 2 (R r ) kRr 2 kr 3 v 2kRr 3kr 2 2 = 0 when r = 0 or r 3 v correctly differentiated. Correct maximum. Correct maximum, with justification for maximum. r = 0 is a minimum (zero velocity) and r 2 R gives maximum velocity. 3 Produced by SINCOS Schedule Page 41 2013 EOY Level 2 Writer: David Fortune 23 Question THREE Evidence Achievement (u) Apply calculus methods in solving problems. 3a h( x) 5 x 3.5 1.5 5x 3.5 dSA 12 x dx Graph x < 0 any straight line with gradient 3 OR Graph for x > 0 an “upside down” parabola with turning point at x = 3. 3c v(t ) Apply calculus methods, using extended abstract thinking, in solving problems. Rate of change correct. At x = 3 rate of change = 36. 3d Apply calculus methods, using relational thinking, in solving problems. Excellence (t) Coordinates correct. ie x = 1 and y = 6 Coordinates are (1,6) 3b Merit (r) 1 3 t t c 12 Correct velocity, showing constant has been calculated. At t = 0 v = 0 so c = 0 1 3 t t 12 1 4 1 2 s (t ) t t c 48 2 v(t ) Graph x < 0 any straight line with gradient 3 AND Graph for x > 0 an “upside down” parabola with turning point at x = 3. [No graph at x = 0.] Correct distance function, showing constant has been calculated. OR Distance in 9th second calculated. Correct distance, showing constants have been calculated. Found max distance but not showed the value found is a maximum OR Found incorrect distance. Found max distance and showed the value found is a maximum using any method. Distance travelled in 10th second 1 3e 1 10 = t 4 t 2 = 62.1 m 2 9 48 Vertical distance = 4 x 3 15 x 2 20 x 7 3x 2 16 x 10 4 x 12 x 36 x 17 dV 12 x 2 24 x 36 dx dV 0 When dx ( x 1)( x 3) 0 x 1 x 3 3 2 Derivative correct and equated to zero. dV 2 24 x 24 so max at x 3 d 2x Max distance = 125 Produced by SINCOS Schedule Page 42 2013 EOY Level 2 Writer: David Fortune 23 ASSESSMENT SCHEDULE 91267 Apply probability methods in solving problems Achievement Achievement with Merit Achievement with Excellence Apply probability methods in solving problems involves: Apply probability methods, using relational thinking, in solving problems involves one or more of: Apply probability methods, using extended abstract thinking, in solving problems involves one or more of: • selecting and using methods • demonstrating knowledge of probability concepts and terms • communicating using appropriate representations. • selecting and carrying out a logical sequence of steps • connecting different concepts or representations • demonstrating understanding of concepts; and also relating findings to a context, or communicating thinking using appropriate mathematical statements. • devising a strategy to investigate or solve a problem • identifying relevant concepts in context • developing a chain of logical reasoning • making a statistical generalisation; and also, where appropriate, using contextual knowledge to reflect on the answer. . Sufficiency for each question: N0: no relevant evidence N1: Attempt at one question with some knowledge shown. N2: 1 u A3: 2 u A4: 3 u M5: 2 r M6: 3 r E7: 1 t E8: 2 t Judgement Statement Score range Produced by SINCOS Not Achieved Achievement Achievement with Merit Achievement with Excellence 0-8 9 - 14 15 - 19 20 -24 Schedule Page 43 2013 EOY Level 2 Writer: David Fortune 23 Question One Evidence Achievement (u) Apply probability methods in solving problems. Merit (r) Apply probability methods, using relational thinking, in solving problems. 1a(i) 0 < z < 2.5 Probability = 0.38493 Correct probability calculated. 1a(ii) z > 3.2 Probability = 0.0025551 0.26 percent Correct probability calculated and answer written as percentage. 1a(iii) z < - 0.4 Probability = 0.34457 Correct probability calculated. 1a(iv) z > 0.6 and z < - 1 Probability = 0.5794 Expected number 695 (or 696) Correct probability calculated. Expected number calculated. 1a(v) 10 % is z = 1.281 Inverse normal value of z found. CAO Found the % value for one grower. Correct supplier with explanation / justification. x 3 1 1.281 Excellence (t) Apply probability methods, using extended abstract thinking, in solving problems. Correct new sd with justification. 1.281 New sd is 1.56 1b 1c For Growquick 87.8 % of potting mix has less than 30% water content. For Readygrow 90.7 % of potting mix has less than 30% water content. Thus preferred supplier is Readygrow. Prob ( People taking less than 10 mins) = 0.054799 40 x 0.054799 0.945201 CAO – no credit. Prob 0.054779 found. Whole group number found (730 people). Correct justified solution. The number taking more than 10 minutes = x = 689 (or 690) OR 40 0.054799 p p population p 40 729.9 0.054799 Other people = 729.9 – 40 = 689 NOTE: There will be differences between probability answers quoted if students use normal distribution tables rather than GC. Either should be accepted. Values used here are all GC values Produced by SINCOS Schedule Page 44 2013 EOY Level 2 Writer: David Fortune 23 Question TWO Achievement (u) Evidence Apply probability methods in solving problems. 2a(i) 2a(ii) A 15-19 year old driver is 5.3 times as likely to have a fatal crash as a 30+ year old. 30+ yo relative risk = 20-29 yo relative risk = 2.87 15-19 yo relative risk = 2.86 Increasing the alcohol amount increases the risk for all drivers by a factor of 2.8, regardless of their age. 2b(i) 100 135 0.741 2b(ii) 53 61 0.869 ie 87 % of those who paid, passed their test. 2b(iii) P (failing) = 35 135 0.259 1 in 4 is 0.25 so statement correct. 2b(iv) Risk (paid and passing) = 0.869 Risk (not paid and passing) = 0.635 Relative risk paid to not paid 1.369 So 37 % more likely to pass if they paid. Ad under rates relative risk – actually 37% more likely rather than 1/3 2b(v) Male Female TOTAL passed failed Tot 38 62 100 18 17 35 56 79 135 Excellence (t) Apply probability methods, using relational thinking, in solving problems. Apply probability methods, using extended abstract thinking, in solving problems. Correct understanding. Correct relative risks calculated. 16.5 = 2.84 5.8 Merit (r) Correct relative risks calculated AND at least one correct statement. Correct probability calculated. Correct percentage calculated. Correct probability calculated. Correct probability with justification statement. A correct probability calculated. Both probabilities calculated correctly. A correct probability calculated. Both probabilities calculated correctly.. Probabilities correctly compared and a justified statement made about the claim being close to correct. Probabilities correctly compared and a justified statement made about the claim being wrong. 18 0.321 56 17 0.215 Risk of female failing = 79 Risk of male failing = So relative risk of male failing to female failing is 1.49 ie males are 50 % more likely to fail than females The claim is wrong. Produced by SINCOS Schedule Page 45 2013 EOY Level 2 Writer: David Fortune 23 Question Three Evidence Achievement (u) Apply probability methods in solving problems. 3a(i) 0.05 0.94 0.047 Correct probability. 3a(ii) 0.05 0.04 0.95 0.92 0.876 Correct percentage. = 87.6 % 3a(iii) Pr (positive test) = 0.05 0.94 0.95 0.05 0.0945 Pr had mGF if they had a positive test = 3a(iv) Pr(positive test) correctly calculated. Merit (r) Apply probability methods, using relational thinking, in solving problems. Excellence (t) Apply probability methods, using extended abstract thinking, in solving problems. Correct probability. 0.05 0.94 0.497 0.0945 P (inconclusive) = 0.05 0.02 0.95 0.0.03 = 0.0295 A relevant correct probability calculated [eg pr(inconclusive)]. 0.0295 x 710 Expected number correctly calculated. x = 24067 people (or 24068) 3b(i) Prob = 0.753 .75 .25 .6 2 .25 .4 .6 = 0.706875 A correct relevant probability calculated. Correct total probability calculated. 3b(ii) 1 0.25 0.4 2 = 0.96 A correct relevant probability calculated. One error in total probability calculated. OR sum of 7 probabilities. Produced by SINCOS Schedule Page 46 Correct total probability calculated with explanation. 2013 EOY Level 2 Writer: David Fortune 23