MATHEMATICS (Project Maths Phase 2)

Size: px

Start display at page:

Download "MATHEMATICS (Project Maths Phase 2)"

Ashley Sparks
5 years ago
Views:

1 PRE-JUNIOR CERTIFICATE EXAMINATION, 2012 MARKING SCHEME MATHEMATICS (Project Maths Phase 2) HIGHER LEVEL Page 1 of 44

2 PAPER 1 Page 2 of 44

3 Question 1 Part (a) Part (b) Part (c) Scale 10C Scale 5A Scale 5C A = { 1, 2,3, 4,6,12} and B = { multiples of 3 less than 20} (a) Find A B. B = { 3, 6,9,12,15,18} B { 3, 6,12} A = Fully correct solution Lists the elements of B and stops Omits one element or one element incorrect in final answer Any correct element listed (b) Write down a proper subset of the set A. Accept any proper subset One correct subset listed (c) Write down all the subsets of ( A B)? { 3,6,12,3,6,3,12,6,12,3,6,12, } { } { } { } { } { } { } { }. (4 Marks) (3 Marks) Fully correct solution At least five correct subsets Any correct subset Page 3 of 44

4 Question 2 Part (a) Part (b) Scale 10C Scale 5B 100 teenagers were asked if they had a Personal Computer, a games console or a laptop at home. 45 had a Personal Computer (PC), 62 had a Games Console (GC) while 66 had a Laptop (L). 32 had a PC and a Games Console, 39 had a Games Console and a Laptop and 29 had a PC and a Laptop, 24 owned all three while 3 owned none of the three. (a) Represent the above information on the following Venn diagram Fully correct solution No more than two incorrect entries Any correct entry (b) Find the probability that a student chosen at random from the group will own exactly two of the items Partial (3 Marks) Correct probability from candidates Venn Diagram Any correct probability from the candidates Venn Diagram Page 4 of 44

5 Question 3 Part (a) Part (b) Scale 5B Scale 5A (a) Draw a Venn diagram to illustrate the set of numbers for the Natural Numbers (N), the Integers (Z) and the Real numbers (R). Partial (3 Marks) Fully correct Venn Diagram Partially correct Venn Diagram (b) Using your Venn diagram or otherwise explain whether the following statement is true or false Z N FALSE. N Z Fully correct Page 5 of 44

6 Question 4 Scale 10C Aoife has an annual salary of 65,000. Income up to the standard rate cut off point of 28,000 is taxed at 23%. If the high rate of tax is 42% and Aoife has tax credits of 3500 calculate her monthly take home pay correct to the nearest euro % = 6440 and = Gross Tax = Tax Payable = = Net Monthly Income = = 3877 ( ) Fully correct solution Correct method with one calculation error Mishandles gross wage Page 6 of 44

7 Question 5 Part (a) Part (b) Scale 10C Scale 10C 12,000 is invested at 4% compound interest for 3 years. Tax of 5% is deducted from the interest in each year except the final year of the investment. (a) Calculate the value of the investment at the end of the first two years correct to the nearest euro. Year 1 = = = = Year 2 = = = = Fully correct solution Correct method with one calculation error Mishandles the 5% tax on interest in both years (b) At the beginning of the third year a further 12,000 is invested at r%. If at the end of the year the investment amounts to calculate the value of r = = = 5% Fully correct solution Correct method with one calculation error Page 7 of 44

8 Question 6 Part (a) Scale 10C Seán is considering purchasing a new television for Christmas. The present cost of the television is 390 excluding VAT of 21%. Seán knows that his local shop will have a 15% sale on the cost of the television exclusive of VAT in January, but also knows that the VAT rate will increase by 2% after the budget in December. By calculating the cost of the television now and in January, decide whether Seán should purchase now or wait until after Christmas. Present Cost = = Sale Price = = = Conclusion: Buy now Fully correct solution Correct solution but no conclusion Mishandles a percentage once only Page 8 of 44

9 Question 7 Part (a) Part (b) Scale 10B Scale 10C (a) Find the value of x if x + 9 = 5 ( ) 2 2 x + 9 = 5 x + 9= 25 x = 16 Partial (6 Marks) Fully correct solution (b) The length of a rectangle of perimeter 24 cm is three times the width. Calculate the area of the rectangle. 3x + 3x+ x+ x= 24 x= 3 cm Area = 3 9= 27 cm 2 Fully correct solution Incorrect answer when solving but continues to find area correctly for incorrect Page 9 of 44

10 Question 8 Part (a) Part (b) Scale 10C Scale 5B (a) Solve the inequality 5 3x + 4< 25, x Z. 9 3x < 21 3 x < 7 Fully correct solution One error in solving inequality (b) Graph your solution on the number line below Partial (3 Marks) Fully correct number line Partially correct number line Page 10 of 44

11 Question 9 Part (a) Part (b) Part (c) Part (d) Scale 5B Scale 10B Scale 10C Scale 10B Four CD s and three DVD s cost a total of 68. One CD and two DVD s cost 27. (a) If both items are less than 12 each, choose suitable values to find the cost of both items. Solution by trialling Partial (3 Marks) Correct solutions found Attempts to find solutions (b) By le x be thee the price of a CD and y be the price of a DVD write down two equations in x and y to represent the above information. 4x + 3y = 68, x + 2y = 27 Both equation correct Partial (6 Marks) One equation correct or partially correct (c) Use the equations above to find the cost of a DVD and a CD. x = 11, y = 8 Fully correct solution based on candidates equations if not oversimplified Only solves for one variable Problem oversimplified by equations formed in (b) Page 11 of 44

12 (d) Verify your solution by plotting the lines below. Partial (6 Marks) Both lines plotted correctly Partially correct Plots intersection point only Page 12 of 44

13 Question 10 Part (a) Part (b) Part (c) Scale 10B Scale 5B Scale 10C A farmer is fencing a small plot of land to grow vegetables as shown below. (a) If the farmer has 26 m of wire to enclose the plot write an expression in x for the length of the plot. 26 2x = 13 x 2 Partial (6 Marks) Correct expression Partially correct expression (b) Show that the area of the plot can be expressed in the form 13x x 2 2 ( 13 ) = 13 x x x x Partial (3 Marks) Correct expression found Any correct valid attempt to find expression for area Page 13 of 44

14 (c) If the plot must have an area between 22 m 2 and 42 m 2, find the possible range of values for the length of the plot. 2 2 x x x x = 0 and = 0 x= 2,11 and x= 6,7 Lenght is between 7 m and 11 m Fully correct solution Correct method but with minor errors Correct range not found or stated Correct range not found and not stated Page 14 of 44

15 Question 11 Part (a) Part (b) Part (c) Part (d) Scale 25C Scale 5A Scale 10B Scale 10C 2 A graph of the function y = x + bx+ c, where bc, Ris shown below (a) Using the information on the graph find the value of a and the value of b. 2 2 ( ) ( ) ( ) ( ) 1 + b 1 + c= 16 and 2 + b 2 + c= 7 b= 2, c= 15 (25 Marks) (20 Marks) Fully correct solution Correct method but with minor errors Correct range not found or stated Correct range not found and not stated Page 15 of 44

16 (b) Find the coordinate of the point R. R( 0, 15) Correct point stated (c) Find the coordinates of the points S and T where the graph crosses the x-axis S ( 5,0 ) T( 3,0) Partial (6 Marks) Both points correct One point correct 2 (d) Hence solve the equation ( t ) ( t ) = 0 for t 0. 3t 1= 5 or 3t 1= t =, 3 3 Fully correct solution Correct method but with minor errors Correct range not found or stated Correct range not found and not stated Page 16 of 44

17 Question 12 Part (a) Part (b) Part (c) Part (d) Part (e) Scale 10B Scale 15C Scale 5A Scale 5B Scale 5B A Gaelic football player takes a free kick off the ground. The balls height above the ground is given by the formula y = 8x x 2, where x is the time in seconds the ball is in flight and y is the height of the ball in metes. (a) How long will it take the ball to hit the ground after it has been kicked? 8 seconds Partial (6 Marks) Correct answer with or without work shown Incorrect answer with work shown otherwise award 0 (b) Choose a suitable range and draw a graph of the height of the ball as it travels. (15 Marks) (13 Marks) Fully correct graph Unsuitable range leads to incorrect graph Graph joined with straight lines Some points found Page 17 of 44

18 (c) Use your graph to estimate the maximum height of the ball Max. height = 16 metres Correct answer with or without work shown on graph (d) If the ball travelled at an average speed of 4 metres per second, calculate the distance the ball travelled. Distance = 4 8 = 32 m Partial (3 Marks) Correct solution with work shown Correct solution without work shown (e) How far from the player was the ball at its maximum height? Distance = 4 4 = 16 m Partial (3 Marks) Correct solution with work shown Correct solution without work shown Page 18 of 44

19 Question 13 Part (a) Part (b) Part (c) Part (d) Scale 10B Scale 5B Scale 10B Scale 5B (a) Write the following formula in terms of V. W 1 = CV 2 2 V = 2W C Partial (6 Marks) Correct answer with work shown Correct solution without work shown (b) A function is defined by f ( x) = 12 3x.If f () t = 3, find the value of t. 12 3t = 3 t = 5 Partial (3 Marks) Correct answer with work shown Correct solution without work shown Page 19 of 44

20 (c) For each of the following mapping diagrams, cirlce the diagrams that represnt a function.: i. ii. iii. iv. (i) and (iv) are functions Partial (6 Marks) Both correct One correct (d) Explain your answer fully. Elements in domain are only mapped to one element in codomain/range Partial (3 Marks) Fully correct explanation Partially correct explanation Page 20 of 44

21 PAPER 2 Structure of the Marking Scheme Page 21 of 44

Question 1 Part (a) Scale 10C* Part (b) Scale 5B Part (c) (i) Scale 10C* (ii) Scale 2B The tyre of a bicycle has a radius of 49 cm. (a) What distance will the tyre cover in 20 complete turns?

22 Question 1 Part (a) Scale 10C* Part (b) Scale 5B Part (c) (i) Scale 10C* (ii) Scale 2B The tyre of a bicycle has a radius of 49 cm. (a) What distance will the tyre cover in 20 complete turns? Give your answer correct to the nearest metre. 2π ( 0.49)( 20) = 62 m Correct answer with work shown Correct answer without work shown. Incorrect relevant formula and continues Neglects multiplication by 20 (b) How many complete revolutions will the tyre make in a 10 km road race? ( π ) = 10,000 m 2.49 = complete revolutions Partial (4 Marks) Correct answer with work shown Any division by answer from part (a) Incorrect or no conversion to metres Answer given as 435 revolution Page 22 of 44

23 (c) An industrial oil tank is in the shape of a cylinder with dimensions as shown. The maximum capacity of the tank is 1000 litres of oil. (i) Calculate x, the length of the tank, correct to the nearest whole number = 1, 000, 000 cm 2 ( ) ( ) = π 160 l l = m l = 12 (i) m During a 3 month period 65% of the oil is used. If oil costs 0.84 per litre calculate the cost of refilling the tank. Correct answer with work shown Correct answer without work shown. Incorrect or no conversion to Incorrect relevant formula and continues Incorrect relevant formula with at least one substitution and stops Any correct relevant step (ii) During a 3 month period 65% of the oil is used. If oil costs 0.84 per litre calculate the cost of refilling the tank % = 650 litres.084 = 546 ( 2 Marks) Partial (1 Marks) Correct answer with work shown Correct answer without work shown Calculation error Any correct use of percentage or any correct multiplication indicated Page 23 of 44

24 Question 2 Part (a) Part (b) Part (c) Scale 5B Scale 10B Scale 10B A class of second year students carried out a survey of cars that collected students from school on a certain day. They noted the colour of each car and recorded their results using a tally count Colour Tally Frequency Red Silver Black Other (a) How many cars collected students from school that day? 100 cars ( 5 Marks) Partial (4 Marks) Correct answer with or without work shown work shown Correct frequencies filled in but incorrect answer Incorrect frequencies filled in but correct answer Incorrect frequencies and incorrect answer Any addition indicated (b) What is the probability that a student was collected in a black car? *Accept answer based on candidates answer in part (a) ( 10 Marks) Partial Correct answer with or without work shown. Incorrect numerator or denominator Probability of any of the other colours fully correct or with correct numerator or denominator Any use of 100 Page 24 of 44

25 (c) The students repeated the survey over a number of weeks. If they recorded a total of 1450 cars how many should they expect to be red? = 319 cars 100 ( 10 Marks) Partial Correct answer with work shown. Incorrect numerator or denominator Probability of any of the other colours fully correct or with correct numerator or denominator used in calcualtion Page 25 of 44

26 Question 3 Part (a) Part (b) Part (c) (i) (ii) (ii) Scale 10B Scale 5B Scale 2B Scale 2B Scale 5B (a) Explain the difference between a survey and a census. A census takes in the whole population whereas a survey only uses a sample of the population. Partial Fully correct explanation(be liberal) Partially correct explanation (b) List one advantage and one disadvantage of carrying out a postal survey? ADVANTAGE: Accept any advantage DISADVANTAGE: Accept any disadvantage ( 5 Marks) Partial (4 Marks) Correct advantage and disadvantage listed Correct advantage or disadvantage listed Page 26 of 44

27 (c) Susan is designing a survey. She designs the following questions to gather information from the public. Examine each question and explain why they may not be suitable for a survey. (i) Are you: YOUNG MIDDLE AGED OLD People might take offence to be labelled as old or middle aged and might not answer the question ( 2 Marks) Partial (1 Marks) (ii) Fully correct explanation(be liberal) Partially correct explanation What is your yearly salary? LOW MEDIUM HIGH People might not answer due to the nature of classing their salary as low ( 2 Marks) Partial (1 Marks) Fully correct explanation(be liberal) Partially correct explanation (iii) From what you have learned about designing surveys, suggest a way that each question above can be asked in a different manner that is more suitable to a survey. Both questions could be asked as ranges e.g yrs or 10,000-30,000 ( 5 Marks) Partial (4 Marks) Suitable suggestion for both questions Suitable suggestion for one questions Page 27 of 44

28 Question 4 Part (a) Part (b) Scale 5A Scale 10B A car dealership offers three models of cars in 4 different colours. (a) If the dealership must have display one of each model and colour, how many cars will be in the showroom? 12 Correct answer with or without work (b) If air-conditioning, alloy wheels and cruise control were added as optional extras, how many cars would the dealer need to display in the showroom? 36 *Accept answer based on candidates part (a) Partial Correct answer with or without work Any correct multiplication shown Page 28 of 44

29 Question 5 Part (a) Part (b) Part (c) Part (d) Scale 5B Scale 5B Scale 15C Scale 15C A student recorded the number of computer games a group of 10 students from two classes bought in a certain month. The results were recorded in the following table: Number of Games Class A Class B (a) Calculate the mean number of games bought in Class A Mean = = 3 10 ( 5 Marks) Partial (4 Marks) Correct answer with work shown Correct answer without work shown towards calculating the mean Calculates the median correctly Identifies the mode correctly (b) Calculate the mean number of games bought in Class A Mean = = 3 10 ( 5 Marks) Partial (4 Marks) Correct answer with work shown Correct answer without work shown towards calculating the mean Calculates the median correctly Identifies the mode correctly Page 29 of 44

30 (c) Choose a suitable graphical method to display both sets of data. Any suitable graphical method with both sets displayed (15 Marks) (14 Marks) (7 Marks) Suitable graph with both sets correctly displayed Suitable graph with both sets displayed with errors Data displayed on separate graphs correctly Axis not labelled Unsuitable graph with both sets displayed with errors Data displayed on separate graphs incorrectly (d) Compare both sets of data in terms of shape of the distribution, the range of value and the median of each set. Correct observation for each (15 Marks) (14 Marks) (7 Marks) Three correct observations Two correct observations One correct observations Page 30 of 44

31 Question 6 Part (a) Part (b) Part (c) Part(d) Scale 10B Scale 2B Scale 5B Scale 5C The length of time taken by a group of students to complete a obstacle course during a school spots day was recorded as follows: BOYS GIRLS (a) Draw an back-to-back stem and leaf plot to display both sets of data. Indicate the key you have used clearly. Boys Girls Key14 1 = Partial Fully correct plot Plot with some errors or omissions (b) Comment on the shape of both distributions. (2 Marks) Partial (1 Marks) Correct observation on both sets of data Correct observation on one set of data Page 31 of 44

32 (c) Calculate the inter quartile range for both sets of data. Boys = 22 Girls = 21 Partial (4 Marks) Both correct with or without work shown One correct with or without work shown (d) On a second run of the course it was found that average time taken for girls to complete the course dropped by 4 seconds while the time taken by the boys remained unchanged. Compare the mean value for both sets of data on the second run of the course. Mean for the boys = seconds Mean for the girls = seconds Observation: Girls were quicker on the second run. (4 Marks) (3 Marks) Both means correct with correct observation Both means correct with incorrect or no observation One mean correct with correct observation Neglects the decrease in girls time and continues correct to end with correct observation One mean correct with no observation Some correct step to calculate the means Neglects the decrease in girls time and continues to end with errors and incorrect observation Page 32 of 44

33 Question 7 Part (a) Scale 2B Part (b) Scale 10C* Part (c) Scale 5C A gardener recorded the height in centimetres of a number of pea plants in his greenhouse after two weeks of growing. He recorded the results in the following table: Height (cm) No. of Plants (a) 60 How many sunflower plants were growing? (2 Marks) Partial (1 Marks) Correct answer with or without work shown Mathematical error (b) Taking mid-interval values, calculate the mean height of the plants correct to one decimal place. Mean = 31 3 cm Correct solution with work shown Correct solution without work shown Incorrect answer with work shown (c) What is the greatest number of plants that could be taller than 25cm? 50 plants (4 Marks) (3 Marks) Correct answer with or without work shown Answer as 36 Answer given as any figure from table other than 3 or 7 Answer given is any combination of figures from table Page 33 of 44

34 Question 8 Part (a) Part (b) Scale 5C Scale 10C The map of Dublin below has points for a road race marked. The race will start at the Red Cow Roundabout (Point A) and travel via Harold s Cross (Point B), Ringsend (Point C), Dublin City Centre (Point D), The Phoenix Park (Point E) and will finish in Castleknock (Point F). Page 34 of 44

35 (a) Write down the co-ordinates of each point on the map. A( 2, 4 ) B( 6,5 ) C( 10,6 ) D( 8,7 ) E( 4,7 ) F ( 2,9) (4 Marks) (3 Marks) All points correct At least 4 points correct Less than 4 points correct (b) The route was design to have two section of equal length. Verify that Harold s Cross is half way between The Red Cow Roundabout and Ringsend. Any suitable method to show AB = BC Any fully correct method with work shown Correct method with some errors Page 35 of 44

36 Question 9 Part (a) Part (b) Scale 5B Scale 10B The lines l, m, and p are shown on the coordinate diagram opposite (a) If the slopes of the three lines shown are 2, 3 and 5 assign each line its correct slope. LINE SLOPE Partial (4 Marks) Three slopes correct One correct slope (b) Prove that two of the lines are perpendicular. Slopes multiplied to give 1 Partial Any fully correct method with work shown Correct method with some errors towards proof Page 36 of 44

37 Question 10 Part (a) Part (b) Part (c) Scale 5B Scale 5A Scale 10B (a) What is meant by the term axiom? Partial (4 Marks) Correct explanation Partially correct explanation (b) Give an example of an axiom you have studied. Partial (4 Marks) Valid example given Partial explanation (c) Divide the line segment [AB] into three equal segments showing all construction lines clearly. Partial Fully correct construction Partially correct construction Page 37 of 44

Question 11 Scale 10C Two tourist buses depart Galway city. One bus travels to the Cliffs of Moher while the other visits Bunratty Castle in Limerick.

38 Question 11 Scale 10C Two tourist buses depart Galway city. One bus travels to the Cliffs of Moher while the other visits Bunratty Castle in Limerick. Both buses finish their journey at the same hotel near the Lakes of Killarney Prove that both buses travel the same distance. Triangles are congruent by RHS Fully correct proof At least two correct steps/ observations about triangle towards proving triangles are congruent Page 38 of 44

39 Question 12 Scale 10C Prove that the angle at the centre of circle standing on a given arc is twice the angle at any point of the circle standing on the same arc. Fully correct proof At least two correct steps with correct diagram At least three correct steps with incorrect diagram Any partially correct diagram Page 39 of 44

Question 13 Scale 10C A set of gaols has dimensions 3.6m 1.7m 0.5m The back support of the goal forms a triangle as shown where CAB = 90 and ABC = 71. Calculate the BC correct to one decimal place.

40 Question 13 Scale 10C A set of gaols has dimensions 3.6m 1.7m 0.5m The back support of the goal forms a triangle as shown where CAB = 90 and ABC = 71. Calculate the BC correct to one decimal place. BC = 1.77 m Correct answer with work shown Correct use of any trigonometric method with a calculation error. Incorrect use of any trigonometric method with or without a calculation error. Any correct addition to diagram Page 40 of 44

Question 14 Part (a) Scale 10C* Part (b) Scale 10C* A goalie kicks out a ball from point A as shown below. The ball reaches its maximum height at the point B and hits the ground at the point D.

41 Question 14 Part (a) Scale 10C* Part (b) Scale 10C* A goalie kicks out a ball from point A as shown below. The ball reaches its maximum height at the point B and hits the ground at the point D. (a) Find the height of the ball above the ground at the point B, correct to the nearest metre = 5 m Correct answer with work shown Correct use of any trigonometric method with a calculation error. Incorrect use of any trigonometric method with or without a calculation error. Page 41 of 44

42 (b) Find the total horizontal distance travelled by the ball, correct to the nearest metre. 5 AC = = = 148 m tan 2 Correct answer with work shown Correct use of any trigonometric method with a calculation error. Finds AC correctly and fails to add 5 Finds AC incorrectly and adds 5 Incorrect use of any trigonometric method with or without a calculation error. Answer given as 5 m Page 42 of 44

Question 15 Part (a) Scale 15C* Part (b) Scale 15C* A tourist is taking photographs of the Leaning Tower of Pisa He stands at the point E, 65.17 m from the point C.

43 Question 15 Part (a) Scale 15C* Part (b) Scale 15C* A tourist is taking photographs of the Leaning Tower of Pisa He stands at the point E, m from the point C. He then moves nearer the tower and takes a picture at the point F a distance of m from the point D. (a) Calculate the height of the Leaning Tower of Pisa on the lower side [BC], correct to two decimal places. BC = tan ( 40.6) = m (15 Marks) (14 Marks) (7 Marks) Correct answer with work shown Correct use of any trigonometric method with a calculation error. Incorrect use of any trigonometric method with or without a calculation error. Page 43 of 44

44 (b) Calculate the difference in height above ground between the low side [BC] and the high side [AD] of the tower, correct to two decimal places. AD = tan = m = 0.83 m (15 Marks) (14 Marks) (7 Marks) Correct answer with work shown Correct use of any trigonometric method with a calculation error. Finds AD correctly but fails to find difference Finds AD incorrectly and continues to find difference Incorrect use of any trigonometric method with or without a calculation error. Page 44 of 44

MATHEMATICS Standard Grade - General Level

MATHEMATICS Standard Grade - General Level General Mathematics - Practice Examination G Please note the format of this practice examination is the same as the current format. The paper timings are the same, as are the marks allocated. Calculators