MATHEMATICS (Project Maths – Phase 2) · PDF file(Project Maths – Phase 2) HIGHER...

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PRE-JUNIOR CERTIFICATE EXAMINATION, 2012 MARKING SCHEME

MATHEMATICS

(Project Maths – Phase 2)

HIGHER LEVEL

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PAPER 1

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Question 1

Part (a) Scale 10C Part (b) Scale 5A Part (c) Scale 5C

{ }1,2,3,4,6,12A = and { }multiples of 3 less than 20B =

(a) Find A B∩ .

Full Credit (10 Marks)

• Fully correct solution

High Partial Credit (8 Marks)

• Lists the elements of B and stops • Omits one element or one element incorrect in final answer

Low Partial Credit (5 Marks)

• Any correct element listed

(b) Write down a proper subset of the set A.


• One correct subset listed

(c) Write down all the subsets of ( )A B∩ ?




• At least five correct subsets


• Any correct subset

{ }{ }

3,6,9,12,15,18

3,6,12

B

A B

=

∩ =

Accept any proper subset

{ } { } { } { } { } { } { } { }3 , 6 , 12 , 3,6 , 3,12 , 6,12 , 3,6,12 , .

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Question 2

Part (a) Scale 10C Part (b) 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




• 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.


• Correct probability from candidates Venn Diagram

Partial Credit (3 Marks)

• Any correct probability from the candidates Venn Diagram

725

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Question 3

Part (a) Scale 5B Part (b) 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).


• 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⊂


• Fully correct

FALSE. N Z⊂

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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.




• Correct method with one calculation error • Mishandles gross wage


• Any correct step

( )

28000 23% 6440 and 37000 .42 15540Gross Tax 21980Tax Payable 21980 3500 18480Net Monthly Income 65000 18480 12 €3877

× = × ==

= − == − ÷ =

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Question 5

Part (a) Scale 10C Part (b) 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.




• 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 €26.175.45 calculate the value of r.




• Correct method with one calculation error



1 12000 0.04 480 0.95 456 12000 12456 2 12456 0.04 498.24 0.95 473.33 12456 €12929

YearYear

= × = × = + == × = × = + =

12929 12000 2492926175.45 24929 1.05 100 100 5%

+ =÷ = × − =

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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.




• Correct solution but no conclusion • Mishandles a percentage once only



Present Cost = 390 ×1.21 = €471.90

Sale Price = 390 × 0.85 = €331.50 1.23 = 407.75

Conclusion: Buy now

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Question 7

Part (a) Scale 10B Part (b) Scale 10C

(a) Find the value of x if 9 5x + =





(b) The length of a rectangle of perimeter 24 cm is three times the width. Calculate the area of the rectangle.




• Incorrect answer when solving but continues to find area correctly for incorrect



( )2 29 5

9 2516

x

xx

+ =

+ ==

2

3 3 24 3 cm3 9 27 cm

x x x x xArea

+ + + = ∴ == × =

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Question 8

Part (a) Scale 10C Part (b) Scale 5B

(a) Solve the inequality 5 3 4 25, x x Z− ≤ + < ∈ .




• One error in solving inequality



(b) Graph your solution on the number line below


• Fully correct number line


• Partially correct number line

9 3 213 7

xx

− ≤ <− ≤ <

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Question 9

Part (a) Scale 5B Part (b) Scale 10B Part (c) Scale 10C Part (d) 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.


• 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.


• Both equation correct


• One equation correct or partially correct

(c) Use the equations above to find the cost of a DVD and a CD.


• Fully correct solution based on candidates equations if not oversimplified


• Only solves for one variable


• Problem oversimplified by equations formed in (b) • Any correct step

Solution by trialling

4x + 3y = 68, x + 2y = 27

x = €11, y = €8

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(d) Verify your solution by plotting the lines below.


• Both lines plotted correctly


• Partially correct • Plots intersection point only

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Question 10

Part (a) Scale 10B Part (b) Scale 5B Part (c) 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.


• Correct expression


• Partially correct expression • Any correct step

(b) Show that the area of the plot can be expressed in the form 13x – x2


• Correct expression found


• Any correct valid attempt to find expression for area

26 2 132

x x− = −

( ) 213 13x x x x− = −

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(c) If the plot must have an area between 22 m2 and 42 m2, find the possible range of values for the length of the plot.




• Correct method but with minor errors • Correct range not found or stated


• Correct range not found and not stated • Any correct step

2 213 22 0 and 13 42 02,11 and 6,7

Lenght is between 7 m and 11 m

x x x xx x

− + = − + == =

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Question 11

Part (a) Scale 25C Part (b) Scale 5A Part (c) Scale 10B Part (d) Scale 10C A graph of the function 2y x bx c= + + , where ,b c R∈ is shown below

(a) Using the information on the graph find the value of a and the value of b.







( ) ( ) ( ) ( )2 21 1 16 and 2 2 72, 15

b c b cb c− + − + = − + + = −= = −

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(b) Find the coordinate of the point R.


• Correct point stated

(c) Find the coordinates of the points S and T where the graph crosses the x-axis


• Both points correct


• One point correct

(d) Hence solve the equation ( ) ( )23 1 2 3 1 15 0t t− + + − = for 0t ≥ .







( )0, 15R −

( ) ( )5,0 3,0S T−

3 1 5 or 3 1 34 4,3 3

t t

t

− = − − =

= −

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Question 12

Part (a) Scale 10B Part (b) Scale 15C Part (c) Scale 5A Part (d) Scale 5B Part (e) 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 – x2 , 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?


• 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.


• Fully correct graph


• Unsuitable range leads to incorrect graph • Graph joined with straight lines


• Some points found

8 seconds

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(c) Use your graph to estimate the maximum height of the ball


• 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.


• Correct solution with work shown


• Correct solution without work shown • Any correct step

(e) How far from the player was the ball at its maximum height?





Max. height = 16 metres

Distance = 4 × 8 = 32 m

Distance = 4 × 4 = 16 m

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Question 13

Part (a) Scale 10B Part (b) Scale 5B Part (c) Scale 10B Part (d) Scale 5B (a) Write the following formula in terms of V.

212

W CV=


• Correct answer with work shown



(b) A function is defined by ( ) 12 3f x x= − .If ( ) 3f t = − , find the value of t.





2WVC

=

12 3 35

tt

− = −=

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(c) For each of the following mapping diagrams, cirlce the diagrams that represnt a function.:

i. ii.

iii. iv.


• Both correct


• One correct

(d) Explain your answer fully.


• Fully correct explanation


• Partially correct explanation

(i) and (iv) are functions

Elements in domain are only mapped to one element in codomain/range

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PAPER 2

Structure of the Marking Scheme

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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.




• 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?




• Any division by answer from part (a) • Incorrect or no conversion to metres • Answer given as 435 revolution

( )( )2 0.49 20 62 mπ =

( )10 1000 10,000 m 2 .49 3248.063248 complete revolutions

π× = ÷ × × =

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(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.

(i) 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 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.

Full Credit ( 2 Marks)



• Correct answer without work shown • Calculation error • Any correct use of percentage or any correct multiplication

indicated

( ) ( )

3

2

1000 1000 1,000,000 cm

1000000 16012.43 m12 m

lll

π

× =

===

1000 × 65% = 650 litres × .084 = €546

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Question 2

Part (a) Scale 5B Part (b) Scale 10B Part (c) 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?


• 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)


• 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

100 cars

19100

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(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?


• 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

211450 319 100

cars× =

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Question 3

Part (a) Scale 10B Part (b) Scale 5B Part (c) (i) Scale 2B (ii) Scale 2B

(ii) Scale 5B

(a) Explain the difference between a survey and a census.


• Fully correct explanation(be liberal)



(b) List one advantage and one disadvantage of carrying out a postal survey?


• Correct advantage and disadvantage listed


• Correct advantage or disadvantage listed

A census takes in the whole population whereas a survey only uses a sample of the population.

ADVANTAGE: Accept any advantage

DISADVANTAGE: Accept any disadvantage

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(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





(ii) What is your yearly salary?

LOW MEDIUM HIGH





(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.


• Suitable suggestion for both questions


• Suitable suggestion for one questions

People might take offence to be labelled as old or middle aged and might not answer the question

People might not answer due to the nature of classing their salary as low

Both questions could be asked as ranges e.g. 18-25yrs or €10,000-€30,000

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Question 4

Part (a) Scale 5A Part (b) 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?


• 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?

*Accept answer based on candidates part (a)


Correct answer with or without work


• Any correct multiplication shown

12

36

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Question 5

Part (a) Scale 5B Part (b) Scale 5B Part (c) Scale 15C Part (d) 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 0 1 2 3 4 5 6 7 Class A 2 3 4 6 7 4 3 1 Class B 1 3 4 5 6 8 1 1

(a) Calculate the mean number of games bought in Class A




• Correct answer without work shown • Any correct step towards calculating the mean • Calculates the median correctly • Identifies the mode correctly

(b) Calculate the mean number of games bought in Class A




• Correct answer without work shown • Any correct step towards calculating the mean • Calculates the median correctly • Identifies the mode correctly

2 3 4 6 7 4 3 1 310

Mean + + + + + + += =

1 3 4 5 7 8 1 1 310

Mean + + + + + + += =

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(c) Choose a suitable graphical method to display both sets of data.


• 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.


• Three correct observations


• Two correct observations


• One correct observations

Any suitable graphical method with both sets displayed

Correct observation for each

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Question 6

Part (a) Scale 10B Part (b) Scale 2B Part (c) Scale 5B Part(d) 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 180 155 165 168 187 181 171 165 183 143 150 177 151 177 142 151 158 141 175 165 164 176 181 142 177 142 164 183 158 158 169 168 150 179 150 156 181 152 165 183 168 174 184 168 156 143 164 172 159 176 173 155 178 141 184 187 153 177 187 166

(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 = 141 2 2 1 6 5 5 1 0 0 0 9 8 8 8 8 5 4 9 8 7 7 7 6 4 3 7 4 4 1 0

14 15 16 17 18

1 2 3 3 1 2 3 6 8 8 8 91 4 4 5 5 5 6 1 2 5 6 7 1 1 3 3 3 7 7


• Fully correct plot


• Plot with some errors or omissions

(b) Comment on the shape of both distributions. Full Credit (2 Marks)

• Correct observation on both sets of data


Correct observation on one set of data

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(c) Calculate the inter quartile range for both sets of data.


• 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.


• 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

Boys = 22

Girls = 21

Mean for the boys = 166.3 seconds

Mean for the girls = 161.3 seconds

Observation: Girls were quicker on the second run.

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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) 0-10 10-20 20-30 30-40 40-50 No. of Plants 3 7 14 21 15

(a) How many sunflower plants were growing? Full Credit (2 Marks)



• Mathematical error

(b) Taking mid-interval values, calculate the mean height of the plants correct to one

decimal place.




• Correct solution without work shown • Incorrect answer with work shown



(c) What is the greatest number of plants that could be taller than 25cm? Full Credit (5 Marks)



• Answer as 36


• Answer given as any figure from table other than 3 or 7 • Answer given is any combination of figures from table

60

Mean = 31 3 cm

50 plants

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Question 8

Part (a) Scale 5C Part (b) 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).

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(a) Write down the co-ordinates of each point on the map.


• 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 fully correct method with work shown


• Correct method with some errors



( ) ( ) ( ) ( ) ( ) ( )2,4 6,5 10,6 8,7 4,7 2,9A B C D E F

Any suitable method to show AB BC=

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Question 9

Part (a) Scale 5B Part (b) 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 5, 3 2 4

and− assign each line its correct

slope.

LINE SLOPE


• Three slopes correct


• One correct slope

(b) Prove that two of the lines are perpendicular.


• Any fully correct method with work shown


• Correct method with some errors • Any correct step towards proof

Slopes multiplied to give – 1

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Question 10

Part (a) Scale 5B Part (b) Scale 5A Part (c) Scale 10B

(a) What is meant by the term axiom?


• Correct explanation



(b) Give an example of an axiom you have studied.


• Valid example given


• Partial explanation

(c) Divide the line segment [AB] into three equal segments showing all construction lines

clearly.


• Fully correct construction


• Partially correct construction

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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.


• Fully correct proof


• At least two correct steps/ observations about triangle


• Any correct step towards proving triangles are congruent

Triangles are congruent by RHS

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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 correct step • Any partially correct diagram

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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 90CAB∠ = ° and 71ABC∠ = ° .

Calculate the |BC| correct to one decimal place.




• Correct use of any trigonometric method with a calculation error.


• Incorrect use of any trigonometric method with or without a calculation error.

• Any correct step • Any correct addition to diagram

1.77 mBC =

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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.








2 27 5 5 m− =

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(b) Find the total horizontal distance travelled by the ball, correct to the nearest metre.





• Finds |AC| correctly and fails to add 5 • Finds |AC| incorrectly and adds 5



• Any correct step • Answer given as 5 m

5 143 5 148 mtan 2

AC = = + =

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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. He then moves nearer the tower and takes a picture at the point F a distance of 56.44 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.








( )65.17 tan 40.6 55.86 mBC = =

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(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.





• Finds |AD| correctly but fails to find difference • Finds |AD| incorrectly and continues to find difference




55.44 tan 45.13 56.69 m56.69 55.86 0.83 mAD = =

− =

MATHEMATICS (Project Maths – Phase 2) · PDF file(Project Maths – Phase 2) HIGHER...

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