Mathematics (Project Maths – Phase 2) · PDF fileMathematics (Project Maths – Phase 2) Paper 1 Ordinary Level Time: 2 hours, 30 minutes 300 marks 1. Section A Concepts and Skills

Leaving Certifi cate Examination, 2013

Sample Paper no. 4

Mathematics (Project Maths – Phase 2)

Paper 1

Ordinary Level

Time: 2 hours, 30 minutes

300 marks

1

Section A Concepts and Skills 100 marks

Answer all four questions from this section.

Question 1 (25 marks)

John and Ali go to Punchestown together to watch the races. John decides to place a wager, and he and Ali go to the betting counter to place his bet. While he’s there, Ali explains that if he places

a bet at odds of 5 _ 4 then he will win €5 for every €4 he puts down, as well as getting back whatever

money he wagered.

(i) If he bets €10 and wins at 5 _ 4 odds, how much money will he receive back, including his €10?

(ii) How much money would he need to wager at 7 _ 1 odds to receive a total of €40, including his original bet?

(iii) At what odds would he need to bet €15 to receive back a total of €480 (inclusive of his original wager)?

(iv) John decides to place an ‘accumulator’ bet over three races, where the winnings from each

race will be carried forward as the wager in the next race. How much will he receive in total if

he places a bet of €10 over three races where the odds are 7 _ 5 , 9 _ 1 and 5 _ 2 respectively?

2


Cans are arranged in a pattern as shown.

10

6

3

1

(a) Draw the next term of the pattern.

(b) Draw a table to represent the number of cans in the fi rst six terms of the pattern.

(c) Describe the sequence of numbers generated by the pattern.

(d) How many cans will there be in the eighth term of the pattern?

3

(e) By trial and error, or otherwise, calculate how many rows there would be in the stack containing 136 cans.


(a) (i) Write 27 and √ __

3 in the form 3n.

(ii) Hence, solve the equation 32x + 1 = ( 27 ___

√ __

3 ) 2 .

(b) Simplify x2 – 2x – 15

__________ 2x2 – 9x – 5

.

4


Let z1 = 5 + 12i and z2 = 2 – 3i

(a) Calculate |z1| and |z2|.

(b) Find the real number k such that |z1| = k|z2|.

(c) p and q are real numbers such that z1 __ z2

= p(q + i). Find the value of p and the value of q.

5

Section B Contexts and Applications 100 marks

Answer both Question 5 and Question 6.


Amy and Darragh have a combined income of €110,000 for the year 2012. They pay tax on the fi rst €65,200 at 20% and 42% on the remainder. They have tax credits of €4,200 each and they both pay PRSI at Class A1 rates.

Income Rates of USC€0–€10,036 2%

€10,036.01–€16,016 4%

Above €16,016 7%

PRSI rates of contribution for 2012Subclass A1

Weekly pay band How much of weekly pay All incomeemployee

All incomeemployer

More than €500First €127 Nil 10.75%

Balance 4.00% 10.75%

(a) Calculate Amy and Darragh’s total tax bill for the year.

(b) Assuming a 52-week year, calculate their combined PRSI contribution.

6

(c) Calculate their combined USC.

(d) Calculate their combined net pay for the year.

Amy and Darragh decide to start contributing to a pension scheme. As they both work for the same company, they decide to join the pension scheme offered by their employer. Since they are paid monthly, their employer will deduct their contribution from their joint income before any other deductions are made. The joint pension contribution is calculated as 7.5% of their gross monthly income.

(e) Calculate their yearly pension contribution.

Their contribution is then invested at 3.5% compound interest, which is added at the end of the year.

(f) Calculate the value of their pension after fi ve years.

7


Josephine and a group of her friends have booked a river cruise that is to last three hours in total. The riverboat travels 15 km upstream and then turns and travels the same distance downstream. The river has a current of 2 km/h downstream.

(a) By letting x be the speed of the riverboat, write an expression in x to represent the speed of the boat travelling upstream.

(b) Write an expression in x to represent the speed of the boat travelling back downstream.

(c) What is the time taken, in terms of x, for the boat to travel up and down the cruise route?

(d) Write an equation in x and solve it to fi nd the speed of the boat.

8

(e) How long does each part of the cruise take in hours and minutes, to the nearest minute?

Including Josephine and her friends, there are 52 passengers on the riverboat, some of whom are children. Based on the fact that an adult ticket for the cruise is €35 and a child’s ticket is €15, Josephine calculates the takings for that day’s cruise to be €1,500.

(f) Find the number of children and the number of adults on the boat.

(g) If all the children on the boat that day are part of family groups and a family ticket for two adults and two children costs €60, what are the actual takings for that day’s cruise?

(h) Calculate the percentage error in Josephine’s estimate.

9

Section C Functions and Calculus (old syllabus) 100 marks



(a) If f(x) = 5x – 8 and g(x) = 13 – 2x, fi nd the value of x for which f(x) = g(x).

(b) Let g(x) = x(x – 2) for x ∈ R.

(i) Find g(0), g(4) and g(–2).

(ii) Show that g(1 + t) = g (1 – t) for t ∈ R.

(iii) Find the derivative, g′(x), and show that g′(x) > 0 for x > 1.

10

(c) If y = 1 – x2

_____ x , fi nd dy

___ dx

.

Show that dy

___ dx

< 0 for all x ≠ 0, x ∈ R.

11


(a) If s = t3 – 4t2, fi nd ds

__ dt

when t = 2.

(b) (i) Differentiate 7x + 3 with respect to x from fi rst principles.

(ii) Find dy

___ dx

if y = ( 1 + 1 __ x ) 10

12

(c) The speed v in metres per second of a body after t seconds is given by v = 2t(6 – t).

(i) Find the acceleration at each of the two instants when the speed is 10 m/s.

(ii) Find the speed at the instant when the acceleration is 0.

13


Sample Paper no. 4

Mathematics(Project Maths – Phase 2)

Paper 2

Ordinary Level


300 marks

14


Answer all six questions from this section.


A bag contains 10 green and 3 red counters. A counter is drawn at random from the bag but it is not

replaced. A second counter is then drawn.

(a) Construct a tree diagram to show the probabilities of all possible outcomes.

(b) Find the probability that the counters choosen will be two different colours.

(c) The fi rst counter was replaced in error and the probability of choosing two counters of

different colours was calculated to be 60 ___ 169 .

Calculate the error, and hence the percentage error, in the probabilities by replacing the fi rst counter.

15


C(–2,–3) is the image of a point A(p,t) under central symmetry in the origin.

(a) Show that 3x – 2y = 0 is the equation of the line AC.

A (p,t)

C (–2,–3)

(b) In the diagram above, if AC is the diameter of a circle c, write down the equation of c.

Equation of circle c

(c) A line k is a tangent to the circle c and the line k touches the circle c at the point A.

Find the equation of the line k.

(d) s is a circle that is the image of c under axial symmetry in k.

Find the equation of the circle s.

16


In the triangle BAD, |AB| = 5 cm, |∠BAC| = 68.18°, |∠BCA| = 29.63° and |CD| = 11 cm.

A C

B

D11 cm

5 cm

68.18° 29.63°

(a) Calculate |BC|, correct to three decimal places.

(b) Calculate |BD|, correct to the nearest centimetre.

17


The table shows the age and annual income (in €1,000 units) of 10 employees in a company.

Age 36 25 44 48 32 50 33 40 60 20

Income (€1,000) 36 29 53 55 34 62 38 46 5 24

(a) On the scatter graph below, the fi rst six data items are plotted. Complete the scatter plot for the last four data items.

Age

Income in €1000

6560555045403530252015105

510

1520

2530354045

50556065

00

(b) Describe what you think the strength of the correlation is between age and income.

(c) Comment on the income of the employee aged 60.

Do you think that this data item should be included in the scatter graph?

(d) How would the strength of the correlation be affected if the income of the 60-year-old was left out of the data?

(e) A new employee joined the company on a salary of €40,000.

Would you estimate that the new employee was around 40 years of age?

18


(a) The letters of the word PROJECT are arranged using all the letters of the word.

How many arrangements are possible if:

(i) There are no restrictions?

(ii) The arrangements begin with the letter P and end in a vowel?

(iii) The two vowels are together?

(b) {5, 6, 7, 8} are a set of numbers.

(i) How many two-digit numbers can be formed from this set, if the digits can be repeated?

(i) How many two-digit numbers can be formed from this set, if the digits cannot be repeated?

19

Question 6

Answer either 6A or 6B.

Question 6A (25 marks)

(a) Explain, with the aid of an example, what is meant by the converse of a theorem.

(b) Show how to construct the centroid G of the triangle ABC.

All construction lines should be shown.

B

A

C

G

(c) The lines that intersect are called ‘median lines’. Show by measuring that the medians divideeach other into a 2 : 1 ratio.

20

Question 6B (25 marks)

A

B

O

C

E

D

A, B, C and D are points on a circle with centre O. [EC] and [ED] are tangents to this circle.

(a) Prove that OCED is a square.

(b) If |AO| = r, fi nd, in terms of p and r, the areas of the semicircles drawn on [EC], [ED] and [DC].

21


Answer Question 7, Question 8 and Question 9.


(a) The amount of petrol bought by motorists is shown in the following stem-and-leaf diagram.

Stem Leaf

12345

4 4 6 91 3 7 7 7 83 6 6 7 90 2 3 3 8 81 3 4 7

Key: 5|1 = 51 litres

(i) Describe the shape of the distribution.

(ii) How many people purchased less than 25 litres?

(iii) What is the median amount purchased?

22

(iv) What is the interquartile range?

(v) Calculate the standard deviation of the distribution.

(b) The data shown below is from Charles Darwin’s study of cross-fertilisation and self-fertilisation. Pairs of seedlings of the same age, one produced by cross-fertilisation and the other by self-fertilisation, were grown under nearly identical conditions. The data shows the heights, in centimetres, of each plant after a fi xed period of time.

Self-fertilised plants

32 44 51 39 45

47 41 47 38 50

45 41 45 50 46

Cross-fertilised plants

59 55 46 58 30

30 58 54 53 53

54 55 55 48 51

(i) Use an appropriate diagram to display the data.

23

(ii) The aim of Darwin’s experiment was to show that cross-fertilised plants grow faster. Based on your diagram, do you think that the experiment demonstrated this?

(iii) How might Darwin have improved the experiment?

24


(a) Find, in terms of p, the volume of a cone of height 9 cm and radius of base 4 cm.

4 cm

9 cm

(b) The cone is immersed in water in a cylinder of radius 6 cm. The depth of the water in the cylinder is 30 cm. Calculate the height h of the rise in the level of the water in the cylinder.

h

25


Roofs of buildings are often supported by timber frames called trusses. One truss is in the shape of a triangle as seen below.

7 m

3 m

B C

A

67°

In the triangular truss, |AB| = 3 m, |BC| = 7 m and |∠ABC| = 67°.

(a) Calculate the length of the timber beam AC, correct to the nearest metre.

(b) Calculate the size of the angle between the beams BC and AC.

26

(c) Calculate the total length of timber required to make a truss.

(d) A carpenter decides to cover the triangular truss with panelling.

Calculate the area of the triangular paneling ABC, correct to one decimal place.

27


Sample Paper no. 5


Paper 1

Ordinary Level


300 marks

28




(a) If a + bi = 3 + i

____ 1 + i

, fi nd the value of a and b, a, b ∈ Z.

(b) Let z = 2 − i be one root of the equation z2 + pz + q = 0, p, q ∈ R. Find the value of p and q.

29


(a) Tick the box across from the function which is symmetric about the y-axis:

y = x + 1

y = x2 + 2x − 3

y = x3 + 2x

y = x2 − 4

(b) Tick the box(es) across from the function(s) which contain the origin:

y = x

y = 3x2 + 2x

y = 3x + 1

y = x + 12

y = x2 + 2x − 3

30


x and y are positive numbers, with 3x − y + 10 = 0 and x2 + y2 = 10.

(a) If you were asked to fi nd x and y, write down two methods that might be used to solve this problem.

(b) Using one of these methods from part (a), fi nd x and y.

(c) Write down the sum of x and y.

31


(a) In fi ve years, the population of fi sh in Lake Springfi eld decreased steadily from 50,000 to 45,000. Find the percentage rate of decrease per year, correct to two places of decimals.

(b) Over the next three years, the population of fi sh in the lake grows at a rate of 0.5%. Find the number of fi sh in the lake at the end of the third year, correct to the nearest whole number.

32




The table below lists the times that Sheila takes to walk the given distances.

Time (minutes) 5 10 15 20 25 30

Distance (km) 1 2 3 4 5 6

Plot the points.

If the relationship between the distances and times is linear, fi nd the equation of the straight line, using any two points. Then use the equation to answer the following questions:

(a) How long will it take Sheila to walk 21 km?

(b) How far will Sheila walk in 7 minutes?

(c) If Sheila were to walk half as fast as she is currently walking, what would the graph of her distances and times look like? Refer to slope in your answer.

33


You are in the lobby of a building waiting for the lift. You are late for a meeting and wonder if it will be quicker to take the stairs. There is a fascinating relationship between the number of fl oors in the building, the number of people in the lift and how often it will stop.

If n people get into a lift at the lobby and the number of fl oors in the building is F, and s is the number of stops, the following equation relates the variables:

s = F − F ( F − 1 _____ F ) n (a) If the building has 16 fl oors and there are nine people who get into the lift, how many times is

the lift expected to stop?

(b) If the lift stopped 12 times and there are 17 fl oors, approximately how many people were in the lift?

34

(c) If there are eight people in the lift, using the axes and scales below, draw the function that relates the number of stops to the number of fl oors. (Hint: Begin the graph at F = 1)

2

2

4

8

F

Num

ber

of sto

ps

Number of floors

4 6 8 10

(d) Is this function linear?

(e) Give a reason for your answer to part (d).

35




(a) If f (x) = 5x – 8 and h(x) = 13 – 2x, fi nd the value of x for which f (x) = h (x).

(b) Differentiate from fi rst principles: x2 + 5x.

36

(c) Let g (x) = (2x + 3)(x2 − 1) for x ∈ R.

(i) For what values of x is the slope of the tangent to the curve of g (x) equal to 10?

(ii) Find the equations of the two tangents to the curve of g (x) which have slope 10.

37


(a) Find ds

__ dt

when s = 6t2 − 3t + 7.

(b) (i) Draw the graph of g (x) = 1 __ x for −3 ≤ x ≤ 3, x ∈ R and x ≠ 0.

(ii) Using the same axes and scales as in part (b) (i), draw the graph of h(x) = x + 1 for −3 ≤ x ≤ 3, x ∈ R.

(iii) Use your graphs to estimate the values of x for which 1 __ x = x + 1.

38

(c) Let f (x) = x3 − 6x2 + 12 for x ∈ R. Find the derivative of f (x).

At the two points (x1, y1) and (x2, y2), the tangents to the curves y = f (x) are parallel to the x-axis, where x2 > x1.

(i) Show that x2 – x1 = 4.

(ii) Show that y2 = y1 – 32.

39


Sample Paper no. 5


Paper 2

Ordinary Level


300 marks

40




The diagram shows two wheels. The fi rst wheel is divided into four equal segments numbered 1, 2, 3 and 4. The second wheel is divided into three equal segments labelled A, B and C. A game consists of spinning the two wheels and noting the segment at the arrow.

1 2

4 3

A

CB

For example, the outcome shown above is (3, B).

(i) List all the possible outcomes of spinning the wheels.

(ii) What is the probability that the outcome is (2, C)?

(iii) What is the probability that the outcome is an odd number with the letter A?

(iv) What is the probability that the outcome includes the letter C?

41


(a) A line l cuts the axes at Q(–2,0) and P(0,4). Plot these points and fi nd the slope of l.

(b) Without using the formula y − y1 = m(x − x

1 ), write down the equation of l.

(c) k is a line through Q perpendicular to l. Find the equation of k.

(d) k cuts the y-axis at R and PQRS in that order, is a rectangle.

Calculate the area of PQRS.

42


The graphs below represent the equations of six circles, A, B, C, D, E and F.

–6 –5 –4 –3 –2 –1 1 2 3 4 5 6 70

(0,0)

–1

1

2

3

4

5

0

–2

–3

–4

–5

(4,3)

A

(0,0)

(4,3)

–2 –1–1

1

2

3

4

5

6

7

8

–2

1 2 3 4 5 6 7 8 9 100

0

B

(0,0)

–4 –3 –2 –1

–1

1

2

3

4

–2

–3

–4

1 2 3 40

C

Radius = 13

–5

–4

–3

–2

–1

1

2

3

4

–1 1 2

(2,–3)

(2,3)

3 4 5 6 70

0

–2–3

D

(–7,0)

–10 –9 –8 –7 –6 –5 –4 –3

E

Radius = 2 2

(3,0)

–1

–1

1

2

3

–2

–3

–4

1 2 3 4 5 6 70

0

F

(a) Fill in the table, matching each equation with its particular graph.

Equation Graph

(x − 3 ) 2 + y 2 = 9

(x − 4 ) 2 + (y − 3 ) 2 = 25

x 2 + y 2 = 25

(x + 7 ) 2 + y 2 = 8

(x − 2 ) 2 + y 2 = 9

x 2 + y 2 = 13

(b) In the table below you are given the centres and radii of various circles.

Write down the equation of the circles in the spaces provided.

Centre and radius Equation

Centre (3,4): Radius = 5

Centre (2,0): Radius = 2

Centre (–3,4): Radius = 4

Centre (0,–4): Radius = 7

Centre (0,0): Radius = √ __

5

Centre (–2,5): Radius = 3 √ __

2

43


The triangle PQR is the image of the triangle ABC under an enlargement of scale factor k.

–11–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

–1

1

2

3

4

–2

–3

–4

0

Q

RC

A P

B

1 2 3 4 5 6 7

(a) Find the value of the scale factor k.

(b) Use the scaled diagram above to construct the centre of enlargement.

(c) Write down the co-ordinates of the centre of enlargement.

(d) In the scaled diagram above, the area of the triangle PQR = x(area of triangle ABC).

Calculate the value of x.

44

(e) What is the relationship between the value for k and x above?


In the representation of the plane below, study the diagram and answer the questions.

The Plane

PF

t

M

D

K

L

A

71°71°

31°

C

JH

G

O

E

B

I

k p

n

l

(a) Identify each of the following from the diagram.

(b) If |∠CJI| = 31°, calculate:

(i) A line:

(ii) A ray:

(iii) A pair of perpendicular lines:

(iv) A pair of parallel lines:

(v) Three points of intersection:

(vi) Four collinear points:

(vii) A parallelogram:

(i) |∠IBO|

(ii) |∠BOJ|

(iii) |∠AIJ|

45


(a) In the space provided below, construct a parallelogram ABCD where |AB| = 8 cm, |∠CAB| = 50° and |AD| = 6 cm.

(b) Construct the diagonals of the parallelogram above, and by measuring, show that the diagonals bisect each other.

46


In the parallelogram PQRS, the points T and W are on the diagonal [PR] such that |∠PQT| = |∠WSR|.

S

R

W

T

P

Q

(i) Prove that |PT| = |WR|.

(ii) Hence or otherwise, show that the triangles PWS and QTR are congruent.

47




The examination results in Maths and Science for a certain class were as follows:

Maths Science

65 25 45 40 45 48

55 53 54 51 62 58

27 52 30 36 47 64

59 67 45 64 55 83

73 83 69 74 65 69

38 86 71 72 67 75

91 85

(a) Construct a back-to-back stem-and-leaf plot of the data above.

(b) How many students took the examinations?

(c) What is the interquartile range of the Science marks?

48

(d) Calculate, correct to one decimal place, the mean and standard deviation of both sets of marks.

(e) State one difference and one similarity between the two distributions.


A plot of land has a triangular shape PQR, as shown.

P

Q

R

40 m

60 m

83°

(a) Find the area of the triangular plot PQR correct to the nearest m2.

49

(b) Find the length of fencing that is required to fence the triangular plot.

P

Q

R

40 m

60 m

83°

(c) If the same length of fencing was used to fence a square plot, calculate the length of the diagonal of the square plot.

50


(a) One of the diagrams A, B, C or D below does not represent the net of a cube. Which one?

A B C D

(b) A solid cone with r = 3 cm and h = h cm and its net are shown in the diagram below.

l

l l

h

r

rr

Given that the total surface area of the net is 24p cm2:

(i) Calculate l, the slant height of the cone.

(ii) Calculate h, the height of the cone.

51

(c) The cone in part (b) is partly fi lled with water so that the slant height of the water in the cone rises to 2 cm as shown. Taking 3.14 as an approximation for p, calculate the volume of water in the cone. Give your answer correct to two decimal places.

3 cm

2 cm

4 cm

52


Sample Paper no. 6


Paper 1

Ordinary Level


300 marks

53




(a) What does it mean to say that ‘a function has degree 2’?

(b) A farmer has 28 metres of fencing with which to create an enclosure for cattle. Sketch two different rectangular enclosures that he could make, showing clearly the dimensions of the sides.

(c) Show that the area is represented by the function A = 14x – x 2.

(d) What is the maximum area the farmer will be able to fence given the information above?

54


(a) What is the mathematical name given to functions that have a variable in the power, e.g. y = 2x?

(b) Solve for x: ( √ __

3 ) 4 – 6x = 27x.


(a) The nth term of a geometric sequence is Tn = 2

n

__ 3n .

(i) Find the fi rst three terms of the sequence.

(ii) Show that the sum of the fi rst 5 terms, S5, is 422 ____

243 .

55

(b) Find how much €5,000 amounts to if invested at a rate of 5% per annum, compounded monthly, for three years.


If ab _____

a – b > 0, how many of the following statements are true:

(i) 0 < b < a (ii) b < a < 0 (iii) a < b < 0

(i) only

(i) and (ii) only

(i) and (iii) only

(i) and (iii) only

All

56




A steel tank is as shown in the diagram. The ends are right-angled triangles having sides 3x, 4x and 5x. The height of the tank is y. The total surface area of the tank is 3,600 cm2.

4x

5x

y

3x

(a) Show that y = 300 – x2

_______ x .

57

(b) A table of results showing the relationship between the height of the tank and the length of the base is shown below:

x (Base) 0 5 10 15 20

y (Height) 0 55 20 5 –5

(i) Explain in your own words the relationship between the height and the base.

(ii) What do the values in the last two columns tell you?

58


Consider the sequence: 7, 13, 23, 37, 55, …

(i) Show that this sequence is quadratic.

(ii) Find the formula for the general term.

(iii) Find the 99th term, T99

.

59




(a) Differentiate with respect to x.

(i) –x 2

(ii) x 4 + x + 1

(b) (i) Find dy

___ dx

when y = 2x _____

4 – x 2 , for x ∈ R and x ≠ ±2.

Show that dy

___ dx

> 0.

60

(ii) Differentiate ( x5 – 1 __ x2 ) 7 with respect to x, x ≠ 0.

(c) The speed, v, in metres per second of an engine moving along a track is related to time, t, in

seconds by v = 1 __ 3 (2t + 5).

(i) Draw the straight line graph of this relation, putting time on the horizontal axis, for 0 ≤ t ≤ 8.

(ii) Use your graph to estimate the speed when t = 2.5 seconds.

(iii) Use your graph to estimate the time at which the speed reaches 6 metres per second.

61


Let f (x) = 1 _____ x – 1 , for x ∈ R and x ≠ 1.

(i) Find the values of f ( 3 __ 2 ) , f(–2), f (0) and f (5).

(ii) Find f ′(x), the derivative of f (x).

(iii) Find the equation of the tangent T to the curve at the point (0,−1).

62

(iv) Find the co-ordinates of the other point on the graph of f (x) at which the tangent to the curve is parallel to T.

63


Sample Paper no. 6


Paper 2

Ordinary Level


300 marks

64




The diagrams below shows various lines l, m, n, p, q and s drawn on co-ordinate planes.

–4 –3 –2 –1–1

1

2

3

4

–2

–3

–4

0

0

1 2 3 4

x

y

l

–4 –3 –2 –1–1

1

2

3

4

–2

–3

–4

0

0

1 2 3 4

x

y

m

–2 –1–1

1

2

3

4

–2

–3

–4

0

0

1 2 3 4 5 6

x

y

n

–2 –1–1

1

2

3

4

5

–2

0

0

1 2 3 4 5 6

x

y p

–4 –3 –2 –1–1

1

2

3

4

–2

–3

–4

0

0

1 2 3 4

x

yq

–4 –3 –2 –1–1

1

2

3

4

–2

–3

0

0

1 2 3

x

y

s

(a) Complete the table below of the properties of the various lines.

Line Sl ope Cuts x-axis at? Cuts y-axis at? Equation

l

m

n

p

q

s

(b) Complete the table below, writing in an equation of a line that is parallel to the original line.

Line Equation of parallel line Line Equation of parallel line

l p

m q

n s

65


(a) Show that (2,3) is on the circle x 2 + y 2 = 13.

(b) Find the equation of the tangent to the circle x 2 + y 2 = 13 at the point (2,3).

(c) Find the equation of the image of x 2 + y 2 = 13 under central symmetry in (2,3).

(d) The line l passes through (2,3) and is perpendicular to the tangent in part (b) above.

Find the equation of l.

66


The triangle ABC is the image of the triangle ARP under an enlargement. |AR| = 6 cm and |RB| = 3 cm.

P

C

B6 cm 3 cmRA

(a) Write down the centre of enlargement.

(b) Find k, the scale factor of the enlargement.

(c) If |AP| = 8 cm, fi nd |AC|.

67


The diagram shows the graph of the curve y = x 2 ___ 10

in the domain 0 ≤ x ≤ 10.

y

x–1

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7 8 9 100

0

(a) Use the grid background to estimate the area under the curve between 0 and 10.

(b) Complete the following table of values for the graph of the curve.

x 0 1 2 3 4 5 6 7 8 9 10

y

(d) Use the completed table and the Trapezoidal Rule to estimate the area under the curve between 0 and 10.

68


(a) A code consists of a four digit number that is formed from the digits 3 to 9 inclusive. No digit can occur more than once in the code.

(i) How many different codes are possible?

(ii) How many of the four-digit codes are greater than 6,000?

(iii) How many of the four-digit codes are divisible by 2?

(iv) How many of the four-digit codes end in an odd number?

(v) Find the probability that a person who does not know the code will guess it on the fi rst guess.

69


(a) In the space provided below, construct a tangent to the circle at the point B.

(b) (i) Construct E, a point on the tangent above, such that |BE| = |AB|.

(ii) If |AB| = √ __

2 , fi nd |AE|.

70


A circle with centre A has two chords [DC] and [BE] which intersect at F as shown.

A

D

B

C

E

F

(i) Prove that the triangle DFB and the triangle EFC are equiangular.

(ii) Hence or otherwise, show that |DF|:|FE| = |BF|:|FC|.

71




Two ships, A and B, leave Cobh at midday. Ship A travels due East and Ship B travels East 50o South as shown in the diagram.

A

50°

B

Cobh

(a) Calculate the distance between Ship A and Ship B, correct to two decimal places, when Ship A is 12 km from Cobh and Ship B is 8 km from Cobh.

72

(b) When Ship A is 12 km from Cobh and Ship B is 8 km from Cobh, Ship A is North x° East from Ship B. (see diagram).

Find the value of x, correct to the nearest degree.

A

50° 90°

x°

B

Cobh

73


On the sea front in Bray, a student decides to set up a cold soft drinks stall. During a two-week period, he decides to see if his sales are affected by the maximum daily temperature. He records his results and these are shown in the table below.

Max. temperature (Co) 21 23 25 25 26 21 19 19 20 17 16 18 23 20

Drinks sold 50 75 79 84 85 55 43 47 50 35 40 45 69 63

(a) Display the data that will allow the examination of the relationship between the maximum

daily temperature and the number of cold soft drinks sold.

(b) The correlation coeffi cient is listed in the following set of numbers.

Identify the correct fi gure.

–1, 0.4, –0.5, 0.9, –0.4

(c) Comment on the relationship between the temperature and the sales of cold soft drinks.

74

(d) If 120 cold soft drinks were sold on a particular day, would that mean that the maximum temperature on that day was very high? Give a reason for your answer.

(e) Calculate the mean maximum daily temperature.

(f) Calculate the mean daily sales of soft drinks.

75


(a) A rectangular piece of wood measures 8 cm by 16 cm and has a thickness of 50 mm. A semicircular section with a radius of 8 cm is removed.

Calculate the area of the face of the remaining piece of wood (take p = 3.142). Give your answer correct to two decimal places.

16 cm

8 cm

(b) Forty rectangular pieces of 8 cm by 16 cm by 50 mm are glued together to form a rectangular block. Calculate the volume of the rectangular block.

16 cm

8 cm

76

(c) Forty semicircular sections as above are cut from the rectangular block.

Calculate the volume of wood remaining in the rectangular block after the sections are cut.

16 cm

8 cm

77


Sample Paper no. 7


Paper 1

Ordinary Level


300 marks

78




A Leaving Certifi cate class is sponsoring a drama production to raise funds for their school. They plan to charge the same admission fee for all seats.

The class incur €700 expenses. They calculate that if 300 tickets are sold, the class will make a profi t of €1,100.

If 500 tickets are sold, what will the profi t be?


(a) Place a tick beside the correct answer.

(i) If n is an integer, then 2n will be:

Odd

Even

Even or odd

(ii) If n is an integer then 3n will be:

Odd

Even

Even or odd

79

(b) Consider the cards shown below.

3a + 1

Card P

2(a – 1)

Card Q

a2 – 2

Card R

6 – a

Card T

(a + 1)2

Card S

(i) If a = 3, which card has the highest value?

(ii) If a = –3, which card has the highest value?

(iii) One of the cards will always produce a positive answer. Write down which card this is and give a reason for your answer.

80


2 4 6

20

40

x

y

Dis

tan

ce f

ro

m

ho

me (

km

)

Time of Day (Hours)

The graph above shows the distance of Shane’s car from his home over a period of time on a given day.

(a) Describe the scenario that is illustrated above.

(b) Calculate the speed of the initial part of Shane’s journey.


Solve for x: 6 __ x +

6 _____

x + 2 =

5 __

2 , x ∈ R.

81




Tom would like to start saving some money, but because he has never tried to save money before, he decides to start slowly. At the end of the fi rst week, he deposits €5 into his bank account, and €5 per week thereafter.

(i) After how many weeks will Tom have €50 in his bank account?

(ii) How many weeks will it take for Tom to save €1,050?

82

Tom has a friend, Dave, who decides that he will save also. He has a different savings plan. He decides that he will start by saving €0.50 at the end of the fi rst week. At the end of the each subsequent week, he will save so that the balance is doubled.

(iii) Fill in the table below, illustrating the progress of Dave’s savings plan.

Week 1 2 3 4 5 6

Amount (€) 0.50 1

Week 7 8 9 10 11 12

Amount (€)

(iv) Comment on the sustainability of Dave’s plan.

83


Glen has €120,000 to invest. The bank has offered him a nominal interest rate of 7.2% per annum compounded monthly.

(i) Calculate the effective rate per annum correct to two decimal places.

(ii) Use the effective rate to calculate the value of Glen’s investment if he invested the money for three years, correct to the nearest cent.

When doing his calculations, Glen misread the details and thought that the interest rate of 7.2% was to be compounded annually.

(iii) Calculate the amount Glen had expected to get.

84

(iv) Calculate the percentage error, correct to two decimal places, between in the calculations based on compounding monthly versus compounding annually.

(v) Suppose Glen invests his money for a total period of four years, but after 18 months makes a withdrawal of €20,000. How much will he receive at the end of the four years, given that the bank reduces the effective rate of interest to 6.1% for the remainder of the term because of the withdrawal? Give the answer correct to the nearest cent.

85




(a) The function f is defi ned by f : R: x → 4x – 5.

(i) Find f (3).

(ii) Hence, fi nd the value of k for which k f (3) = f (10).

(b) Let g(x) = 1 __ x , for x ∈ R and x ≠ 0.

(i) Find g ( 1 _ 4 ) , g ( 1 _ 2 ) , g(2), g(4).

(ii) Under central symmetry in the origin, fi nd the image of each of the points (1,1) and ( 4, 1 _ 4 ) .

86

(iii) Draw the graph of g(x) = 1 __ x for – 4 ≤ x ≤ 4.

(iv) Find the derivative of g(x).

87


(a) Differentiate x(5 – 3x2) with respect to x.

(b) Find dy

___ dx

of the following:

(i) y = (1 – x 2)2

(ii) y = 1 – x 2

_____ x

88

(c) The height h metres of a balloon is related to the time t seconds by h = 120t – 15t2.

(i) Find the height after 2 seconds.

(ii) Find the maximum height reached by the balloon.

89


Sample Paper no. 7


Paper 2

Ordinary Level


300 marks

90




The points P(1,1) and Q(7,5) are the endpoints of the diameter of a circle k.

(a) Find the co-ordinates of the centre and length of the radius of k.

(b) Find the equation of k.

(c) Find the equation of the tangent to k at the point Q(7,5).

91

(d) The circle s is the image of the circle k under Sy, axial symmetry in the y-axis.

Find the equation of s.


(a) A line l cuts the axes at (3,0) and P(0,6). Find the slope of l.

(b) Find the equation of l.

92

(c) k is a line whose equation is 6x − 2y − 3 = 0. Sketch the line in the diagram provided in part (a) and write down the point of intersection between l and k.

(d) Find the area of the triangle formed by the two lines l and k and the x-axis.


(a) In the diagram shown below, O is the centre of the circle with radius length of 4 cm.

P and Q are points on the circle such that |∠POQ| = 60°.

4 cm

O

P

Q

60°

93

Calculate the area, correct to one decimal place, of the shaded region between [PQ] and the circle.

(b) The sketch shows a fi eld ABCD as shown.

A

B C

D

10 m

15 m 17 m17 m 14 m 15 m17 m

20 m

12 m

9 m

At equal intervals of 10 m along [BC], perpendicular measurements of 15 m, 17 m, 17 m, 14 m, 15 m, 17 m, 20 m, 12 m and 9 m are made to the top boundary [AD].

Use the Trapezoidal Rule to estimate the area of ABCD.

94


Two non-biased six-sided dice are thrown.

(a) Use the two-way table below to show the sample space.

1 2 3 4 5 6

1

2

3

4

5

6

(b) Use the two-way table to fi nd the probability of throwing two numbers:

(i) which are the same.

(ii) which give a total of 10.

(iii) which are both odd.

(iv) which add to a prime number.

95

(c) Create a two-way table to show the sample space of the outcomes of throwing a dice and tossing a coin.

(i) Find the probability of getting an odd number and a head when you toss the coin and throw the dice.


In the diagram given, |AC| = 11 cm, |∠DAB| = 122° and |∠ECB| = 140°.

Calculate the area of the triangle ABC correct to the nearest cm2.

96


(a) Explain what a theorem is.

(b) The diagram shows the circle k, with centre O and the angles A and B as shown.

OB

B

A

A

(i) Mark into the diagram another angle which is equal to the angle A. Give a reason why you marked in that angle.

(ii) Mark into the diagram another angle which is equal to the angle B.Give a reason why you marked in that angle.

(iii) Explain why |∠A| + |∠B| = 90°.

97


Find the value of the angles A, B, C and D in the following diagram.

BDC

A

47°29°

Angle A:

Reason:

Angle B:

Reason:

Angle C:

Reason:

Angle D:

Reason:

98




A group of students in class wanted to test to see if the probability of getting a particular number when you roll a dice is actually

1

_ 6 . They performed an experiment in a series of steps.

Step 1

They rolled the dice 120 times and recorded the results in the table below.

Number on dice

Actual number of times the number comes up

Theoretical number of times the number comes up

1 24 20

2 11

3 22

4 20

5 25

6 15

(a) Use the information given to complete the table above.

Step 2

They said that they wanted to display their results on a chart or graph. One student displayed the results as a pie chart and another said it would be better to display the results as a bar chart. Both are shown below.

12

3

45

6

Freq

uenc

y

1 2 3 4 5 6

20

Number on diceIndicates theoretical result

30

25

20

15

10

0

5

99

(b) Which chart is best in showing a picture of the data collected? Give a reason for your choice.

Step 3

The students concluded that rolling the dice 120 times was not enough, so they decided to repeat the experiment. They rolled the dice 250 times, 500 times and 1,000 times and recorded their results as follows:

Freq

uenc

y

1 2 3 4 5 6

50

40

30

20

10

0

Number on dice

250 Times

41.66

Freq

uenc

y

1 2 3 4 5 6

95

90

85

80

75

70

Number on dice

500 Times

83.3

Freq

uenc

y

1 2 3 4 5 6

200

150

100

50

0

Number on dice

1,000 Times

166

(c) (i) From each of the charts in Step 3, estimate the probability of getting a 5 when you roll a dice.

100

(ii) Do you think that Step 3 gave defi nite evidence that the probability of getting a 5 when

you roll the dice is 1

_ 6 ? Tick Yes or No and give a reason for your answer.


Pádraic noticed that the more he practised his place-kicking (kicking the ball from the ground) during training, the more his scoring rate increased during a competitive game. He set about doing an experiment to see if his scoring rate did actually increase if he increased the number of practice kicks he took during training.

He recorded the number of practice kicks each Thursday night and then recorded his scoring rate for the following Sunday. The data he gathered is shown in the table below:

No. of practice kicks (Thursday) Percentage scoring rate (Sunday)

Week 1 10 30

Week 2 20 65

Week 3 15 48

Week 4 30 73

Week 5 15 45

Week 6 12 37

Week 7 25 73

Week 8 40 87

Week 9 26 75

(a) Draw a scatter graph of the data on the diagram provided below.

Yes No

Reason:

101

(b) Comment on the correlation between the number of practice kicks and the percentage scoring rate.

(c) Calculate the mean number of practice kicks over the ten-week period.

(d) Find the median percentage scoring rate.

(e) Pádraic concluded that the more practice kicks he took during Thursday’s training, the higher his percentage scoring rate would be during Sunday’s game. If you were his manager, what advice would you give to Pádraic?

102


(a) The shape given below represents the markings on the fl oor of a gym. The shape consists of an arc AB with A and B being joined to O. |AO| = |OB| = 3 m, and |∠AOB| = 45°.

Calculate the perimeter of the shape, correct to two decimal places.

45°

3 m 3 m

O

A B

(b) (i) Find the volume of a solid sphere of diameter 3 cm. Give your answer in terms of p.

103

(ii) A cylindrical vessel with an internal diameter of 18 cm contains water. The surface of the water is 11 cm from the top of the cylinder. How many spheres, each with a diameter of 3 cm, must be placed in the water in order to bring the surface of the water to 1 cm from the top of the cylinder? Assume that all the spheres are under the water level.

11 cm

18 cm

1 cm

18 cm

104

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Mathematics (Project Maths – Phase 2) · PDF fileMathematics (Project Maths – Phase 2) Paper 1 Ordinary Level Time: 2 hours, 30 minutes 300 marks 1. Section A Concepts and Skills