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2019.1 L.16 1/16

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Pre-Leaving Certificate Examination, 2019 Mathematics Paper 1 Ordinary Level Time: 2 hours, 30 minutes 300 marks

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Pre-Leaving Certificate Examination, 2019 Mathematics, Paper 1 – Ordinary Level 2019.1 L.16 2/16

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Instructions

There are two sections in this examination paper.

Section A Concepts and Skills 150 marks 6 questions

Section B Contexts and Applications 150 marks 3 questions

Answer all nine questions.

Write your answers in the spaces provided in this booklet. You may lose marks if you do not do so. You may ask the superintendent for more paper. Label any extra work clearly with the question number and part.

The superintendent will give you a copy of the Formulae and Tables booklet. You must return it at the end of the examination. You are not allowed to bring your own copy into the examination.

You may lose marks if your solutions do not include supporting work.

You may lose marks if you do not include appropriate units of measurement, where relevant.

You may lose marks if you do not give your answers in simplest form, where relevant.

Write the make and model of your calculator(s) here:

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Section A Concepts and Skills 150 marks

Answer all six questions from this section.

Question 1 (25 marks)

The table below shows the population of Ireland according to the most recent censuses.

Census Year 2006 2011 2016

Population 4 239 848 4 588 252 4 761 865

(a) What is the population of Ireland according to the most recent census? Write this number in the form a × 10n, where 1 ≤ a < 10 and n ∈ ℕ,

correct to two significant figures.

(b) Show that the average annual increase in population between 2011 and 2016 was 0·757%, correct to three decimal places.

(c) Find the expected increase in the population of Ireland by the time of the next census in 2021, assuming the population grows at the same rate year-on-year.

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(a) z1 = 2 − 3i is a complex number, where i2 = −1.

(i) Let z2 = iz1 and z3 = iz2. Find z2 and z3, in the form a + bi, where a, b ∈ ℝ.

(ii) Plot z1, z2 and z3 on the Argand diagram. Label each point clearly.

(iii) Use your diagram to describe what happens when a complex number is multiplied by i.

(b) Let w = 2 − 4i, where i2 = −1. Find the real number k such that

k(ww ) = 15,

where w is the complex conjugate of w.

z2 = z3 =

Re( )z

Im( )z

�2�4�6 2 4 6

2

�2

4

�4

6

�6

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(a) (i) Solve for x: 3(5x + 2) − 14x = 8 − 13(5 − x).

(ii) Hence, or otherwise, solve the inequality 3(5x + 2) − 14x ≥ 8 − 13(5 − x), where x ∈ ℕ, and show the solution set on the number line below.

(b) (i) Write 33+x

− 12 4−x

as a single fraction.

(ii) Hence show that 33+x

− 12 4−x

= 23 has no real solutions.

0

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The function f ׀→ x3 − 2x2 − 4x − 3 is defined for x ∈ ℝ.

(a) (i) Find f (−1) and f (3).

(ii) Find the co-ordinates of the point at which the graph of f cuts the y-axis.

(b) The diagram below shows the graph of the function f (x) = x3 − 2x2 − 4x − 3, where x ∈ ℝ.

(i) Mark on the diagram the range of values of x for which f (x) is decreasing.

(ii) On the same diagram, sketch the graph of the function f ′(x), the derivative of f (x).

1

x

yf x( )

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(c) Find f ′(x), the derivative of f (x). Hence find the co-ordinates of the local minimum turning point of the function f (x).

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Niamh lives in England and is part of a lottery syndicate which wins £3·77 million sterling. Each week, 15 people pay £2, 14 people pay £4 and 6 people pay £5 into the syndicate. The prize money is divided in proportion to how much each person pays.

(a) (i) Find the amount of money paid into the syndicate each week.

(ii) Given that Niamh contributes £5 per week, find the amount of prize money she receives.

(b) Niamh returns to Ireland and exchanges £150 000 sterling for euro. The exchange rate for the transaction is €1 = £0·87 sterling. Find, correct to the nearest euro, the amount that she receives.

(c) Niamh wishes to exchange the remainder of her prize money for US dollars. On a given day, the exchange rate for euro to sterling is €1 = £0·87 and for euro to dollars is €1 = $1·19. Find the sterling to dollar exchange rate and write your answer in the form £1 = $·. Hence calculate the amount that Niamh can expect to receive in dollars.

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Mia earns a gross salary of €2185 every fortnight. She pays income tax, universal social charge (USC) and pay-related social insurance (PRSI) on this salary.

(a) The standard rate of income tax is 20% and the higher rate is 40%. Mia has tax credits of €3302 per annum and a standard rate cut-off point of €35 308.

(i) Find Mia’s tax credit and standard rate cut-off point per fortnight [1 year = 26 fortnights].

(ii) How much income tax does Mia pay each fortnight, correct to the nearest cent?

(b) Mia pays USC on her gross salary. Her total USC amounts to €72⋅27 each fortnight. She pays 0·5% on the first €462 she earns, 2% on the next €303 and x% on the balance.

Find x%, the highest percentage rate at which Mia pays USC.

(c) Mia also pays PRSI of 4% on her gross salary. Write Mia’s take-home pay each fortnight as a percentage of her gross salary.

Give your answer correct to one decimal place.

Rate (%) Amount (€)

0·5

2

x

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Section B Contexts and Applications 150 marks

Answer all three questions from this section.


E. coli is a bacterium that exists naturally in our bodies. However, some strains can cause sickness if contaminated food or polluted water is consumed.

A microbiologist wishes to examine the growth of E. coli bacteria under different sets of conditions. Three samples of meat are prepared in Petri dishes and examined under a microscope every hour to observe the numbers of bacteria present.

(a) The first sample of meat was fully cooked, allowed to cool and then stored in the fridge. The numbers of bacteria observed in the sample form an arithmetic sequence.

(i) Complete the table below to show the growth rate of bacteria over the first 6 hours.

After n hours 1 2 3 4 5 6

Number of bacteria 2000 4000 6000

(ii) The number of bacteria present after n hours is given by the formula Tn = pn + q, where p, q ∈ ℤ. Find the value of p and the value of q.

(iii) Using your formula, or otherwise, find the expected number of bacteria after 12 hours.

(b) The second sample of meat was partially cooked and then stored in the fridge. The numbers of bacteria observed form a quadratic sequence with a second difference of 2000.

(i) Complete the table below to show the growth rate of bacteria over the first 6 hours.


Number of bacteria 2000 4000

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(ii) The number of bacteria present after n hours is given by the formula Tn = 1000n2 + bn + c, where b, c ∈ ℤ. Find the value of b and the value of c.

(iii) After how many hours does the number of bacteria in the sample exceed 134 000?

(c) The third sample of meat was left uncooked and kept out of the fridge. The numbers of bacteria present after n hours form an exponential sequence, given by the formula Tn = abn, where a, b ∈ ℕ.

Find the value of a and the value of b. Hence complete the table to show the growth rate of bacteria in this sample.


Number of bacteria 2000 4000

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A climber fires a distress flare up into the air from an elevated position on the side of a mountain. The estimated height of the flare above sea-level is given by the function:

h(t) = 25t − t2 + 650

where h is the height of the distress flare in metres above sea-level and t is the time in seconds after it is fired.

(a) Use the height function, h(t), to find the height above sea-level at which the climber fires the distress flare.

(b) (i) Use calculus to find, in terms of t, the rate at which the height of the distress flare is changing after t seconds.

(ii) Use your answer to part (b)(i) to find the maximum height of the distress flare above the position of the climber.

(c) The ground at the base of the mountain is 400 m above sea-level. How long does it take the distress flare to hit the ground after it is fired?

Give your answer in seconds, correct to two decimal places.

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(d) Find h″(t), the second derivative of h(t). Hence explain the significance of your answer.

(e) (i) Use the height function, h(t), to complete the table, showing the estimated height of the distress flare above sea-level over time.

t (seconds) 0 5 10 15 20 25 30

h(t) (metres) 800

(ii) Use the data in the table to draw the graph of the function h(t) on the axes below for 0 ≤ t ≤ 30, where t ∈ ℝ.

(iii) Use your graph to estimate the time interval for which the height of the distress flare is at least 75 m above the position of the climber.

1

1

1

500

550

650

750

850

600

700

800

Time (seconds)

He

igh

ta

bo

ve

sea

-le

ve

l(m

)

5 10 15 20 25 30

x

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When thunderstorms occur, people often try to establish how far away they are from where lightning strikes the ground.

One technique, called the “flash-to-bang” method, involves counting the number of seconds, t, that pass between the flash of lightning being seen and the clap of thunder being heard and then dividing that number by 5. The result denotes how far away, in miles, from where lightning strikes.

(a) (i) Write down a formula to estimate the distance, in miles, from where lightning strikes. State clearly the meaning of any letters used in your formula.

(ii) Use your formula to find how far away does lightning strike if there is a 20-second gap between the flash of lightning and the clap of thunder.

(b) In Ireland we usually use the metric system. The speed of sound is approximately 340 metres per second depending on air temperature.

(i) Find the number of seconds that would pass between the flash of lightning and the clap of thunder for the lightning strike to be 1 kilometre away.

Give your answer correct to the nearest whole number. [Hint: How long does it take for the sound of thunder to travel 1 kilometre?]

(ii) Hence, or otherwise, write down a formula to estimate the distance, in kilometres, from where lightning strikes.

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(c) During a thunderstorm, two students standing at the same location counted 15 seconds between the flash of lightning and the clap of thunder. One student used the mile calculation formula while the other student used the kilometre calculation formula.

(i) Estimate the distance from where the lightning struck using both methods.

(ii) Given that the conversion between miles and kilometres is 1 mile = 1·609344 km, state whether these two methods agree. Explain your answer.

(d) The speed of sound at a particular air temperature is given by the formula:

c(T) = 331 + 0·6 × T

where c is the speed of sound, in metres per second, and T is the air temperature measured in degrees Celsius.

(i) Find c, the speed of sound when the air temperature is 0 °C.

(ii) Find the air temperature when the speed of sound is 340 metres per second.

(iii) Write down a formula to estimate the distance from where lightning strikes in terms of air temperature and the time gap between the flash of lightning and the clap of thunder.

(iv) Hence, or otherwise, find how far away lightning strikes if the air temperature is 18 °C and there is a 9-second gap between the flash of lightning and the clap of thunder.

Give your answer in kilometres, correct to one decimal place.

Mile calculation: Kilometre calculation:

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2019.1 L.16 16/16

Pre-Leaving Certificate Examination, 2019 – Ordinary Level Mathematics – Paper 1 Time: 2 hours, 30 minutes 06349d72-b3d1-4020-a7ac-9e1d8cc9ef89

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