Mathematics

[Pages:16]L.16

NAME SCHOOL

TEACHER

Pre-Leaving Certificate Examination, 2019

Mathematics

Paper 1 Ordinary Level

Name/versio Printed:

Checked: To:

Updated: Name/versio Complete (y/

Time: 2 hours, 30 minutes

300 marks

School stamp Running total

For examiner

Question

Mark

1

2

3

4

5

6

7

8

9

Total

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06349d72-b3d1-4020-a7ac-9e1d8cc9ef89

Grade

Instructions

There are two sections in this examination paper.

Section A

Concepts and Skills

Section B

Contexts and Applications

150 marks 150 marks

6 questions 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 Population

2006 4 239 848

2011 4 588 252

2016 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 ,

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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Question 2 (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 .

z2 =

z3 =

(25 marks)

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

Im(z) 6 4

(iii) Use your diagram to describe what

happens when a complex number is multiplied by i.

2

-6 -4 -2 -2 -4

Re(z) 246

-6

(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.

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Question 3 (a) (i) Solve for x:

3(5x + 2) - 14x = 8 - 13(5 - x).

(25 marks)

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

0

(b) (i)

Write

x

3 +

3

-

4 2x -

1

as

a

single

fraction.

(ii)

Hence

show

that

x

3 +

3

-

4 2x -

1

=

3 2

has

no

real

solutions.

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Question 4 The function f x3 - 2x2 - 4x - 3 is defined for x . (a) (i) Find f (-1) and f (3).

(25 marks)

(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 .

y

f (x)

1 x

(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).

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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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Question 5

(25 marks)

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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