Paper 1 Ordinary Level - School of Maths - Home

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Pre-Leaving Certificate Examination, 2017

Triailscrúdú na hArdteistiméireachta, 2017

Mathematics

Paper 1

Ordinary Level

2½ hours

300 marks

For examiner

Question Mark

1

2

3

4

5

6

7

8

9

Total

Grade

*B6*

Name:

School:

Address:

Class:

Teacher:

Running total

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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. There is space for extra work at the back of the booklet. You may also 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 will lose marks if you do not show all necessary 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 down 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) (a) Jack is making a door frame for his house. He measures the width of the frame as 88 cm. The frame is actually 89 cm wide.

(i) Calculate Jack’s error in his measurement.

(ii) Calculate the percentage error in Jack’s measurement, correct to three significant figures.

(b) The height of the door frame is 210 cm. The lengths of timber Jack

needs are only sold in 5 m lengths.

(i) If Jack has to make 5 door frames, how many lengths of timber will he require?

(ii) The lengths of timber cost €24 per length. VAT is charged at 21%. Calculate the cost of the door frames.

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Question 2 (25 marks) (a) The complex number, 1z is shown on the Argand diagram.

(i) Write down 1z in the form bia + , where ∈ba, ℤ.

(ii) Given that 12 2zz = , write down 2z in the form bia + , where ∈ba, ℤ.

(iii) Show 2iz on the Argand diagram.

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(b) (i) Find 1z , the modulus of 1z .

(ii) Investigate if 2121 zzzz +=+ .

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Question 3 (25 marks) (a) Factorise fully each of the following expressions.

(i) 152 2 −+ xx

(ii) 123 2 −y

(iii) afbebfae −+−

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(b) (i) Express 2

3

1

2

++

+ xx as a single fraction where 2,1 −−≠x .

(ii) Hence, or otherwise, solve the equation; 22

3

1

2 =+

++ xx

, where ∈x ℝ.

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Question 4 (25 marks) (a) Solve for x.

( ) ( ) 2212365 −=−−+ xx (b) Solve the inequality 33222 +≤−− xx where ∈x ℝ and show the solution set on a number

line. (c) Solve the equation 3742 2 −− xx , giving the solutions correct to two decimal places.

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Question 5 (25 marks) The function ( ) 32 ++= bxaxxg is shown on the Cartesian plane, where ∈ba, ℤ. The points ( )7,4−A and ( )3,2 −−B are on ( )xg as shown. (a) (i) Given that ( ) 74 =−g and ( ) 32 −=−g , form two equations in a and b.

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(ii) Solve the equations to find the value of a and b. (b) Find the slope of the tangent to the curve at the point B.

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Question 6 (25 marks) A farmer is fencing in a rectangular field. A wall bounds the field along one of its lengths and this side does not require fencing. The farmer has 400 m of wire to fence in the area. (a) Show that the area of the field is given by

Area2

400 2xx −= m2.

(b) Given that one possible area of the field is 18,750 m2, find the value of x, the length of the field.

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(c) Find the dimensions of the field that will give the maximum area.

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Section B Contexts and Applications 150 marks

Answer all three questions from this section. Question 7 (45 marks) USA athlete Michelle Carter won a Gold medal in the shot put in the Rio Olympics in 2016. The height, h, of her winning shot put throw, measured in metres, can be modelled using the equation

( ) 68.17277.02 ++−= ttth where t is measured in seconds. (a) At what height above the ground was the shot put when it was released by Michelle? Explain your answer. (b) What was the height of the shot put after 0.7 seconds, correct to one decimal place?

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(c) Find the maximum height reached by the shot put, correct to two decimal places. (d) (i) Draw a graph of the function ( ) 68.17277.02 ++−= ttth in the domain 7.10 ≤≤ t where ∈x ℝ.

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(ii) After how many seconds does the shot put hit the ground?

(e) Given that the average speed of the shot put over the length of the throw is 12.22 m/s, calculate the distance Michelle Carter threw to win gold in Rio.

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Question 8 (45 marks) (a) Jack and Jill are playing a numbers game with arithmetic sequences of numbers. Jack gives Jill the list of numbers below with some of the numbers missing. He tells her that they are the first 5 terms in an arithmetic sequence.

x ,9, w, y, 24.

(i) Jill states that the common difference for the sequence must be 5. Is she correct? Explain your answer.

(ii) Jill then calculated the missing values. What were the values of x, w, and y that she calculated?

(iii) Jill states that the sequence can be represented by the rule 15 −= nTn . Is Jill correct?

Explain your answer.

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(iv) Jack asks Jill to find the value of T57 for his sequence. What value did she calculate?

(v) Jack added up a number of terms in his sequence. If the sum of these n terms is Sn = 837, how many terms did Jack add up?

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(b) Jill the gives Jack the following sequence of numbers

5, 10, 17, 26, 37, 50.

(i) Jack states that Jill is cheating and her sequence is not arithmetic. Is Jack correct? Explain your answer fully.

(ii) Find the formula for Tn for Jill’s sequence of numbers

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Question 9 (60 marks)

(a) Paul earns a gross salary of €54,000 per annum. He has a standard cut off rate of €28,000. The standard rate of tax is 21% and the high rate of tax is 42%. Paul has tax credits of

€2,400 per annum.

(i) Calculate Paul’s gross tax.

(ii) Calculate Paul’s net monthly income, correct to the nearest euro.

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(b) (i) Each month Paul saves 5% of his monthly salary for his holiday fund. How much will Paul save between 1st of August and the 31st of May the following year?

(ii) Paul decides to go to Las Vegas on his holidays when the exchange rate is

€1=$1.35. How much is Paul’s holiday fund worth in US dollars?

(iii) On arrival in Las Vegas Paul notices that his hotel offers a rate of €1=$1.40, but charges a commission of 2% on the transaction. Would Paul have received more or less dollars in his hotel?

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(c) Paul decides to invest €15,000 in a 6.5 year bond. When the bond matures he receives

€19,355.66.

(i) How much interest did he receive in total?

(ii) What was the total percentage increase earned for his investment, correct to the nearest whole number?

(iii) Calculate the AER for this bond, correct to three decimal places.

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