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Pre-Junior Certificate Examination, 2013

Triailscrúdú an Teastais Shóisearaigh, 2013

_______________

Project Maths

Paper 1

Higher Level

2½ hours

300 marks

For examiner

Question Mark 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15

Total

*PM60*

Name:

School:

Address:

Class:

Teacher:

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Instructions There are 15 questions on this examination paper. Answer all questions. Questions do not necessarily carry equal marks. To help you manage your time during this examination, a maximum time for each question is suggested. If you remain within these times, you should have about 10 minutes left to review your work. Write your answers in the spaces provided in this booklet. There is space for extra work at the back of the booklet. Label any extra work clearly with the question number and part. Marks will be lost if all necessary work is not clearly shown. Answers should include the appropriate units of measurement, where relevant. Answers should be given in simplest form, where relevant. Write the make and model of your calculator(s) here:

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Question 1 (Suggested maximum time: 10 minutes) 180 students were surveyed about internet sites they used regularly: 60 used Twitter 76 used Facebook 81 used YouTube x used Facebook and YouTube 30 used Facebook and Twitter 15 used Twitter and YouTube 22 used none of these sites 8 used all three regularly (a) Represent the above information on the Venn diagram. (b) Calculate the value of x.

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(c) What is the probability that a student chosen at random used Facebook only? (d) What is the probability that a student chosen at random used Twitter or YouTube? Question 2 (Suggested maximum time: 5 minutes) Solve the following equation ( ) ( )3 1 5 5 12x x− − − = .

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Question 3 (Suggested maximum time: 10 minutes) The USC is calculated on gross income. The rates for USC are: − 2% on the first €10,036 of gross annual income − 4% on the next €5,980 − 7% on the balance (a) Christine earned €34,000 in 2012. Find her USC for that year. (b) Robert paid €455 more USC than Christine did in 2012. Calculate Robert’s gross pay for 2012.

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Question 4 (Suggested maximum time: 10 minutes) The table below shows the growth of new bacteria cells every hour for 5 hours. (a) Complete the table of differences or changes below and say whether

the growth of the cells is linear, quadratic or exponential. Give a reason for your choice.

What type of function: Reason for your choice: (b) Students were asked to write down a formula to describe the growth of the bacteria cells.

Using the variables, T per time in hours and B for the number of new bacteria cells, there are three student’s attempts:

Garry’s attempt: ( )3 2 TB =

Caroline’s attempt: ( )2 3 TB =

Tony’s attempt: 2 3 1B T T= + − One of these formulas is correct. Identify which one is correct and justify your choice.

Time (T hours)

Number of new bacteria cells (B)

First Difference (First Change)

Second Difference (Third Change)

0 2

1 6

2 18

3 54

4 162

5 486

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Question 5 (Suggested maximum time: 10 minutes) (a) The area of a rectangle is given by the expression 22 7 15x x+ − . If the length of one of the

sides is 5x + , write down an expression in x for the length of the other side. (b) If the area of the above rectangle above is 24 square units, use 22 7 15x x+ − to find the dimensions of the rectangle.

22 7 15x x+ −5x +

24

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Question 6 (Suggested maximum time: 10 minutes) (a) Fiona invests €5,000 in a saving scheme with an investment agency that gives 5% interest at the end of each year. The investment agency deducts a tax of 25% on the interest earned. Calculate how much money will Fiona have in the savings account at the end of the first year? (b) John invested a sum of money in the same saving scheme and at the end of the first year his

investment had earned € 270 once the tax had been deducted. Calculate how much money John invested.

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Question 7 (Suggested maximum time: 5 minutes) Solve the following equation: 4 3 2 1 1

3 2 2x x− −− =

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Question 8 (Suggested maximum time: 15 minutes) A company is designing a rectangular memory card. The length of the perimeter of the face of the card is 100 mm. (a) If the length of the card is x mm long write down an expression in x for the width of the card. (b) Hence, or otherwise, show that the area of the memory card is given by the formula 250A x x= − . (c) Let f be the function 2: 50f x x x→ − in the domain 0 50x≤ ≤ . Evaluate ( )f x when 0,10, 20, 30, 40, 50x = .

x

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(d) Hence, draw the graph of f for 0 50,x x≤ ≤ ∈ . (e) Use your graph from part (b) to estimate:

(i) the area of the card when 15x = mm. (ii) the maximum possible area. (f) Technicians require the area of the chip to be at least 200 mm. Find the range of all possible values of x that meet this requirement.

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Question 9 (Suggested maximum time: 10 minutes) Factorise each of the following: (i) 24 10p pq− (ii) 23 2 3 2m n nm n− − + (iii) 2 24 25p q−

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Question 10 (Suggested maximum time: 10 minutes) (a) The annual running cost for a small golf club in 2011 was €18,000. If this cost is shared equally between the members write down an expression in x for the 2011 membership fee. (b) In 2012 the running cost for the golf club was €17,500 and the club admitted 10 new members. If this cost is again shared equally between the members write down an expression in x for the membership fee. (c) If the membership fee in 2012 was €50 euro cheaper than the membership fee in 2011,

write down an equation in x and solve it to find out the number of members in 2011.

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Question 11 (Suggested maximum time: 10 minutes) Here are 5 graphs of different quadratic functions labelled a, b, c, d and e: In the table below match each function to its graph and in the section provided give a reason for each choice.

Function Label Reason

( ) 2f x x=

( ) ( )22f x x= −

( ) 2 2f x x= −

( ) 2f x x= −

( ) 2 3f x x= − +

a

b c

d e

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Question 12 (Suggested maximum time: 5 minutes) (a) Solve the inequality 6 3 3 15,x x− < + ≤ ∈ . (b) Graph your solution on the number line below. (c) A teacher asked her class if it was possible to graph the inequality: 1 5,x x≤ ≤ ∈ .

John, a student in her class, came up with the following solution:

Explain, giving a reason, why this answer is incorrect?

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Question 13 (Suggested maximum time: 15 minutes) Kevin and Ailbhe are regular joggers. Kevin runs every day for 40 minutes and Ailbhe runs every day for 16 minutes. Both of them decide to run in the next Cork marathon. Kevin decides to increase by 5 minutes every day the amount of time he spends running and Ailbhe decides to increase by 8 minutes every day the amount of time she spends running. (a) Complete the table below to show the amount of time they each spend running for the next

10 days. (b) On the grid below graph the daily running times for both runners.

Day Minutes Kevin spends running Minutes Ailbhe spends running

0 40 16

1

2

3

4

5

6

7

8

9

10

Tim

e sp

ent r

unni

ng

Days

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(c) After how many days will Kevin and Ailbhe be training for the same amount of time? (d) Write down a formula that allows you to work out the amount of time Kevin spends jogging on a particular day. State clearly the meaning of any letters you use in the formula. (e) Kevin wants to build up his training time to 160 minutes per day. Using the formula

from (d), or otherwise, work out how many days it will take Kevin to reach this time.

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(f) Write down a formula that allows you to work out the amount of time Ailbhe spends jogging on a particular day. (g) Use your formulas from (e) and (f) to verify the answer that you gave to part (c) above. (h) After 14 days Ailbhe decides to stop increasing her training time by 8 minutes and just train every day at the 14-day training time. If Kevin continues to increase his training times, on which day and will he pass Ailbhe‘s training time?

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Question 14 (Suggested maximum time: 5 minutes) The quadratic equation 2 0x px q+ + = has roots 5− and 6. Find the values of p and q.

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Question 15 (Suggested maximum time: 10 minutes) (a) Solve, correct to two decimal places, 23 5 1 0x x− − = . (b) Hence, or otherwise, find, correct to one decimal place, the values of t for which ( ) ( )23 1 2 5 1 2 1 0t t− − − − = .

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