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TABLE of CONTENTS Pre-Leaving Certificate Examination - HIGHER LEVEL
MOCK PAPER 1
MOCK PAPER 5 01
Paper 1
229
Paper 2
33
Paper 2
257
MOCK PAPER 2
Paper 2
MOCK PAPER 3
65
Paper 1
285
93
Paper 2
309
MOCK PAPER 7
117
Paper 1
333
149
Paper 2
357
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Paper 1
MOCK PAPER 6
PL
Paper 1
Paper 2
E
Paper 1
MOCK PAPER 4
MOCK PAPER 8
Paper 1
181
Paper 1
381
Paper 2
205
Paper 2
405
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*P6*
Pre-Leaving Certificate Examination
E
Mathematics Higher Level
PL
Paper 1
2 hours 30 mins
SA M
300 marks
Name:
School:
Address: Class:
Teacher:
For examiner
Question 1 2
Mark
3 4 5 6 7 8 9
10
Total
Running total
1
Grade
Instructions There are two sections in this examination paper.
Section A Section B
Concepts and Skills Contexts and Applications
150 marks 150 marks
6 questions 4 questions
Answer questions as follows: • any five questions from Section A – Concepts and Skills
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• any three questions from Section B – Contexts and Applications
Write your details in the box on the front cover.
Write your answers in blue or black pen. You may use pencil in graphs and diagrams only.
PL
Anything that you write outside of the answer areas may not be seen by the examiner.
Write all answers into this booklet. There is space for extra work at the back of the booklet. If you need to use it, 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.
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You will lose marks if your solutions do not include relevant supporting work.
You may lose marks if the appropriate units of measurement are not included, where relevant. You may lose marks if your answers are not given in simplest form, where relevant.
Write the make and model of your calculator(s) here:
2 2
Section A
Concepts and Skills
150 marks
Answer any five questions from this section.
Question 1 (a)
Given
(30 marks) 4𝑥𝑥𝑥𝑥 2 + 8𝑥𝑥𝑥𝑥 + 3 = 𝑎𝑎𝑎𝑎(𝑥𝑥𝑥𝑥 + 𝑏𝑏𝑏𝑏)2 + 𝑐𝑐𝑐𝑐
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Find the values of the constants 𝑎𝑎𝑎𝑎, 𝑏𝑏𝑏𝑏 and 𝑐𝑐𝑐𝑐.
3 3
(i)
Show that 𝑘𝑘𝑘𝑘 2 − 4𝑘𝑘𝑘𝑘 𝑘 12 > 0.
(ii)
Find the set of possible values of 𝑘𝑘𝑘𝑘.
E
The equation 𝑥𝑥𝑥𝑥 2 + 𝑘𝑘𝑘𝑘𝑥𝑥𝑥𝑥 + (𝑘𝑘𝑘𝑘 + 3) = 0, where 𝑘𝑘𝑘𝑘 is a constant, has two distinct real roots.
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(b)
4 4
Question 2 (i)
A complex number 𝑧𝑧𝑧𝑧 can be written in the form 𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏. Given 𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 = 18, write an equation in terms of 𝑎𝑎𝑎𝑎 and 𝑏𝑏𝑏𝑏.
(ii)
If arg(𝑧𝑧𝑧𝑧) = , find the values of 𝑎𝑎𝑎𝑎 and 𝑏𝑏𝑏𝑏 and hence write 𝑧𝑧𝑧𝑧 in the form 𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏.
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𝜋𝜋𝜋𝜋
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(a)
(30 marks)
4
5 5
(i)
Write the complex number −4 + 4√3𝑏𝑏𝑏𝑏 in polar form.
(ii)
Find the three complex numbers 𝑧𝑧𝑧𝑧 for which 𝑧𝑧𝑧𝑧 3 = −4 + 4√3𝑏𝑏𝑏𝑏.
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(b)
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Express each root in polar form.
6 6
Question 3
(30 marks)
𝑓𝑓𝑓𝑓(𝑥𝑥𝑥𝑥) = 2𝑥𝑥𝑥𝑥 3 − 𝑥𝑥𝑥𝑥 2 + 2𝑥𝑥𝑥𝑥 𝑥 16 is a cubic function. Show that (𝑥𝑥𝑥𝑥 𝑥 2) is a factor of 𝑓𝑓𝑓𝑓(𝑥𝑥𝑥𝑥).
(b)
Given that 𝑓𝑓𝑓𝑓(𝑥𝑥𝑥𝑥) = (𝑥𝑥𝑥𝑥 𝑥 2)(2𝑥𝑥𝑥𝑥 2 + 𝑏𝑏𝑏𝑏𝑥𝑥𝑥𝑥 + 𝑐𝑐𝑐𝑐), find the values of 𝑏𝑏𝑏𝑏 and 𝑐𝑐𝑐𝑐.
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(a)
7 7
(c)
There are two points on 𝑓𝑓𝑓𝑓(𝑥𝑥𝑥𝑥) where the slope of a tangent to the curve is 10.
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Find the coordinates of these two points.
8 8
Question 4 (a)
(30 marks)
The first term of an arithmetic series is 𝑎𝑎𝑎𝑎 and the common difference is 𝑑𝑑𝑑𝑑.
The 18th term of the series is 25 and the 21st term of the series is 32·5.
Use this information to write down two equations in terms of 𝑎𝑎𝑎𝑎 and 𝑑𝑑𝑑𝑑.
(ii)
Show that 𝑑𝑑𝑑𝑑 = 2 · 5 and find the value of 𝑎𝑎𝑎𝑎.
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(i)
(iii)
The sum of the first 𝑛𝑛𝑛𝑛 terms of the series is 2750. Find the value of 𝑛𝑛𝑛𝑛.
9 9
A geometric sequence is as follows: l n 𝑥𝑥𝑥𝑥 , l n 𝑥𝑥𝑥𝑥 2 , l n 𝑥𝑥𝑥𝑥 4 , l n 𝑥𝑥𝑥𝑥 8 , … Find 𝑟𝑟𝑟𝑟, the common ratio between the terms.
(ii)
If 𝑇𝑇𝑇𝑇8 − 𝑇𝑇𝑇𝑇6 = 45, find the value of 𝑥𝑥𝑥𝑥. Give your answer to 1 decimal place.
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(i)
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(b)
10 10
Question 5
(30 marks)
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Find the coordinates of the point where 𝐶𝐶𝐶𝐶 crosses the 𝑦𝑦𝑦𝑦-axis.
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(a)
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The graph shows a sketch of the curve 𝐶𝐶𝐶𝐶 with the equation 𝑦𝑦𝑦𝑦 = (2𝑥𝑥𝑥𝑥 2 − 5𝑥𝑥𝑥𝑥 + 2)(𝑒𝑒𝑒𝑒 −𝑥𝑥𝑥𝑥 )
(b)
Show that 𝐶𝐶𝐶𝐶 crosses the 𝑥𝑥𝑥𝑥-axis at 𝑥𝑥𝑥𝑥 = 2 and find the 𝑥𝑥𝑥𝑥-coordinate of the other point where 𝐶𝐶𝐶𝐶 crosses the 𝑥𝑥𝑥𝑥-axis.
11 11
𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
. Give your answer in simplest form.
(c)
Find
(d)
Hence, find the coordinates of the turning points of C.
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𝑑𝑑𝑑𝑑𝑥𝑥𝑥𝑥
12 12
Question 6 Given that 𝑦𝑦𝑦𝑦 = 2𝑥𝑥𝑥𝑥 , express 4𝑥𝑥𝑥𝑥 − 2𝑥𝑥𝑥𝑥𝑥𝑥 = 3 as an equation in terms of 𝑦𝑦𝑦𝑦.
(ii)
Hence, find the value of 𝑥𝑥𝑥𝑥, correct to 2 decimal places.
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(i)
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(a)
(30 marks)
13 13
Prove using induction that 4𝑛𝑛𝑛𝑛 + 6𝑛𝑛𝑛𝑛 𝑛 1 is divisible by 3 for all 𝑛𝑛𝑛𝑛 𝑛 𝑛.
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(b)
14 14
Section B
Contexts and Applications
150 marks
Answer any three questions from this section. Question 7
(50 marks)
A heated metal ball is dropped into a liquid. As the ball cools, its temperature, 𝑇𝑇𝑇𝑇 °C, 𝑡𝑡𝑡𝑡 minutes after it enters the liquid, is given by 𝑇𝑇𝑇𝑇 = 400 𝑒𝑒𝑒𝑒 −0·05𝑡𝑡𝑡𝑡 + 25, 𝑡𝑡𝑡𝑡 𝑡 0
Find the temperature of the ball as it enters the liquid.
(b)
Find the value of 𝑡𝑡𝑡𝑡 for which 𝑇𝑇𝑇𝑇 = 300, giving your answer to 1 decimal place.
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(a)
15 15
Find the rate at which the temperature of the ball is decreasing at the instant when 𝑡𝑡𝑡𝑡 = 50. Give your answer in °C per minute to 3 significant figures.
(d)
Calculate the average temperature of the ball between 10 minutes and 30 minutes after entering the liquid. Give your answer to the nearest degree.
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(c)
16 16
If the ball is left in the water until it has cooled fully, what is the lowest temperature to which it will fall?
(f)
When the ball is cooling, the metal will contract, and so the volume of the ball will decrease. If the volume of the ball is decreasing at a rate of 0·5π cm3 per minute, find the rate of change of the radius of the ball at a time when the radius is 6 cm.
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(e)
17 17
Question 8 (a)
(50 marks)
Dan has won a prize in a lottery game. When he goes to collect his prize, he is offered one of the following options: Option A: Receive a payment of €2 200 at the beginning of each month for 25 years, starting immediately. Option B: Receive a single lump sum payment immediately.
Dan is unsure of which option to take. He initially opts for the monthly payments and puts them in a bank while he decides what to do. The bank is offering a rate of interest which corresponds to an annual equivalent rate (AER) of 2·8%. Dan allows the monthly repayments to build up in the bank account over a 6-month period. Find the amount in the bank account at the end of the 6 months.
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(i)
18 18
At the end of the 6 months, Dan requests to have the remaining monthly payments paid immediately as a lump sum. Based on an AER of 2·8%, calculate how much Dan would expect to receive as the lump sum.
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(ii)
19 19
(b)
Dan used some of his winnings to buy a new car. He paid €54 000 for the car, which dropped in value each year.
The dealership advise Dan that the car will have a value of €39 015 at the end of two years.
(i)
Assuming that the annual percentage loss remains constant, find the annual depreciation rate and hence deduce the value of the car at the end of the first, third and fourth years and enter these values in the table.
Age in years
1
€54 000
2
3
4
€39 015
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Value
0
(ii)
The dealership have an offer where they will take Dan’s car back as a trade-in against a new car at the end of five years, for a value of €25 000. Is this a good deal for Dan? Justify your answer. Yes, it is a good deal.
No, it is not a good deal.
Justification:
20 20
(c)
A local garage has a scheme to cover the maintenance of the car. The cost is €600 for the first year, and for every following year the cost increases by 12%. Dan thinks that the annual payments form a geometric sequence. Explain why he is correct, and write down the values for 𝑎𝑎𝑎𝑎, the first term, and 𝑟𝑟𝑟𝑟, the common ratio.
(ii)
If Dan signs up to this scheme and stays with it for 10 years, calculate the total amount he will have paid to the local garage by the end of the 10th year.
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(i)
21 21
Question 9 (a)
(50 marks)
A human's respiratory cycle is the length of time elapsed from the beginning of one breath to the beginning of the next breath. For a person at rest, the velocity 𝑣𝑣𝑣𝑣, in litres per second, of airflow during a respiratory cycle is given by 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎 sin �
2𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡� 5
where 𝑎𝑎𝑎𝑎 ∈ ℝ and t is the time in seconds and the angle is measured in radians.
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The graph of this function is shown below.
(i)
Explain why 𝑎𝑎𝑎𝑎 = 0 · 8.
22 22
Use the function to find the time taken for one respiratory cycle (the period) and hence write the coordinates of the points 𝑄𝑄𝑄𝑄 and 𝑅𝑅𝑅𝑅.
Period =
𝑅𝑅𝑅𝑅 =
Describe what is happening in a person’s respiratory cycle at the point 𝑃𝑃𝑃𝑃 on the graph.
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(iii)
𝑄𝑄𝑄𝑄 =
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(ii)
(iv)
Why do you think, in this case, the velocity of airflow for the respiratory system is best described by a sine function and not as a cosine function?
23 23
(b)
Breathing techniques can be a useful tool in reducing stress and anxiety. By breathing more slowly and more deeply, the body’s nervous system receives signals to calm down.
(i)
One suggested breathing technique is to slowly inhale for 6 seconds and then slowly exhale for 6 seconds. The velocity of air flow for the respiratory system when practising this 6 – 6 breathing technique is
The 4-7-8 breathing technique aims to reduce anxiety or help people get to sleep. It involves breathing in for 4 seconds, holding the breath for 7 seconds and exhaling for 8 seconds.
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(ii)
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Find the value of 𝑏𝑏𝑏𝑏.
𝑣𝑣𝑣𝑣 = 0 · 8 sin(𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡), where 𝑏𝑏𝑏𝑏 ∈ ℝ.
Use this information to sketch a graph of the respiratory cycle for a person practising the 4-7-8 breathing technique.
24 24
(c)
Brian is an athlete, and when he is racing the velocity of airflow for his respiratory system can be modelled by: 𝑣𝑣𝑣𝑣 = 1 · 2 sin �
4𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡� 3
where t is the time in seconds and the angle is in radians.
Find the acceleration of Brian’s airflow, in terms of 𝑡𝑡𝑡𝑡.
(ii)
Hence, determine if the velocity of Brian’s airflow is increasing or decreasing at 𝑡𝑡𝑡𝑡 = 12 seconds.
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(i)
25 25
Question 10 (a)
(50 marks)
A package dropped from an aircraft moves with velocity: 𝑡𝑡𝑡𝑡
𝑣𝑣𝑣𝑣(𝑡𝑡𝑡𝑡) = 75 �1 − 𝑒𝑒𝑒𝑒 − 10 �
where 𝑡𝑡𝑡𝑡 is the time in seconds from when the package was released.
Calculate the velocity, to 1 decimal place, of the package after 1 second.
(ii)
Calculate, to 2 decimal places, the time taken for the package to reach a velocity of 15 m/s.
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(i)
(iii)
Find an expression for the acceleration of the package, in terms of t.
(iv)
Show mathematically that, as time goes on, the package will stop accelerating.
26 26
The velocity of the package during the first 14 seconds of its motion is shown in the graph.
(i)
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(b)
𝑡𝑡𝑡𝑡
Use the function 𝑣𝑣𝑣𝑣(𝑡𝑡𝑡𝑡) = 75 �1 − 𝑒𝑒𝑒𝑒 − 10 � to complete the table below. Give each value to 1 decimal place. 𝒕𝒕𝒕𝒕 (sec)
0
2
13·6
4
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𝒗𝒗𝒗𝒗 (m/s)
0
(ii)
6
33·8
8
10
12
14
Given that the distance travelled by the package is equal to the area under the curve, use the trapezoidal rule to find an estimate for the distance travelled during the first 14 seconds of the package’s motion.
27 27
Using integration, find the exact distance travelled by the package between 𝑡𝑡𝑡𝑡 = 0 and 𝑡𝑡𝑡𝑡 = 14. Give your answer to 1 decimal place.
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(iii)
28 28
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Page for extra work Label any extra work clearly with the question number and part.
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Page for extra work Label any extra work clearly with the question number and part.
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