L.20
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Pre-Leaving Certificate Examination, 2016
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Mathematics
Name/vers
Complete
Paper 2
Higher Level
Time: 2 hours, 30 minutes
300 marks
For examiner
Question
Mark
1
2
School stamp
3
4
5
6
7
8
9
Running total
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Total
Page 1 of 19
Grade
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 will lose marks if all necessary work is not clearly shown.
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:
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Mathematics
Paper 2 – Higher Level
Section A
Concepts and Skills
150 marks
Answer all six questions from this section.
Question 1
(25 marks)
D
The points A, B, C, and D are the vertices of a square, as shown.
The equation of AC is 3x − 7y + 2 = 0 and the co-ordinates of B
are (2, −3).
(a)
(i)
Find the slope of AC.
C
A
B (2, –3)
(ii)
Hence, find the equation of BD.
(b)
Find the co-ordinates of the point of intersection of the two diagonals, [ AC ] and [ BD ],
and hence, find the co-ordinates of D.
(c)
Hence, or otherwise, find the area of the square ABCD.
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Mathematics
Paper 2 – Higher Level
Question 2
(a)
(25 marks)
The function f : x ‫ →׀‬−1 − 3cos 6x is defined for x ∈ ℝ.
Write down the period and range of f and hence, sketch the graph of f in the domain 0 ≤ x ≤ π.
Period =
Range =
y
x
p
(b)
A triangle has sides of length a, b and c. The angle opposite the side of length a is A.
(i)
Prove that a2 = b2 + c2 − 2bc cos A.
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Mathematics
Paper 2 – Higher Level
(ii)
In the case that angle A is obtuse, show that a2 > b2 + c2.
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Mathematics
Paper 2 – Higher Level
Question 3
(25 marks)
(a)
Write down the centre of the circle c: x 2 + y 2 − 10x − 4y + k = 0 and find the radius-length
in terms of k.
(b)
The midpoint of a chord of c is (4, 1) and its length is 4 2 .
(c)
(i)
Find the radius-length of c.
(ii)
Hence, find the value of k.
c
Find the slopes of the tangents to c from the point (−2, 1).
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Mathematics
Paper 2 – Higher Level
Question 4
(25 marks)
A soft toys manufacturer continuously carries out quality control testing to uncover
defects in its products. After testing a large sample of its products, the manufacturer
found the probability of a soft toy having poor stitching is 0·03 and that a soft toy
with poor stitching has a probability of 0·7 of splitting open. A soft toy without poor
stitching has a probability 0·02 of splitting open.
(a)
(b)
(i)
Represent this information on a tree diagram, showing clearly the probability associated
with each branch.
(ii)
Use your tree diagram, or otherwise, to find the probability that a soft toy, chosen
at random, has exactly one of these defects.
The manufacturer also found that the colours used in the soft toys can fade. The probability
of this defect occurring is 0·05 and it is independent of poor stitching or splitting open.
A soft toy is chosen at random.
(i)
Find the probability that the soft toy has none of these three defects.
(ii)
Find the probability that the soft toy has exactly one of these three defects.
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Mathematics
Paper 2 – Higher Level
Question 5
(25 marks)
Two events A and B are such that P(A) = 0·33, P(B) = 0·45 and P(A ∩ B) = 0·13.
(a)
(i)
Find P(A ∪ B), the probability that either A or B occurs.
(ii)
Find the conditional probability P(A′| B′).
(iii) State whether events A and B are independent and justify your answer.
(b)
Event C is such that P(C) = 0·2.
Events A and C are mutually exclusive, while events B and C are independent.
(i)
Represent the probability of each event on a Venn diagram.
(ii)
Hence, find P((B ∪ C)′), the probability that neither B nor C occurs.
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Mathematics
Paper 2 – Higher Level
Question 6
(a)
(25 marks)
ABC is a triangle such that | AB | = 8 cm, | AC | = 6 cm and | BC | = 12 cm.
(i)
Given the line segment [ AB ] below, construct the triangle ABC.
(ii)
On the same diagram, construct the circumcentre and the circumcircle of the triangle ABC,
using only a compass and a straight edge. Show all construction lines clearly.
B
A
(iii) Under what condition(s) does the circumcentre of a triangle lie inside the triangle?
Justify your answers.
Condition(s):
Justification:
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Mathematics
Paper 2 – Higher Level
(b)
In the diagram, PQ is parallel to RS.
PS and QR intersect at O, which lies on the circle.
| PO | = 3 cm, | OS | = 5 cm and | ∠SOR | = 90°.
Let | RS | = x.
Q
S
5
O
x
3
P
R
(i)
Prove that | PO |.| RS | = | OS |.| PQ |.
(ii)
Hence, find the area of the circle, in terms of x.
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Mathematics
Paper 2 – Higher Level
Section B
Contexts and Applications
150 marks
Answer all three questions from this section.
Question 7
(60 marks)
Intelligence Quotient (IQ) is a score derived from standardised tests designed to assess intelligence.
(a)
The following table shows the countries with the highest recorded mean IQ scores.
Countries with the Highest Recorded Mean IQ Scores
Rank
Country
IQ
Rank
Country
IQ
Rank
Country
IQ
1
Hong Kong
107
10
Sweden
101
19
Australia
98
2
South Korea
106
10
Switzerland
101
19
Denmark
98
3
Japan
105
12
Belgium
100
19
France
98
4
Taiwan
104
12
China
100
19
Mongolia
98
5
Singapore
103
12
New Zealand
100
19
Norway
98
6
Austria
102
12
United Kingdom
100
19
United States
98
6
Germany
102
16
Hungary
99
25
Canada
97
6
Italy
102
16
Poland
99
25
Czech Republic
97
6
Netherlands
102
16
Spain
99
25
Finland
97
(i)
Using a calculator, or otherwise, calculate the mean and standard deviation for the
above set of data, correct to one decimal place.
(ii)
Explain why standard deviation is the most appropriate measure of variability by which
to analyse the above data.
(iii) Suggest one limitation of using standard deviation in this case. Give a reason
for your answer.
Limitation:
Reason:
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Mathematics
Paper 2 – Higher Level
(b)
The mean IQ score of people who have been tested in Ireland is 96 and the standard deviation
is 15. Assume that IQ scores are normally distributed and the number of people who have
been tested is so large that it can be assumed to represent a population.
(i)
Find the probability that an IQ score chosen at random from this population is greater
than 100.
Random samples of size 50 are repeatedly selected from this population and the mean of each
sample is calculated and recorded. 500 such sample means are recorded.
(ii)
Describe the expected distribution of all possible sample means from this population.
You should refer to the shape of the distribution and to its mean and standard deviation.
(iii) Find the number of sample means you would expect to be greater than 100.
The sample mean is used to estimate the population mean.
(iv) Find a 95% confidence interval for the population mean and interpret this interval
in the context of the question.
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Mathematics
Paper 2 – Higher Level
(c)
The principal in a large school in Ireland claims that the students in her school are above
average intelligence. A sample of 50 students from the school are chosen at random and
their IQ scores are determined. The mean IQ score of the sample is calculated to be 102.
(i)
Conduct a hypothesis test at the 5% level of significance to decide whether there is
sufficient evidence to conclude that the principal’s claim is legitimate. Write the null
hypothesis and the alternative hypothesis and state your conclusion clearly.
(ii)
Find the p-value of the test you performed in part (ii) above and explain what this value
represents in the context of the question.
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Mathematics
Paper 2 – Higher Level
Question 8
(a)
(40 marks)
The surface of the oceans are curved. Although we tend to think
about water forming large flat sheets, the surface of a large body
of water is not actually flat at all – it follows the curvature of
the Earth.
The diagram below represents part of a circular cross-section of the Earth. From point A on
the top of a mountain, 4⋅83 km above sea level, an observer measures the angle of depression
to the ocean horizon, H, as 2⋅23°.
A
2⋅23°
4⋅83 km
H
r
O
(i)
Find | ∠AHO | and give a reason for your answer.
(ii)
Hence, use a suitable trigonometric ratio to show that, correct to the nearest kilometre,
the radius of the Earth is 6373 km.
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Mathematics
Paper 2 – Higher Level
(b)
S
In astronomy, the same concept can be used to determine the
distance between the Earth and objects in space.
Let O be the centre of the Earth, E be a point on the equator and
S represent the nearest point of the object in space, as shown
(diagram not to scale).
Given the Earth is positioned in such a way that | ∠OES | = 90°,
then  = | ∠OSE | is called the equatorial parallax of the object
in space. In the case of the sun,  has been observed as 0⋅00244°.

E
O
Using the result from part (a) (ii) above, or otherwise, find, correct to the nearest kilometre,
the minimum distance between the surface of the Earth and the surface of the sun.
(c)
An observer on the equator measures the angle from one visible edge
of the sun to the other opposite visible edge of the sun, as shown
(diagram not to scale).
| ∠PEQ | = 32′ 4″.
E
P
C
32′ 4″
Q
(i)
Prove that | ∠PEC | = | ∠QEC |.
(ii)
Hence, using your answer to part (b) above, find the radius of the sun, correct to the
nearest kilometre.
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Mathematics
Paper 2 – Higher Level
Question 9
(a)
(b)
(50 marks)
The area of an equilateral triangle is 9 3 cm2.
(i)
Find the length of each side of the triangle.
(ii)
Find the area of the largest circle that fits in the triangle, in terms of π.
A circle, centre O, has a radius of 90 cm, as shown.
[ OM ] intersects the chord [ AB ] at M.
| OM | = 30 cm.
(i)
Find | AB |, in surd form.
(ii)
Find | ∠AOB |, correct to two decimal places.
O
A
M
B
(iii) Hence, find the area of the shaded region, correct to the nearest cm2.
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Mathematics
Paper 2 – Higher Level
(c)
The diagram shows a horizontal cylindrical oil tank of length 2⋅2 m
and radius 0⋅9 m.
(i)
Find, in litres, the capacity (volume) of the oil tank.
Give your answer correct to two significant figures.
‘Dipping’ is a method by which the level of the oil remaining in the tank can be checked.
A graduated stick is used to measure the level of oil from the bottom of the tank. It was found
that the depth of oil in the tank was 60 cm.
(ii)
Find, in litres, the volume of oil in the tank.
After a fill of oil is delivered, the tank was dipped again and the depth of oil
in the tank was found to be 160 cm.
(iii) Find, in litres, the volume of oil delivered, correct to the nearest whole number.
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Mathematics
Paper 2 – Higher Level
You may use this page for extra work.
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Mathematics
Paper 2 – Higher Level
You may use this page for extra work.
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Mathematics
Paper 2 – Higher Level
Pre-Leaving Certificate, 2016 – Higher Level
Mathematics – Paper 2
Time: 2 hours, 30 minutes
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Mathematics
Paper 2 – Higher Level