L.20
2020-L020-1-EL-01/28
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Pre-Leaving Certificate Examination, 2020
Mathematics
Paper 2
Higher Level
Time: 2 hours, 30 minutes
300 marks
For examiner
Question
Mark
1
2
CANDIDATE DETAILS
EXAM ID
3
Optional 4 or 5-digit number
(only if provided by your school)
4
5
NAME
6
7
SCHOOL
8
9
TEACHER
Total
ce3d33ba-d261-49d6-b596-ff99c66c321c
2020-L020-1-EL-01/28
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Do not write on this page
Pre-Leaving Certificate Examination, 2020
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-02/28
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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
3 questions
Answer all nine questions.
Write your Exam ID, Name, School’s Name and Teacher’s Name in the grid on the front cover.
Write your answers in blue or black pen. You may use pencil in graphs and diagrams only.
This examination booklet will be scanned and your work will be presented to an examiner on screen.
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.
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:
Pre-Leaving Certificate Examination, 2020
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-03/28
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Section A
Concepts and Skills
150 marks
Answer all six questions from this section.
Question 1
(25 marks)
The points A(−2, −3), B(4, 9) and C(−4, 3) are shown in the diagram below.
y
B
C
x
A
(a)
Find the equation of the line through the midpoint of AB which is perpendicular to AB.
Pre-Leaving Certificate Examination, 2020
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-04/28
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(b)
Use your answer to part (a) to find the co-ordinates of the circumcentre of the triangle ABC.
Pre-Leaving Certificate Examination, 2020
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-05/28
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Question 2
(25 marks)
The circle has centre C(7, −8) and passes through the point P(2, −2).
(a)
(i)
Find the equation of circle .
(ii)
Q is the point on the circle that is closest to the -axis.
Find, in surd form, the co-ordinates of Q.
Pre-Leaving Certificate Examination, 2020
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-06/28
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(b)
The point R is also on the circle . The length of the chord PR is 10 units.
The diagram shows R1 and R2 , the two possible positions of R.
Find the possible equations of PR.
y
x
R2
P
C
R1
s
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-07/28
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Question 3
(a)
(25 marks)
The probability that a certain rugby player scores from each place kick he attempts is 85%.
During a particular match, he takes five place kicks.
Find, correct to four decimal places, the probability that:
(i)
He scores on exactly three of the five attempts;
(ii)
He scores for the third time on his fifth attempt;
(iii)
He scores on at least three attempts during the match.
Pre-Leaving Certificate Examination, 2020
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-08/28
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(b)
A, B and C are three events. A and B are independent.
1
1
1
5
P(A) = , P(A ∩ B) =
, P(C) = and P(B ∪ C) = .
3
12
2
8
Find P(B ∩ C) and investigate whether events B and C are mutually exclusive.
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-09/28
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Question 4
(a)
(25 marks)
Let sin A =
(i)
10
, where 0 < A <
π
.
4
Find sin 2A and cos 2A in the form
sin 2A =
(ii)
1
p
, where , ∈ ℕ.
q
cos 2A =
By expressing sin 3A in the form sin (2A + A), find the exact value of sin 3A.
a b
Give your answer in the form
, where , , ∈ ℕ.
c
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-10/28
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(b)
Express 2cos2 + 3sin − 3 = 0 as a quadratic equation in sin
and hence find all the values of , where 0 ≤ ≤ 2π and is in radians.
Pre-Leaving Certificate Examination, 2020
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-11/28
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Question 5
(a)
(25 marks)
A bank issues a unique four-digit PIN code to customers to use with their debit or credit cards.
The code is chosen at random from the digits 0 to 9, inclusive. A code cannot begin with zero
but digits may be repeated. For example, 1
is a valid code.
9
9
5
Find the number of possible four-digit PIN codes in which no digit is repeated as a percentage
of the total number of possible codes. Give your answer correct to one decimal place.
(b)
A PIN code in which no digit is repeated is issued to a customer.
(i)
How many different PIN codes which are even numbers greater than 3000 are possible?
(ii)
Find the probability that all of the digits in the PIN code issued are in ascending order,
e.g. 3469 or 2789.
Pre-Leaving Certificate Examination, 2020
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-12/28
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(c)
Six students compare the months in which they celebrate their birthdays.
Assuming that all months are equally likely, find the probability that no two students were born
in the same month. Give your answer correct to four decimal places.
Pre-Leaving Certificate Examination, 2020
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-13/28
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Question 6
(a)
(25 marks)
The lengths of the sides of the triangle ABC are 9 units, 12 units and 18 units, as shown
in the diagram. Each side is divided into three segments of equal length.
C
12
A
9
18
B
(i)
Find the perimeter of the shaded region in the diagram above.
(ii)
If the area of the triangle ABC is 48 square units, find the area of the shaded region.
Pre-Leaving Certificate Examination, 2020
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-14/28
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(b)
In the diagram, [ CD ] is parallel to [ AB ] and [ AC ] is perpendicular to [ AB ].
[ AD ] and [ BC ] intersect at the point O.
| AB | = 11 units, | CD | = 9 units and | AC | = 12 units.
C
12
A
9
D
O
11
B
(i)
Prove that the triangles ABO and CDO are similar.
(ii)
Find | AD | and hence find | OD |.
| AD | =
| OD | =
Pre-Leaving Certificate Examination, 2020
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-15/28
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Section B
Contexts and Applications
150 marks
Answer all three questions from this section.
Question 7
(a)
(50 marks)
An inverted right circular cone with its axis vertical is filled with water to a depth of 15 cm
above its vertex, as shown. The semi-vertical angle of the cone is 30°.
r
15
15 cm
cm
30
30°
(i)
Find , the radius of the circular surface of the water in the cone.
Give your answer in the form
b , where , b ∈ ℕ.
(ii)
Hence find the volume of water in the cone, in terms of π.
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-16/28
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(b)
A solid sphere of radius is placed in the cone.
The water rises so as to just cover the sphere,
which touches the sides of the cone, as shown.
R
d cm
30
30°
Find , the depth of the water, and R, the radius of the circular surface of the water,
in terms of .
(i)
=
(ii)
R=
Hence find , the radius of the sphere, correct to two decimal places.
This question continues on the next page.
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-17/28
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(c)
A buoy at sea consists of a cone mounted on a heavy cylindrical base which floats with
the cone uppermost. The buoy has an overall height of 6 m, and the cone and the cylinder
have equal volumes and equal radii.
m
66 m
(i)
Find the vertical height of the cone.
(ii)
The diameter of the cone and cylinder is 2⋅5 metres.
Find the total volume of the buoy, in terms of π.
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-18/28
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(iii)
The buoy floats with its axis vertical and two-thirds of its volume submerged below
the waterline.
Find the height of the vertex of the cone above the waterline.
Give your answer correct to two decimal places.
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-19/28
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Question 8
(a)
(50 marks)
Farmed salmon are harvested when they grow to a certain length.
The lengths of the salmon produced in a particular off-shore fish farm are normally distributed
with a mean of 32⋅8 cm and a standard deviation of 2⋅4 cm.
(i)
Find the proportion of salmon which are more than 35 cm in length.
(ii)
On further analysis, it was determined that 60% of the salmon produced in the fish farm
have lengths of between 31⋅8 cm and cm.
Find the value of .
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-20/28
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(b)
The owners of the fish farm have introduced new practices to produce salmon in larger,
less densely populated cages which allow the fish to follow their natural shoaling behaviour.
In a random sample of 250 salmon produced in this way, it was found that their lengths
were normally distributed with a mean of 33⋅2 cm and the same standard deviation.
(i)
Test the hypothesis, at the 5% level of significance, that the mean length of the salmon
produced has not changed. State the null hypothesis and your alternative hypothesis.
Give your conclusion in the context of the question.
(ii)
Find the -value of the test you performed in part (b)(i) and explain what this value
represents in the context of the question.
-value:
Explanation:
This question continues on the next page.
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-21/28
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(c)
Farmed trout are produced in freshwater fish farms. In a particular fish farm, the lengths
of the trout produced are normally distributed with 97⋅5% of them having lengths of less
than 34⋅2 cm and 67% of them having lengths of greater than 26⋅6 cm.
Find the mean and standard deviation of the lengths of the trout produced in this fish farm.
Give your answers correct to two decimal places.
Mean =
Standard deviation =
Pre-Leaving Certificate Examination, 2020
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-22/28
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Question 9
(a)
(50 marks)
A television mast is held in a vertical position
by two metal supports, [ AE ] and [ CD ], and
a wire cable [ AF ], as shown.
| AD | = 2⋅4 m, | AF | = 3⋅7 m, | DF | = 1⋅8 m,
| CE | = 2⋅9 m and | ∠BCE | = 48°.
F
3.7 m
E
1 .8 m
D
2.4 m
2.9 m
48
A
B
(i)
Find | BE |, correct to two decimal places.
(ii)
Find | ∠ADF |, correct to the nearest degree.
(iii)
Hence find | EF |, correct to two decimal places.
C
This question continues on the next page.
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-23/28
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(b)
The summer season in a certain holiday resort runs from 15 April to 30 September, inclusive.
The number of visitors to the resort (in thousands), n( ), can be approximated by the function:
 π 
n( ) = 4⋅8 − 2⋅6 cos  t  ,
 84 
 π 
where is the number of days after 15 April and  t  is expressed in radians.
 84 
(i)
Find the number of visitors to the resort on 13 May (28 days after 15 April).
(ii)
Find the largest number of visitors to the resort and the date on which this occurs.
(iii)
Find the period and the range of n( ).
Hence draw a rough sketch of n( ) on the axes below.
n (t )
t
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-24/28
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(iv)
Find the two dates on which the number of visitors to the resort is approximately 3851.
(v)
Find the rate at which the number of visitors to the holiday resort is changing
on 19 August (126 days after 15 April).
Explain your answer in the context of the question.
Rate:
Explanation:
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-25/28
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You may use this page for extra work.
Label any extra work clearly with the question number and part.
Q. & part
Pre-Leaving Certificate Examination, 2020
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-26/28
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You may use this page for extra work.
Label any extra work clearly with the question number and part.
Q. & part
Pre-Leaving Certificate Examination, 2020
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Mathematics, Paper 2 – Higher Level
2020-L020-1-EL-27/28
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You may use this page for extra work.
Label any extra work clearly with the question number and part.
Q. & part
Pre-Leaving Certificate Examination, 2020 – Higher Level
Mathematics – Paper 2
Time: 2 hours, 30 minutes
ce3d33ba-d261-49d6-b596-ff99c66c321c
2020-L020-1-EL-28/28
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