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2016.1 L.20 1/20 of 19

L.20

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Pre-Leaving Certificate Examination, 2016

Mathematics

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

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Pre-Leaving Certificate, 2016 2016.1 L.20 2/20

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MathematicsPaper 2 – Higher Level

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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Section A Concepts and Skills 150 marks

Answer all six questions from this section.

Question 1 (25 marks)

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.

(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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D

B (2, –3)

C

A


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(a) The function f : x 3 − 1− →׀cos 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.

(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.

Period = Range =

�

x

y


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(ii) In the case that angle A is obtuse, show that a2 > b2 + c2.

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(a) Write down the centre of the circle c: x2 + y2 − 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 24 .

(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).

c


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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) (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.

(b) 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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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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(a) 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.

BA

(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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(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.

(i) Prove that | PO |.| RS | = | OS |.| PQ |.

(ii) Hence, find the area of the circle, in terms of x.

S

x

R

Q

P

O

3

5


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Section B Contexts and Applications 150 marks

Answer all three questions from this section.


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: page running


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(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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(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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(a) 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°.

(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.

r

2⋅23°

O

H

A

4⋅83 km


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(b) 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°.

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″.

(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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E32′ 4″ C

P

Q

O

S

E


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(a) 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.

(b) 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.

(iii) Hence, find the area of the shaded region, correct to the nearest cm2.

O

A M B


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(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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You may use this page for extra work.


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You may use this page for extra work.

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Pre-Leaving Certificate, 2016 – Higher Level

Mathematics – Paper 2 Time: 2 hours, 30 minutes

Name/vers Updated: Mathematics Name/vers Complete

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