Paper 2
Higher Level
Monday 13 June Morning 9:30 - 12:00
300 marks
Examination number
Centre stamp
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Grade Running total
For examiner Question Mark
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5
6
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8
9
Total
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2016. M30
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State Examinations Commission
Leaving Certificate Examination 2016
Mathematics
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Section . (onepts and Skills 150 marks
Answer all six questions from this section.
Question 1
The points A(6, —2), B(5, 3) and C(-3, 4) are shown on the diagram. (a) Find the equation of the line through e
B which is perpendicular to AC.
(25 marks)
................................. ........................................................,. (b) Use your answer to part (a) above to find the co-ordinates of tJe ortho centre of the
triangle ABC. JT
(b) C is the centre of a circles, of radius 5 cm. The line 1: 3x + 4y - 21 = 0 is a tangent to s and passes through X, as shown. Find the equation of one such circles.
Leaving Cerqficate 2016
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- Mathematics Paper 2- Higher Level
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Question 3 cos 7A+cosA
(a) Show that cot3A. sin 7A— sin A
(25 marks)
(b) Given that cos 20 = find cos 0 in the form ±where a, b E N.
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- Mathematics Paper 2—Higher Level
Ii "-1 A C
Question 4
(25 marks)
The diagram shows a semi-circle standing on a diameter [AC], and [BD] I [AC].
(a) (i) Prove that the triangles ABD and DBC are similar.
(ii) If JAB I = x, IBCI = 1, and IBDI =y, write in terms of x.
Leaving Certificate 2016
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Page 6 of 19 Mathematics Taper 2— Higher Level
(b) Use your result from part (a)(ii) to construct a line segment equal in length (in centimetres) to the square root of the length of the line segment [TU] which is drawn below.
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Mathematics Paper 2— Higher Level
Way 1 Wy 2 \Vav 3 \%av 4
/
X 'Z .1
Jo I i
Dai i d
=Hit
Y- = Miss
(ii) Hence or otherwise find the probability that they will win the competition.
lo ..................ji•....
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3 r
Question 5 (25 marks) (a) (i) In an archery competition, the team consisting of John, David, and Mike will win
1st prize if at least two of them hit the bullseye with their last arrows. Xrom past experience, they know that the probability that John, David, and Mike will hit the
bullseye on their last arrow is -, , and respectively. Complete the table below to show all the ways in which they could win 1st prize.
(b) Two events, A and B, are represented in the diagram. P(A nB) = 01, P(B\A) = 0-3 and P(A\B) =x. Write P(A) in terms of x and hence, or otherwise, find the value of for which the events A and B are independent.
.
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Question 6 (25 marks)
A local sports club is planning to run a weekly lotto. To win the Jackpot of €1000, contestants must match one letter chosen from the 26 letters in the alphabet and two numbers chosen, in the correct order, from the numbers 0 to 9. In this lotto, repetition of numbers is allowed (e.g. M, 3, 3 is an outcome). (a) Calculate the probability that M,.3, 3 would be the winning outcome in a particular week.
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(b) If a contestant matches the letter only, or the letter and one number (but not both numbers), they will win €50. Using the table below, or otherwise, find how much the club should expect to make or lose on each play, correct to the nearest cent, if they charge €2 per play.
Eerit
ill Ik i ck pol
Payout (x) C Probability (P(x)) x. P(x)
100 OCO o
Match 1eU.r aii J
firstumber onl\ Mthiètter•and se&ndnurnberoni\
-
L'OL&l 3J0 3 3e - OS
Fall io \\ a
- 4- )f& —
i.J....
(c) The club estimates that the average number of plays per week will be 845. If the club wants to make an average profit of €600 per week from the lotto, how much should the club charge per play, correct to the nearest cent?
Leaving Certificate 2016
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Mathematics Paper 2—Higher Level
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9
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: Ls(_Ic tion i Contexts .1110I 1)IICUiOfl 50 marks
Answer all three questions from this section.
Question 7 (55 marks)
A glass Roof Lantern in the shape of a pyramid has a rectangular base CDEF and its apex is at B as shown. The vertical height -of the pyramid is JAI, where A is the point of intersection of the diagonals of the base as shown in the diagram. Also ICDI = 25 m and lCF =3 m.
(a) (i) Show that IACI 1 - 95 m, correct to two decimal places. B
. ........................ .
..... . - cll~ xK)'
F(" "E
(ii) The angle of elevation of B from C is 500 (i.e. ZBCAj = 50 0), Show that IABI = 23 m, correct to one decimal place.
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Mathematics Paper 2—Higher Level
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(b) Another Roof Lantern, in the shape of a pyramid, has a square base CDEF. The vertical height IABI = 3 m, where F A is the point of intersection of the diagonals of the base as shown. The angle of elevation of B from C is 60 0 (i.e. JZBCAJ = 60°). Find the length of the side of the square base of the lantern. Give your answer in the form 'J7im, whei a E N.
E
C I
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Mathematics Paper 2 Higher Level
Question 8 (45 marks)
The height of the water in a port was measured over a period of time. The average height was found to be 1-6 m The height measured in metres, h(t), was modelled using the function
h(t) = 1•6 + i•5 cos (6 0 where t represents the number of hours since the last recorded high tide and ( t) is expressed in radians.
(a) Find the period and range of h(t).
(b) Find the maximum height of the water in the port.
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(c) Find the rate at which the height of the water is changing when t = 2, correct to two decimal places. Explain your answer in the context of the question.
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4,os' Explanation:
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Mathematics Paper 2— Higher Level
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(d) (i) On a particular day the high tide occurred at midnight (i.e. t = 0). Use the function to complete the table and show the height, h(t), of the water between midnight and the following midnight.
It) = 16 + 15 cos
Time \1idniht i ii ( iii 9 a. in. 12 noon . pn P p.m. 9 m \lidni1it
0 3 9 'S / 2.1 29
31 i' •1 i'i., 301 ('i, 1 i'd, 3iJ, fly
4z )OL-(ii) Sketch the graph of h(t) between midnight and the following midnight. j,cjt) V"
r i :,+ J%su ••4•
2'O
I)
O 1T/2
, ,f 0.
(D6
Time 3.I.
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Mathematics Paper 2 —Higher Level
(e) Find, from your sketch, the difference in water height between low tide and high tide.
A / e. , ,
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Mathematics Paper 2—Higher I evel
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Question 9 (50 marks)
Data on earnings were published for a particular country. The data showed that the annual income of people in full-time employment was normally distributed with a mean of €39 400 and a standard deviation of €12 920.
(a) (i) The government intends to impose a new tax on incomes over €60 000. Find the percentage of full-time workers who will be liable for this tax, correct to one decimal place.
(ii)
__I 1
eamers. The government will also provide a subsidy to the lwest 10 Oocome Find the level of income at which the government will stop paying the subsidy, correct ' S5 to the nearest euro.
(iii) Some time later a research institute surveyed a sample of 1000 full-time workers, randomly selected, and found that the mean annual income of the sample was €38 280. Test the hypothesis, at the 5 % level of significance, that the mean annual income of full-time workers has changed since the national data were published. State the null hypothesis and the alternative hypothesis. Give your conclusion in the context of the question.
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• Paper 2— Higher Level
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(d) The research institute also carried out a survey into the use of agricultural land. n farmers were surveyed. If the margin of error of the survey was 45 %, find the value of n.
*
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Mathematics Paper 2—Higher Level
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