LeavingCertificateMockExams2020

Maths(HigherLevel)

Paper2

Monday9thMarch2020

9.30a.m.–12.00p.m.

Time:2½hrs

Name:________________________________________________

Instructions

Therearetwosectionsinthisexaminationpaper.

SectionAConceptsandSkills 150marks6questionsSectionBContextsandApplications 150marks3questions

Answerallninequestions.

Writeyouranswers inthespacesprovidedinthisbooklet.Youmaylosemarks ifyoudonotdoso.Thereisspaceforextraworkatthebackofthebooklet.Youmayalsoaskthesuperintendentformorepaper.Labelanyextraworkclearlywiththequestionnumberandpart.

ThesuperintendentwillgiveyouacopyoftheFormulaeandTablesbooklet.Youmustreturnitattheendoftheexamination.Youarenotallowedtobringyourowncopyintotheexamination.

You may lose marks if your solutions do not include supporting work.Youmaylosemarksiftheappropriateunitsofmeasurementarenotincluded,whererelevant.Youmaylosemarksifyouranswersarenotgiveninsimplestform,whererelevant.

Writethemakeandmodelofyourcalculator(s)here:

SECTIONA ConceptsandSkills 150marksQuestion1 25MarksTwocircles𝑐!and𝑐"aregivenbytheequations

𝑐!: 𝑥" + 𝑦" − 4𝑥 − 6𝑦 − 12 = 0𝑐": (𝑥 − 11)" + (𝑦 − 15)" = 100

(a) Prove that the two circles touch externally.

(b) Find the point of contact of the two circles.

(c) Find the equation of the common tangent of the two circles.

Question2 25Marks𝑙 is the line 2𝑥 − 3𝑦 + 11 = 0 and 𝑘 is the line 2𝑥 − 3𝑦 − 9 = 0 (a) (i) Find the perpendicular distance between the two lines?

(ii) Find the equation of the line which lies midway between ‘l’ and ‘k’ and parallel to both lines.

(b) Another line ‘𝑛’ has equation √3𝑥 − 𝑦 + 2 = 0. Find the equation of the line which passes through the point (1, −2) and makes an angle of 60# with ‘𝑛’.

Question3 25Marks The diagram to the right shows the spire in O’Connell street which stands vertically (121m) tall. 𝐴, 𝐵 and C are three points on horizontal ground with |∠𝐵𝐶𝐴| = 32#. Find the distance

𝑥 = |𝐴𝐵| to the nearest meter.

𝑇 Spire (121𝑚) 𝐴 60# C 𝑥 32# 𝐵

30#

Question4 25Marks Dr Murphy is a genetic engineer who studies the presence of a disease which can be present due to genetic or random reasons (i.e. it is possible to have the disease without having the gene). Statistically 45% of the people who carry the gene have the disease and 2% of people have the disease among those who do not have any genetic issue. If 3% of the entire population carry the gene, find (a) What is the probability that a random member of the population has the disease and

(b) What is the probability that a random sufferer from the disease carries the gene?

(c) Given that Ireland has a population of approximately 4,500,000, find the number of people who have the disease nation-wide.

Question5 25Marks A boat moves up and down on a wave in the ocean. The height of the boat above the horizon is given by the formula 𝑓(𝑥) = 𝑎 sin 𝑏𝑥 𝑎, 𝑏 ∈ ℝ 𝑥 in secs. The period of the waves are 8 secs and the maximum height of the wave above the horizon is 2.5 m (a) Find the values of ‘a’ and ‘b’

𝑌/𝑚 𝑦 = 𝑓(𝑥) x/s

(b) Find the first time (𝑥 > 0) for which the boat is 2𝑚 above the horizon?

(c) How quickly (the speed) is the boat rising at this time (to 2 decimal places).

horizon

Question6 25Marks A local GAA lottery consists of choosing 5 numbers from a possible 18. A person wins some money if they have three or more numbers correct. (a) Find the probability of having three numbers correct in a random draw?

(b) Find the probability of having four numbers correct?

It cost €2 per line to play the game and the prices are as follows

3 numbers 4 numbers 5 numbers Prize €4 €20 €1200

(c) In a random draw find the expected win or loss when a person plays one line in the lottery (Note that you do not get your money back, even if you win something)

SECTIONA Contexts&Applications 150marksQuestion7 55MarksThe average age of competitors in a city marathon is 41.5 years with standard deviation 11.5 years. Assuming that the age of a runner is normally distributed, find (a) The probability that a randomly selected runner is over 65 years old?

(b) Find the probability that a randomly selected run is younger than 20 years old.

(c) A group of 10 typical competitors from the same club are competing in the marathon. Find the probability that more than 2 of them are below the age of 20 years old.

The mean finishing time for women in the race is 4.5 hours with standard deviation 1.1 hours. A random sample of finishing times for 225 men yields a sample mean of 4.4 hours. (d) Do you feel, at a 5% level of significance, that the finishing time for men is different than for women?

(e) Find the p-value for the survey conducted in part (d)?

(f) A new type of running shoe is thought to produce faster finishing times than other types. The company claim that 35% of the people running in the marathon are using this type of running shoe. In a random sample of 100 people it is noted that 31 are wearing that type of footwear. Construct a 95% confidence interval for the portion of the population who are wearing that type of shoe and consider, at a 5% level of significance whether or not you feel that the company’s claim is correct.

Question8 45Marks (a) In the diagram to the right the area of the triangle is 15√3 sq. units The two sides shown are 5 units and 12 units respectively? (i) Find two possible values of the angle 𝐴?

A 5 12

(ii) Find the length of the third side of the triangle in each situation.

(b) Johnny Sexton takes a penalty kick in a very important game in the 6 nations. The position of the ball in relation to the goals is shown in the diagram to the right. The ball is 35m from the goal-line and 10m from the side-line. The width of the pitch is 70m. If the width of the goal is 𝑤 = 5.6𝑚 find the value of 𝛼 if he is to score (i.e. the ball goes between the posts)

w 70m 𝛼 35𝑚 10𝑚

Question9 50Marks The diagram to the right shows the results of a group of 20 people in a particular test and the age of each person. (a) Draw a scatter plot in the space provided below to show the data. Age (yrs)

50

40

30

20

10

10 20 30 40 50 60 70 80 90 100

Results (%) (b) Draw the “line of best fit” on the graph

Results(x) Age(y) 1 90 34 2 50 19 3 78 28 4 40 17 5 72 38 6 81 40 7 74 27 8 94 39 9 73 38

10 63 23 11 93 37 12 44 17 13 84 30 14 77 29 15 60 21 16 89 37 17 41 17 18 68 23 19 100 37 20 82 34

(c) Complete the following table of statistical measures

�̅� 𝑦w 𝜎$ 𝜎% CorrelationCoefficient

(d) Find the equation of the line of best fit?

(e) If the ages of “all” people taking the test is known to be normally distributed with mean 34 years and 7.2 years, do you feel that the group of people sampled above is representative of the population as a whole?

Additional Paper

Additional Paper

END OF TEST PAPER