*W8* Pre-Junior Certificate Examination, 2017 Triailscrúdú an Teastais Shóisearaigh, 2017 Mathematics Paper 2 Higher Level 2½ hours 300 marks Name: School: Address: Class: Teacher: For examiner Question Mark 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Total Grade Running total Page 1 of 24
Instructions There are 14 questions on this examination paper. Answer all questions. Questions do not necessarily carry equal marks. To help you manage your time during this examination, a maximum time for each question is suggested. If you remain within these times you should have about 10 minutes left to review your work. 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 you do not show all necessary work. You may lose marks if you do not include the appropriate units of measurement, where relevant. You may lose marks if you do not give your answers in simplest form, where relevant. Write the make and model of your calculator(s) here: Page 2 of 24
Question 1 (Suggested maximum time: 7 minutes) (a) A circular helipad (a helicopter landing area) has a radius of 11 m. Find, correct to the nearest m 2, the area of the helipad. (b) Find, correct to the nearest m 2, the shaded area of the helipad in the diagram below. Page 3 of 24
Question 2 (Suggested maximum time: 7 minutes) A class of second year students carried out a survey of cars that collected students from school on a certain day. They noted the colour of each car and recorded their results using a tally count. (a) Complete the frequency table below. Colour Tally Frequency Red Silver Black Other (b) How many cars collected students from school that day? (c) What is the probability that a student was collected in a silver car? (d) The students repeated the survey over a number of weeks. If they recorded a total of 1500 cars, how many should they expect to be red? Page 4 of 24
Question 3 (Suggested maximum time: 8 minutes) The diagram below shows the side view of a set of concrete steps. Each step is square and the steps are all of equal size. (a) Calculate the value of x if the overall height of the steps is 75 cm. (b) Calculate the total volume of the set of concrete steps if the steps are 80 cm wide. (c) Calculate the mass of the steps, in kilograms, if the mass of 1 cm 3 of concrete is 2.3 grams. Page 5 of 24
Question 4 (Suggested maximum time: 18 minutes) (a) The Student Council is writing a questionnaire for the students in your school. You have been asked to write questions that will generate four different categories of data: numerical discrete, numerical continious, categorical nominal and categorical ordinal. Fill in a question for each category of data. The questions can relate to any aspect of a student s school life. (i) Numerical discrete Question: (ii) Numerical continuous Question: (iii) Categorical nominal Question: (iv) Categorical ordinal Question: Page 6 of 24
(b) In the space below, give one example of a question that would not be acceptable in this questionnaire. Give a reason why the question is not suitable. Question: Reason: (c) The Student Council want the sample to be a simple random sample of 60 students from the total school population of 600 students. Explain what a simple random sample is and outline how the Student Council should find a simple random sample of 60 students from the school population of 600 students. Simple Random Sample: Method: Page 7 of 24
Question 5 (Suggested maximum time: 10 minutes) A grain silo is made up of a cylinder surmounting a cone. The cylinder has height 14 m and radius 3 m. The cone s height h is twice its radius. (a) Find the volume of the cylinder in terms of π. (b) Find the height of the cone and hence find the volume of the full grain silo in terms of π. (c) One quarter of the volume of the full grain silo is emptied from the full silo. Calculate the new height of the grain in the silo. Page 8 of 24
Question 6 (Suggested maximum time: 5 minutes) Draw a line parallel to the line l through the point A. l Page 9 of 24
Question 7 (Suggested maximum time: 16 minutes) (a) 120 gamers were asked if the had downloaded the new Pokemon Go gaming app. The two-way table shows some information about these students. App Downloaded App Not Downloaded Total Female 27 40 Male 17 Total 90 120 (i) (ii) Complete the two-way table. What percentage of the gamers were female and had downloaded the app? If I pick a gamer at random, what is the probability that I pick: (iii) a male gamer? (iv) a gamer who has downloaded the app? (v) If a gamer is picked who has downloaded the app, what is the probability that the gamer is female? Page 10 of 24
(b) The 80 gamers who had downloaded the app kept a record of the number of kilometres they walked in a month when they were playing the game. The results are in the table below: Kilometres 0 4 4 8 8 12 12 20 Frequency 20 16 35 9 (i) Using mid-interval values calculate the mean number of kilometres walked in a month by each gamer. (ii) In which interval will you find the median? Explain your choice. Page 11 of 24
Question 8 (Suggested maximum time: 5 minutes) If k 1, k 2 and k 3 are parallel lines, find the measure of the angles p, q and r. Page 12 of 24
Question 9 (Suggested maximum time: 10 minutes) The table opposite gives the equations of six lines. (a) Which lines are perpendicular? Give a reason to support your answer. Answer: Reason: Line 1 y = 4 x + 7 Line 2 y = 3x 5 Line 3 3 x + y = 4 Line 4 y = 2 x + 3 Line 5 x + 2 y = 4 Line 6 y = 3x 2 (b) Which line has the greatest slope? Answer: (c) Which line has the highest y-intercept? Write down the co-ordinates of the y-intercept. Answer: (d) The diagram below represents one of the given lines. Which line does it represent? Give a reason to support your answer. Answer = Line Page 13 of 24
(e) Draw a sketch of Line 6 on the axes shown. (f) The table shows some values of x and y for the equation of one of the lines. Which equation do they satisfy? Justify your choice. x y 1 2 3 4 5 10 Line : Reason: Page 14 of 24
Question 10 (Suggested maximum time: 8 minutes) (a) Prove that the three angles in a triangle add up to 180. Diagram: Given: To Prove: Construction: Proof: Page 15 of 24
(b) The measurements of the angles of a certain triangle are consecutive even integers. Find their measurements. (c) Prove that p + q + r = 360. Page 16 of 24
Question 11 (Suggested maximum time: 16 minutes) (a) Plot the following points on the diagram below. A B C ( 5, 4) ( 2, 3) ( 7, 0) (b) By using the slope formula show that the points A, B and C are collinear. Page 17 of 24
(c) Find AB and BC and verify that 3 AB = BC. (d) Find the image of A ( 5, 4) under a central symmetry in the point ( 2, 3) B. (e) Find the midpoint between A ( 5, 4) and ( 7, 0) C. (f) A student correctly found that the answers to (d) and (e) were the same. Explain how the student, using this fact, was able to answer (c) without using the distance formula. Page 18 of 24
Question 12 (Suggested maximum time: 8 minutes) (a) Construct a right angled triangle containing an angle α, such that cosα = 0 5. (b) Find, from your triangle, tan α in surd form. Page 19 of 24
Question 13 (Suggested maximum time: 12 minutes) As part of their job, taxi drivers record the number of kilometres they travel each day. A random sample of the mileages recorded by taxi drivers Brendan and Asif are summarised in the back-toback stem and leaf diagram below. Totals Brendan Asif Totals (9) (11) (6) (6) (4) (2) (2) (1) (2) 8 7 7 4 3 2 1 1 0 9 9 8 7 6 5 4 3 3 1 1 8 7 4 2 2 0 9 4 3 1 0 0 6 4 1 1 2 1 7 1 9 9 3 43 42 Total days Key: 0 18 4 means 180 km for Brendan and 184 km for Asif. Total days 18 19 20 21 22 23 24 25 26 4 4 5 7 5 7 8 9 9 0 2 2 4 4 8 2 3 5 6 6 7 9 1 1 2 4 5 5 8 1 1 3 4 6 6 7 8 2 4 8 9 4 Some of the quartiles for these two distributions are summarised in the table below. (4) (5) (6) (7) (7) (8) (4) (1) (0) Brendan Asif Lower Quartile Q1 191 Median Q2 218 Upper Quartile Q3 221 (a) (b) Complete the above table. Asif and Brendan are having a friendly argument about which of them is busier. Asif claims that he is the busiest. Using the statistics above or any other statistics, can you find evidence to support or contradict Asif s claim? Page 20 of 24
(c) Outliers are values that lie outside the limits: Q 1. ( Q Q ) and Q + 1. ( Q Q ) where: Q1 = the lower quartile Q2 = the median quartile and Q3 = the upper quartile 1 5 3 1 3 5 Calculate these limits and check if Brendan s lowest mileage of 180 kilometres and his highest mileage of 269 kilometres are outliers? 3 1 Page 21 of 24
Question 14 (Suggested maximum time: 10 minutes) Two identical buildings P and Q, each of height h, are standing on opposite sides of a level road. They are 40 m apart. The point A, on the road directly between the two buildings, is a distance x from building Q. The angle of elevation from A to the top of building Q is 60. (a) Show that h = x tan 60. (b) From A the angle of elevation to the top of building P is 30. Write down a second expression for h in terms of x. (c) By using your answers in (a) and (b) above, find the value of h. Page 22 of 24
You may use this page for extra work. Page 23 of 24
You may use this page for extra work. Page 24 of 24