2017 PAPER 1 SOLUTIONS. Junior Cert Higher Level

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2 2017 PAPER 1 SOLUTIONS Junior Cert Higher Level

3 2017 JCHL Paper 1 Question 1 (a) (i) 15 Marks A person s Body Mass Index (BMI) is given by the following formula: BMI = w h 2 where w is their weight in kg, and h is their height in metres. Geri is an athlete. Her weight is 77 5 kg and her height is 1 63 m. Work out Geri s BMI. Give your answer correct to one decimal place. BMI = w h 2 BMI = BMI = BMI 29.2 (ii) Geri loses some weight during her training. Her height stays the same. Her new BMI is Work out her new weight. Give your answer in kg, correct to one decimal place. BMI = w h 2 24 = w = w = w w 63.8 kg

4 2017 JCHL Paper 1 Question 1 (b) 10 Marks Alex and Jo have the same weight. Alex is 10 cm taller than Jo. Put a tick ( ) in the correct box to show if Alex s BMI is greater than Jo s or less than Jo s. Justify your answer fully. Alex s BMI is: greater than Jo s less than Jo s (tick one box only) Less than Jo s Justification: Alex is dividing the same top line as Jo by a bigger number. w h < w h 2

5 2017 JCHL Paper 1 Question 2 (a) (i) 15 Marks A sports shop buys t-shirts for 25 and sells them for 49. Find the mark up for the t-shirts (profit as a percentage of cost price). Profit = = 24 % Mark Up = Profit = Cost Price = 96% (ii) Find the margin for the t-shirts (profit as a percentage of selling price). Give your answer correct to the nearest percent. % Margin 24 Profit = = Selling Price = 48.9%

6 2017 JCHL Paper 1 Question 2 (b) 5 Marks The shop also sells runners, at a mark up of 50%. Find the margin for these runners. Give your answer correct to the nearest percent. Mark Up = 50% Profit = 50% Cost P = 0.5C Selling Price (S) C + P = 1.5C % Margin = Profit Selling Price 100 P S = 0.5C 1.5C = 1 3 = 33%

7 2017 JCHL Paper 1 Question 3 (a) 5 Marks Write 868 million in the form a 10 n, where n Z and 1 a < 10, a R. 868 million = 868,000,000 =

8 2017 JCHL Paper 1 Question 3 (b) 10 Marks for (b) and (c) During the Apollo-11 mission, it took approximately 1 3 seconds for a radio signal to travel km. Find the average speed of the radio signal, in km per minute. Give your answer in the form a 10 n, where n Z and where 1 a < 10 is correct to two decimal places. Speed = Distance Time = 380, = km/s = km/min = km/min

9 2017 JCHL Paper 1 Question 3 (c) 5 Marks In 2016, a spacecraft flew around Jupiter, 868 million km from earth. Find how many minutes it would take a radio signal to travel 868 million km. Assume that the radio signal would travel at the same speed as your answer to part (b). Speed = Distance Time = = 49.6 minutes The Juno Spacecraft cost over $1.1 Billion

10 2017 JCHL Paper 1 Question 4 (a) 10 Marks for (a) and (b) (i) Fruitex and Juicy are two drinks. A shop buys cartons of Fruitex from the UK. In December 2015, the exchange rate was 1 = The shop bought Fruitex for 380. Find the price of the Fruitex in euro ( ). Give your answer correct to the nearest cent. 1 = x = 380 Juicy x = 380 x = x

11 2017 JCHL Paper 1 Question 4 (b) (i) 10 Marks for (a) and (b) (i) Fruitex and Juicy are each made from mixing fruit juice and water. In Fruitex, the ratio of fruit juice to water is 3: 7. Find how many litres of fruit juice are in 20 litres of Fruitex. Fruit Juice: Water 3: 7 Juicy = 10 parts = 2 litres in 1 part Add the ratios together to find how many parts there are. Divide the total amount of Fruitex by the sum of the ratios. Fruit Juice: 3 parts 3 2 = 6 Multiply the amount given out per 1 part by the number of parts of the juice. There is 6 litres of Fruit Juice in Fruitex.

12 2017 JCHL Paper 1 Question 4 (b) (ii) 5 Marks 20 litres of Fruitex is mixed with 40 litres of Juicy. In this 60-litre mixture, the ratio of fruit juice to water is 7: 8. Find the ratio of fruit juice to water in Juicy. Give your answer in its simplest form. Divide the 60 litre mixture into its juice and water parts. Fruit Juice: Water 7: = 15 parts = 4 litres in 1 part Fruit Juice: 7 parts 7 4 = 28 There is 28 litres of fruit juice in the 60 litre mixture. There is 6 litres of fruit juice in the 20 litres of Fruitex which leaves 28 6 = 22 litres of fruit juice in Juicy = 18 litres of water in Juicy Ratio of Fruit Juice: Water in Juicy: = 22 : 18 = 11 : 9 Water: 3 parts 8 4 = 32

13 2017 JCHL Paper 1 Question 5 (a) (i) 15 Marks Pete and Maeve are saving to buy an Xbox. Pete has saved 20 to begin with. He saves a further 12 each week. Find the total amount of money Pete will have saved after 5 weeks = = 80 (a) (ii) Write an expression in n for the total amount of money Pete will have saved after n weeks. Pete s total savings after n weeks: 20 + n 12 = n (b) Maeve has saved 15 to begin with. She saves a further 6 each week. Write an expression in n for the total amount of money Maeve will have saved after n weeks n 6 = n

14 2017 JCHL Paper 1 Question 5 (c) 10 Marks Pete will give one quarter of his savings to buy the Xbox. Maeve will give two thirds of her savings to buy the Xbox. The Xbox costs 200. After how many weeks will they have enough money saved to buy the Xbox? Pete = n Maeve = n n n = n n = 200 3n + 4n = n = 185 n = n 26.4 weeks They will have enough money saved after 27 weeks to buy the Xbox. Note that the marking scheme allowed numerous interpretations. Many people calculated the time taken for one quarter of Pete s savings to be 200 (65 weeks) and the time calculated for two thirds of Maeve s savings to be 200 (48 weeks).

15 2017 JCHL Paper 1 Question 6 (a) 15 Marks Each of the students in sixth year in a particular school has WhatsApp (W), Instagram (I), or Snapchat (S). The numbers who have each app are as follows: 36 students have WhatsApp 40 students have Instagram 54 students have Snapchat 14 students have WhatsApp and Instagram 24 students have Instagram and Snapchat x students have WhatsApp and Snapchat, but not Instagram 8 students have all three apps. Use this information to fill in the Venn diagram below, in terms of x x 10 8 x x students have WhatsApp and Instagram # W I \ S = 6 24 students have Instagram and Snapchat # I S \ W = 16

16 2017 JCHL Paper 1 Question 6 (b) 5 Marks There are 80 students in total in sixth year in the school. Find the value of x. The sum of all the sections of the Venn Diagram is equal to the total number of students. 22 x x x = 80 x = 12 x = x 10 8 x x 54 Alternate Method: We know that there are 36 in the WhatApp Circle so: x = 80 x = 12 x = 12

17 2017 JCHL Paper 1 Question 6 (c) 15 Marks The table below shows four statements. Each statement is written in mathematical notation and in English. Complete the table. 8 students have WhatsApp, Instagram and Snapchat 24 = # I S 10 students have Instagram but NOT WhatsApp or Snapchat #S > #W

18 2017 JCHL Paper 1 Question 7 (a) 5 Marks Give an example of two non-empty sets A and B for which A B = A. A = 1,2 B = 1,2,3 A B = A A B (b) Give an example of two non-empty sets P and Q for which P\Q = P. P = 1,2 Q = 3 P \Q = P P Q

19 2017 JCHL Paper 1 Question 8 (a) 15 Marks for (a) and (b) A school can get its electricity from one of two companies, Buzz or Lecky. The graphs below show the cost of the electricity per month from each company, if the school uses x units of electricity. The cost from Buzz is b x, and the cost from Lecky is l x. One of the companies charges a fixed fee each month, plus a fee for each unit of electricity used. State which company charges no fixed fee. Give a reason for your answer, based on the graph. The fixed fees are the costs on the y axis when x = 0. So what will it cost even if 0 units are used. We can see that the fixed costs for BUZZ is 50 and the fixed cost for Lecky is 0. Lecky has no fixed fee. You only pay for units used.

20 2017 JCHL Paper 1 Question 8 (b) 15 Marks for (a) and (b) Write down the domain and the range of the function b x, as shown on the diagram. Domain are the input numbers of the function. l x and b x have the same domain, the numbers from 0 to x 1000 Range is the output numbers of the function. The range of b x are the numbers from 50 to b x 325

21 2017 JCHL Paper 1 Question 8 (c) (i) 5 Marks for (i) and (ii) Use the graphs to estimate the set of values of x R for which b x < l x. b x < l x from x = 100 to x = < x < JCHL Paper 1 Question 7 (c) (ii) Explain what your answer to part (c)(i) means about the cost of electricity from Buzz and Lecky. Buzz will be cheaper than Lecky of the school uses between 100 and 800 units of electricity.

22 2017 JCHL Paper 1 Question 8 (d) (i) 10 Marks for (i) and (ii) Find the slope of the graph of b x. Slope = Rise Run Run 1000 Rise 275 Slope = Slope = Slope = Slope = JCHL Paper 1 Question 7 (d) (ii) Explain what your answer to part (d)(i) means about the cost of electricity from Buzz. The cost of electricity rises by for every one unit increase of usage.

23 2017 JCHL Paper 1 Question 9 (a) Solve the equation x 2 2x 4 = 0. Give your answers in the form a ± b, where a, b N. x 2 2x 4 = 0 a = 1 b = 2 c = 4 x = b ± b2 4ac 2a x = 2 ± x = 2 ± x = 2 ± 20 2 x = 2 ± x = 1 ± 5 10 Marks

24 2017 JCHL Paper 1 Question 9 (b) 5 Marks Given that d 2 = d, multiply out and simplify c + d 2. c + d 2 = c + d c + d = c c + d + d c + d = c 2 + c d + c d + d = c 2 + 2c d + c d

25 2017 JCHL Paper 1 Question 9 (c) 10 Marks The table below shows whether each of the given numbers is an element of the natural numbers (N), the integers (Z), the rational numbers (Q), and the irrational numbers (R\Q). Complete the table by writing Yes or No into each box. One row is already done for you: 16 is an element of N, Z, and Q, but not of R\Q. No No No Yes No No Yes No No Yes Yes No

26 2017 JCHL Paper 1 Question Marks Write each of the following in the form 2 n, where n Q. Rules Applied (a) = 2 18 a p a q = a p+q Page 21 of Tables (b) = = = 2 3 a p q = a pq (c) 8 1 q a = aq 8 = = = = 2 3 a p q = a pq

27 2017 JCHL Paper 1 Question 11 (a) 10 Marks In a particular linear sequence, the second term is 40 and the sixth term is 116. Fill in the boxes below to show the rest of the first six terms of this sequence = =

28 2017 JCHL Paper 1 Question 11 (b) 10 Marks Orla is asked to write down a quadratic sequence. She writes down the following: 5, 6, 9, 14, 22, 30, 41 Exactly one of the terms in Orla s sequence is incorrect. Write down the correct quadratic sequence in the spaces below. You may only change one of the terms in Orla s sequence. Sequence 5, 6, 9, 14, 22, 30, 41 1 st Difference 1, 3, 5, 8, 8, 11 2 nd Difference 2, 2, 3, 0, 3, Incorrect The 2 nd Difference should be 2 across the entire sequence and so the entry of 22 is incorrect

29 2017 JCHL Paper 1 Question 12 (a) 5+5 Marks Factorise n 2 11n n 2 11n + 18 n 9 n 2 Quadratic n 2 11n + 18 n 9 n JCHL Paper 1 Question 12 (b) 9n 2n 2n 9n 11n Factorise fully wy y 1 + w. wy y 1 + w = wy y + w 1 = y w w 1 = y + 1 w 1 Factorise by Grouping

30 2017 JCHL Paper 1 Question 12 (c) 10 Marks Find the value of 5 7, when x = 4. 3x 2 6x x 2 7 = 6x = = = = = 5 60 = 1 12

31 2017 JCHL Paper 1 Question 12 (d) 5 Marks Use factorisation to simplify 4e 2 9 2e 2 + 3e 9. Combination Factorising Factorise the top and bottom. 4e 2 9 2e 2 + 3e 9 = 2e + 3 2e 3 2e 3 e + 3 = 2e + 3 e + 3 Difference of two squares. Quadratic

32 2017 JCHL Paper 1 Question 12 (e) 10 Marks A rectangle has sides of length x 3 units and ax 2 + bx + c units, where a, b, c Z. The area of the rectangle is 2x 3 13x + 25x 12 square units. Find the value of a, the value of b, and the value of c. 2x 2 7x + 4 x 3 2x 3 13x x 12 2x 3 + 6x 2 7x x 12 7x x 4x x 12 0 a = 2 b = 7 c = 4

33 2017 JCHL Paper 1 Question Marks for all Q13 Draw each of the following three functions in the domain 2 x 2, for x R. Function: y = x 1 Must find at least 2 points as the function is a straight line. x = 2 y = 2 1 y = 3 2, 3 x = 2 y = 2 1 y = 1 2,1

34 2017 JCHL Paper 1 Question Marks for all Q13 Draw each of the following three functions in the domain 2 x 2, for x R. Function: y = 2 x 2 Draw a table to find all the points in the domain. x y = 2 x 2 y x, y , , , , , 2

35 2017 JCHL Paper 1 Question Marks for all Q13 Draw each of the following three functions in the domain 2 x 2, for x R. Function: y = 2 x Draw a table to find all the points in the domain. x y = 2 x y x, y , , , , , 4

36 2017 JCHL Paper 1 Question 14 (a) 10 Marks for (a) and (c) The table below shows some information about regular polygons. These are shapes where all of the angles are the same size. The sum of the angles increases in a linear pattern. Complete the column in the table above showing the sum of the angles in each of these shapes

37 2017 JCHL Paper 1 Question 14 (b) 5 Marks Find a formula for the sum of the angles in a regular polygon with n angles. Remember that these values follow a linear pattern n = n = n = n = = 180 n = 180 n = 180 n = 180 n 2 Formula for the sum of the angles = 180 n 2

38 2017 JCHL Paper 1 Question 14 (c) 5 Marks for (d) Complete the column in the table above showing the size of each angle in each of these shapes. Remember that, in each polygon, all of the angles are the same size JCHL Paper 1 Question 14 (d) Find a formula for the size of each angle in a regular polygon with n angles. Formula for the size of each angle Sum of Angles = Number of Angles 180 n 2 = n

39 2017 JCHL Paper 1 Question 15 5 Marks The three curves A, B, and C are shown in the co-ordinate diagrams below. Two of the curves show a function of x. Put a tick ( ) in the correct box to show which curve does not show a function of x. Give a reason for your answer. Some x values have more than one y value. It fails the vertical line test. There are vertical lines that cut curve C more than once. C is a relation.