J.18/20. Pre-Junior Certificate Examination, Maths Higher Level. Marking Scheme. Paper 1 Pg. 2. Paper 2 Pg. 38. Page 1 of 56

J.18/20 Pre-Junior Certificate Examination, 2017 Maths Higher Level Marking Scheme Paper 1 Pg. 2 Paper 2 Pg. 38 Page 1 of 56

Name/version: Printed: Whom: exams Checked: Pre-Junior Certificate Examination, 2017 To: Ret d: Updated: Whom: Mathematics Name/version: Complete (y/n): Whom: 2005 Print Stamp.doc Higher Level Paper 1 Marking Scheme (300 marks) Structure of the Marking Scheme Students responses are marked according to different scales, depending on the types of response anticipated. Scales labelled A divide students responses into two categories (correct and incorrect). Scales labelled B divide responses into three categories (correct, partially correct, and incorrect), and so on. These scales and the marks that they generate are summarised in the following table: Scale label A B C D No. of categories 2 3 4 5 5 mark scale 0, 5 0, 3, 5 0, 2, 4, 5 0, 2, 3, 4, 5 10 mark scale 0, 4, 7, 10 0, 4, 6, 8, 10 15 mark scale A general descriptor of each point on each scale is given below. More specific directions in relation to interpreting the scales in the context of each question are given in the scheme, where necessary. Marking scales level descriptors A-scales (two categories) incorrect response (no credit) correct response (full credit) B-scales (three categories) response of no substantial merit (no credit) partially correct response (partial credit) correct response (full credit) 2014 (Paper 1) Scale label A B C No of categories 2 3 4 5 mark scale 0, 5 0, 3, 5 0, 2, 10 mark scale 0, 10 0, 5, 10 0, 3, 7 15 mark scale 0, 15 0, 10, 15 0, 10, C-scales (four categories) response of no substantial merit (no credit) response with some merit (low partial credit) almost correct response (high partial credit) correct response (full credit) D-scales (five categories) response of no substantial merit (no credit) response with some merit (low partial credit) response about half-right (middle partial credit) almost correct response (high partial credit) correct response (full credit) In certain cases, typically involving incorrect rounding, omission of units, a misreading that does not oversimplify the work or an arithmetical error that does not oversimplify the work, a mark that is one mark below the full-credit mark may also be awarded. Such cases are flagged with an asterisk. Thus, for example, scale 10C* indicates that 9 marks may be awarded. The * for units to be applied only if the student s answer is fully correct. The * to be applied once only within each section (a), (b), (c), etc. of all questions. The * penalty is not applied to currency solutions. Unless otherwise specified, accept correct answer with or without work shown. Accept students work in one part of a question for use in subsequent parts of the question, unless this oversimplifies the work involved. 2017.1 J.18/20_MS 2/72 Page 2 of 71 exams

Summary of Marks 2017 JC Maths (Higher Level, Paper 1) Q.1 (a) 5C (0, 2, 4, 5) Q.7 (a) (i) 5A (0, 5) (b) 5C (0, 2, 4, 5) (ii) 5C (0, 2, 4, 5) (c) 5B (0, 3, 5) (b) 10D* (0, 4, 6, 8, 10) (d) (i) 5A (0, 5) 20 (ii) 5C (0, 2, 4, 5) 25 Q.8 (a) (i) 5C (0, 2, 4, 5) (ii) 5C (0, 2, 4, 5) Q.2 (a) (i) 10D (0, 4, 6, 8, 10) (b) (i) 5C (0, 2, 4, 5) (ii) 5B (0, 3, 5) (ii) 5C (0, 2, 4, 5) (iii) 5B (0, 3, 5) (iii) 5C* (0, 2, 4, 5) (b) (i) (ii) 10C (0, 4, 7, 10) 30 25 Q.9 (a) 10D (0, 4, 6, 8, 10) (b) (i) 5B (0, 3, 5) Q.3 (a) 5C (0, 2, 4, 5) (ii) 5C (0, 2, 4, 5) (b) (i) 5C (0, 2, 4, 5) (c) 10D (0, 4, 6, 8, 10) (ii) 5C (0, 2, 4, 5) 30 (c) 5B (0, 3, 5) 20 Q.10 (a) 5B (0, 3, 5) (b) 5C (0, 2, 4, 5) Q.4 (a) 5D (0, 2, 3, 4, 5) (c) 10C (0, 4, 7, 10) (b) 5C (0, 2, 4, 5) 20 (c) (i) 5B (0, 3, 5) (ii) 5C (0, 2, 4, 5) 20 Q.11 (a) 5B (0, 3, 5) (b) 5C (0, 2, 4, 5) (c) 5C (0, 2, 4, 5) (d) 5B (0, 3, 5) Q.5 (a) 5C (0, 2, 4, 5) (e) 5C (0, 2, 4, 5) (b) Values 10D (0, 4, 6, 8, 10) (f) 10C (0, 4, 7, 10) Graph 5D (0, 2, 3, 4, 5 ) 35 (c) (i) 5B* (0, 3, 5) (ii) 5B* (0, 3, 5) 30 Q.12 (a) 10C (0, 4, 7, 10) (b) (i) 5C (0, 2, 4, 5) (ii) 5C (0, 2, 4, 5) Q.6 (a) (i) 5C (0, 2, 4, 5) 20 (ii) 5C (0, 2, 4, 5) (b) 10D (0, 4, 6, 8, 10) (c) 5C (0, 2, 4, 5) 25 Assumptions about these marking schemes on the basis of past SEC marking schemes should be avoided. While the underlying assessment principles remain the same, the exact details of the marking of a particular type of question may vary from a similar question asked by the SEC in previous years in accordance with the contribution of that question to the overall examination in the current year. In setting these marking schemes, we have strived to determine how best to ensure the fair and accurate assessment of students work and to ensure consistency in the standard of assessment from year to year. Therefore, aspects of the structure, detail and application of the marking schemes for these examinations are subject to change from past SEC marking schemes and from one year to the next without notice. 2017.1 J.18/20_MS 3/72 Page 3 of 71 exams

exams Pre-Junior Certificate Examination, 2017 Mathematics Higher Level Paper 1 Marking Scheme (300 marks) General Instructions 1. There are 12 questions on this examination paper. Answer all questions. 2. Questions do not necessarily carry equal marks. 3. Marks will be lost if all necessary work is not clearly shown. 4. Answers should include the appropriate units of measurement, where relevant. 5. Answers should be given in simplest form, where relevant. Q.1 (Suggested maximum time: 10 minutes) (25) 1(a) Write down the fraction that is half-way between 2 1 and 3 2. (5C) 1 1 2 1 (difference) ( ) 2 2 3 2 1 4 3 ( ) 2 6 1 1 ( ) 2 6 1 12 Half-way fraction 1 1 + 2 12 6 + 1 12 7 12 or 2017.1 J.18/20_MS 4/72 Page 4 of 71 exams

2017 JC Maths [HL] Paper 1 Q.1 (cont d.) 1(a) (cont d.) Half-way between Half-way between 1 2 and 2 3 3 4 and 6 6 3 2 1 6 7 2 6 7 1 2 6 7 12 Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. finds difference [ans. 6 1 ] and stops. High partial credit: (4 marks) 3 1 Finds 2 or similar and stops or fails 6 to finish correctly. 7 1 Finds or 2 12 6 and stops or fails to finish correctly. 1(b) Write the following four numbers in order, from the smallest to the biggest. 1, 1 3 10 1, 0 14, 11 5%. (5C) 8 1 8 0 125 1 3 10 1 0 13 0 14 0 14 11 5% 0 115 Order: 11 5% // 0 115 1 // 0 125 8 1 3 10 1 // 0 13 0 14 Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) One or two numbers in correct order. High partial credit: (4 marks) All numbers in the correct order, but not written in ascending order. 2017.1 J.18/20_MS 5/72 Page 5 of 71 exams

2017 JC Maths [HL] Paper 1 Q.1 (cont d.) 1(c) Mount Everest is the highest mountain peak on Earth, located in the Mahalangur mountain range in Nepal and Tibet. Its summit is 8800 metres above sea level, correct to the nearest 100 metres. Complete the inequality below, where h is the height of Mount Everest. (5B) 8,750 m h < 8,849 m Scale 5B (0, 3, 5) Partial credit: (3 marks) One correct answer. * No deduction applied for the omission of or incorrect use of units ( m ). 1(d) On Budget Day in October 2016, Ireland s National Debt had fallen to 184,932,672,554. (i) Express this figure correct to six significant figures. (5A) 184,932,672,554 184,933,000,000 Scale 5A (0, 5) Partial credit: (0 marks) Hit or miss. * No deduction applied for the omission of or incorrect use of units involving currency. (ii) At the peak of the recession in 2012, Ireland s National Debt was 215,539 million. Using your answer to part (i), find the percentage decrease in our National Debt. Give your answer correct to one decimal place. 215,539 million 215,539,000,000 Decrease in debt 215,539,000,000 184,933,000,000 30,606,000,000 % Decrease in debt 30,606,000,000 215,539,000,000 14 199750...% 14 2% 100 1 ** Accept students answers from part (d)(i) if not oversimplified. (5C) Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. finds correct Decrease in debt. High partial credit: (4 marks) Correct expression for % Decrease in debt, but fails to finish or finishes incorrectly. Incorrect decrease, from substantially correct work, but finishes correctly. * No deduction applied for the omission of or incorrect use of symbol ( % ). 2017.1 J.18/20_MS 6/72 Page 6 of 71 exams

2017 JC Maths [HL] Paper 1 Notes: 2017.1 J.18/20_MS 7/72 Page 7 of 71 exams

2017 JC Maths [HL] Paper 1 Q.2 (Suggested maximum time: 10 minutes) (30) 2(a) U is the set of natural numbers less than 20. P is the set of divisors of 15. Q is the set of divisors of 18. R is the set of prime numbers less than 20. (i) Use this information to fill in the Venn diagram below. (10D) P {1, 3, 5, 15} Q {1, 2, 3, 6, 9, 18} R {2, 3, 5, 7, 11, 13, 17, 19} U P 15 4 8 10 12 14 16 5 1 7 3 6 9 18 2 11 13 17 19 R Q Scale 10D (0, 4, 6, 8, 10) Low partial credit: (4 marks) Some work of merit, e.g. at least four correct elements of P, Q or R stated. One to four elements correctly placed. Middle partial credit: (6 marks) Five to eight elements correctly placed. High partial credit: (8 marks) Nine to twelve elements correctly placed. (ii) Hence, or otherwise, write down the highest common factor of 15 and 18. (5B) From Venn diagram Factors of 15 and 18 (P Q) {1, 3} HCF of 15 and 18 3 or Alternative method 15 15 3 5 5 5 1 15 3 5 18 18 2 9 9 3 3 3 3 1 18 2 3 3 HCF of 15 and 18 1 3 3 2017.1 J.18/20_MS 8/72 Page 8 of 71 exams

2017 JC Maths [HL] Paper 1 Q.2 (cont d.) 2(a) (ii) (cont d.) ** Accept students answers from part (a)(i) if not oversimplified. Scale 5B (0, 3, 5) Partial credit: (3 marks) Some work of merit, e.g. writes down (P Q) {1, 3} and stops or continues incorrectly. Finds correctly 15 and/or 18 as a product of prime numbers, but fails to find or finds incorrect HCF of the two numbers. (iii) Calculate #(P Q R). P Q R {1, 2, 3, 5, 6, 7, 9, 11, 13, 15, 17, 18, 19} (P Q R) {4, 8, 10, 12, 14, 16} #(P Q R) 6 ** Accept students answers from part (a)(i) if not oversimplified. Scale 5B (0, 3, 5) Partial credit: (3 marks) Some work of merit, e.g. writes down P Q R or (P Q R) and stops or continues incorrectly. (5B) 2(b) U is the universal set. A and B are two subsets of U. #U 28, #A 18 and #B 8. (10C) (i) Find, with the aid of the Venn diagram, the minimum value of #(A B). Let #(A B) 0 #(A B) 18 + 8 26 min. #(A B) 28 26 2 U [28] A [18] [18] [0] [8] B [8] (ii) Find, with the aid of the Venn diagram, the maximum value of #(A B). Let #(A B) 8 #(B / A) 0 and #(A / B) 18 8 10 #(A B) 10 + 8 + 0 18 max. #(A B) 28 18 10 U [28] A [18] [10] [8] [0] B [8] Scale 10C (0, 4, 7, 10) Low partial credit: (4 marks) Some work of merit in one part, e.g. identifies #(A B) 0 [min. value] or #(A B) 8 [max. value]. Finds #(A B) 26 [min. value]. Finds #(A B) 18 [max. value]. High partial credit: (7 marks) One part fully correct. Two parts correct, but answers swapped. Finds correct #(A B) in both parts, but fails to finish or finishes incorrectly. 2017.1 J.18/20_MS 9/72 Page 9 of 71 exams

2017 JC Maths [HL] Paper 1 Q.3 (Suggested maximum time: 10 minutes) (20) 3(a) A supermarket sells a single 32 5 g packet of cheese and onion flavoured crisps for 0 545, excluding VAT. If the current rate of VAT on crisps is 23%, calculate the price of a packet of crisps including VAT. Give your answer correct to the nearest cent. VAT 23 0 545 100 0 12535 CRISPS (5C) Price (inc. VAT) 0 545 + 0 12535 0 67035 0 67 or 67c or Price (inc. VAT) 123 0 545 100 0 67035 0 67 Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Any work of merit, e.g. writes down VAT 0 545 23% or Price (inc. VAT) 0 545 123%. Correct formula for VAT, but error(s) calculating value. High partial credit: (4 marks) Finds correct VAT and stops. One error calculating Price (inc. VAT), but finishes correctly. 3(b) The same supermarket is also offering the following multipack deals: Special Offer 1 6-pack (6 25 g) of cheese and onion flavoured crisps only 1 99 Special Offer 2 14-pack (14 25 g) of variety flavoured crisps only 3 99 (i) Which special offer is better value? Give a reason for your answer. Answer Special offer 2 (5C) Reason Special offer 1 Cost per packet 1 99 6 0 331666... or 33 166666...c Special offer 2 3 99 Cost per packet 14 0 285 or 28 5c ** Accept other appropriate answers. 2017.1 J.18/20_MS 10/72 Page 10 of 71 exams

2017 JC Maths [HL] Paper 1 Q.3 (cont d.) 3(b) (i) (cont d.) Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. calculates cost per 25g packet for Special offer 1 or Special offer 2. Correct answer but no reason given. High partial credit: (4 marks) Finds cost per packet for both offers, but no conclusion or incorrect conclusion given. (ii) Paula says The price per gram when purchasing a 6-pack of cheese and onion crisps is almost 50% cheaper than the price per gram when purchasing a single packet. Is she correct? Give a reason for your answer. Answer no (5C) Reason Single packet 0 67 Cost per g 32 5 0 020615... or 2 061538...c Special offer 1 Cost per g 0 331666... 25 0 013266... or 1 326666...c % Difference 0 020615... 0 013266... 0 020615... 35 648799... 35 65% ** Accept students answer from parts (a) and (b)(i) if not oversimplified. Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. calculates cost per g for single packet and/or cost per g per packet in special offer 1. Correct answer but no justification given. High partial credit: (4 marks) Finds cost per g for both offers and gives correct conclusion, but fails to find or find incorrect % Difference. Finds % Difference, but no conclusion or incorrect conclusion given. 100 1 (c) Give two limitations of special offers in general. (5B) Limitations Any 2: customers are encouraged to buy more than they need // customers are encouraged to buy products / goods they don t need // customer may want to buy only one variety // may contribute to impulse buying / buying on the spur of the moment // may contribute to a person getting into financial difficulties // etc. ** Accept other appropriate answers. Scale 5B (0, 3, 5) Partial credit: (3 marks) One correct limitation. 2017.1 J.18/20_MS 11/72 Page 11 of 71 exams

2017 JC Maths [HL] Paper 1 Q.4 (Suggested maximum time: 10 minutes) (20) 4(a) For each of the following, write down the value of x, where 21 x 27 and x N. (5D) Description Value of x A square number 25 A multiple of 7 21 A factor of 66 22 A cube number 27 A prime number 23 Scale 5D (0, 2, 3, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. one correct value of x. Middle partial credit: (3 marks) Two or three correct values of x. High partial credit: (4 marks) Four correct values of x. 4(b) 27 + 75 12 k 3, where k Z. Find the value of k. (5C) 27 + 75 12 9 3 + 25 3 4 3 3 3 + 5 3 2 3 6 3 Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. writes down one or more of 27 9 3, 75 25 3 or 12 4 3. Uses decimal versions to find k 3. High partial credit: (4 marks) Finds 3 3 + 5 3 2 3, but fails to finish or finishes incorrectly. 4(c) (i) Simplify ( 3 x ) 6. ( 3 x ) 6 (x 3 1 ) 6 x 3 6 x 2 (5B) Scale 5B (0, 3, 5) Partial credit: (3 marks) Some work of merit, e.g. writes down ( 3 x ) 6 (x 3 1 ) 6 and stops or continues incorrectly. 2017.1 J.18/20_MS 12/72 Page 12 of 71 exams

2017 JC Maths [HL] Paper 1 Q.4 (cont d.) 3y (10y) 4(c) (ii) Simplify. 5 2y 2 3 2 3y (10y) 2y 5 3 2 3y 1,000y 5 2y 5 3,000y 5 2y 1,500 3 (5C) Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. writes down (10y) 3 1,000y 3. High partial credit: (4 marks) 5 3,000y Finds 5, but fails to finish or 2y finishes incorrectly. 2017.1 J.18/20_MS 13/72 Page 13 of 71 exams

2017 JC Maths [HL] Paper 1 Q.5 (Suggested maximum time: 15 minutes) (30) A school has a rectangular vegetable garden. The wall of the school building forms one side of the garden and a fence, of length 16 m, is used to form the remaining three sides, as shown. The width of the garden is x m. 5(a) Show that the enclosed area of the garden, A m 2, is given by the function A(x) 16x 2x 2. x (5C) Width x m Length 16 (x + x) 16 2x m Area w l x(16 2x) 16x 2x 2 m 2 Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. writes down l + x + x 16 or similar and stops. Finds l 16 2x and stops or continues incorrectly. High partial credit: (4 marks) Finds x(16 2x), but fails to finish or finishes incorrectly. 5(b) On the axes below, draw the graph of the function A(x) 16x 2x 2 for 0 x 8, where x R. Values Table (1st method) x 0 1 2 3 4 5 6 7 8 16x 0 16 32 48 64 80 96 112 128 2x 2 0 2 8 18 32 50 72 98 128 y A 0 14 24 30 32 30 24 14 0 (10D) or Table (2nd method) A(x) 16x 2x 2 y A(0) 0 0 0 A(1) 16 2 14 A(2) 32 8 24 A(3) 48 18 30 A(4) 64 32 32 A(5) 80 50 30 A(6) 96 72 24 A(7) 112 98 14 A(8) 128 128 0 2017.1 J.18/20_MS 14/72 Page 14 of 71 exams

2017 JC Maths [HL] Paper 1 Q.5 (cont d.) 5(b) (cont d.) or Substitution method A(x) 16x 2x 2 A(0) 16(0) 2(0) 2 0 A(1) 16(1) 2(1) 2 14 A(2) 16(2) 2(2) 2 24 A(3) 16(3) 2(3) 2 30 A(4) 16(4) 2(4) 2 32 A(5) 16(5) 2(5) 2 30 A(6) 16(6) 2(6) 2 24 A(7) 16(7) 2(7) 2 14 A(8) 16(8) 2(8) 2 0 Scale 10D (0, 4, 6, 8, 10) Low partial credit: (4 marks) One or two points, (x, y) or (y, x), correctly identified. Middle partial credit: (6 marks) Three, four or five points, (x, y) or (y, x), correctly identified. High partial credit: (8 marks) Six, seven or eight points, (x, y) or (y, x), correctly identified. 2017.1 J.18/20_MS 15/72 Page 15 of 71 exams

2017 JC Maths [HL] Paper 1 Q.5 (cont d.) 5(b) (cont d.) Graph (5D) Points (0, 0), (1, 14), (2, 24), (3, 30), (4, 32), (5, 30), (6, 24), (7, 14), (8, 0) 40 2 Area of Vegetable Garden, A (m ) 32 30 26 20 10 Ax () x 0 225 5 75 0 1 2 3 4 5 6 7 8 Width, x (m) ** Accept values calculated from previous work (nine co-ordinates needed). ** If no points are worked out, but all points are correctly graphed, award marks for the graph in both parts. Scale 5D (0, 2, 3, 4, 5) Low partial credit: (2 marks) One or two points, (x, y) or (y, x), correctly plotted, labelled or not. Points incorrectly calculated, but correctly plotted to form a line. Middle partial credit: (3 marks) Three, four or five points, (x, y) or (y, x), correctly plotted, labelled or not (joined or not). High partial credit: (4 marks) Six, seven or eight points, (x, y) or (y, x), correctly plotted, labelled or not (joined or not). All points, (x, y) or (y, x), correctly plotted, but joined together with inappropriate curve. 2017.1 J.18/20_MS 16/72 Page 16 of 71 exams

2017 JC Maths [HL] Paper 1 Q.5 (cont d.) 5(c) Use your graph in part (b) to answer the following questions. (i) Estimate the maximum possible area of the garden. (5B*) From graph Maximum area 32 m 2 ** Accept students answers based on fully plotted graph in part (b). Scale 5B* (0, 3, 5) Partial credit: (3 marks) Some work of merit, e.g. draws vertical line from 4 m and intersects graph with or without horizontal line from point of intersection to the y-axis. Answer outside tolerance (±1 grid box), but inside tolerance of ±2 grid boxes. Gives 4 m as final answer. * Deduct 1 mark off correct answer only for the omission of or incorrect use of units ( m 2 ) - apply only once to each section (a), (b), (c), etc. of question. (ii) Find the range of values of x for which the area of the garden is greater than 26 m 2. (5B*) From graph Area (> 26 m 2 ): 2 25 m x 5 75 m ** Accept students answers based on fully plotted graph in part (b). Scale 5B* (0, 3, 5) Low partial credit: (3 marks) Some work of merit, e.g. draws horizontal line from 26 m 2 and intersects graph with or without vertical line(s) from points of intersection to the x-axis. Answer outside tolerance (±1 grid box), but inside tolerance of ±2 grid boxes. One correct x-intercept. * Deduct 1 mark off correct answer only for the omission of or incorrect use of units ( m ) - apply only once to each section (a), (b), (c), etc. of question. 2017.1 J.18/20_MS 17/72 Page 17 of 71 exams

2017 JC Maths [HL] Paper 1 Q.6 (Suggested maximum time: 15 minutes) (25) 6(a) p 2a + c. (i) Write a in terms of p and c. (5C) p 2a + c 2a p c a 1 p c p c (p c) or or. 2 2 2 2 Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. one correct manipulation. High partial credit: (4 marks) 2a p c, but fails to finish or finishes incorrectly. (ii) Hence, find the value of a when p 5 and c 5 1. (5C) a 1 (p c) 2 1 1 (5 ) 2 5 1 25 1 ( ) 2 5 24 10 12 2 or 2 or 2 4 5 5 Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. some correct substitution into equation. High partial credit: (4 marks) Fully correct substitution into equation, but fails to finish or finishes incorrectly. 2017.1 J.18/20_MS 18/72 Page 18 of 71 exams

2017 JC Maths [HL] Paper 1 Q.6 (cont d.) 6(b) Solve the equation x 2 6x + 4 0. Give each answer in surd form. x 2 6x + 4 0 x 2 6x + 9 5 0 (x 3) 2 5 0 (x 3) 2 5 x 3 ± 5 x 3 5 x 3 + 5 (10D) or x 3 5 x 3 5 or x b ± b 4ac 2a x ( 6) ± ( 6) 2(1) 6 ± 36 16 2 6 ± 20 2 6 ± 4 5 2 6 ± 2 5 2 3 ± 5 2 4(1)(4) x 3 + 5 or x 3 5 Scale 10D (0, 4, 6, 8, 10) Low partial credit: (4 marks) Some work of merit, e.g. writes down b formula and identifies a, b and c. Finds (x 3) 2 5 0 and stops. Middle partial credit: (6 marks) Fully correct substitution into b formula, ( 6) ± ( 6) 2 4(1)(4) i.e. x 2(1) and stops. Both factors correct (not looking for 0 ). High partial credit: (8 marks) Each of the following steps correct: correct formula and correctly identifies a, b and c, fully correct substitution, works out solution to general surd form [ans. 3 ± 5]. Finds only one value of x [ans. 3 + 5 or 3 5], but fails to find or finds incorrectly other value of x. Finds x 3 ± 5, but fails to finish or finishes incorrectly. 2017.1 J.18/20_MS 19/72 Page 19 of 71 exams

2017 JC Maths [HL] Paper 1 Q.6 (cont d.) 6(c) Write the following as a single fraction in its simplest form. 2x 3 4x 7. (5C) 2 5 2x 3 2 4x 5 7 5(2x 3) 2(4x 7) (2)(5) 10 x 15 8x + 14 10 2x 1 10 Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. identifies 10 as the denominator with some (incorrect) numerator. Correct numerator, i.e. multiplies 5(2x 3) 2(4x 7), but no denominator. High partial credit: (4 marks) 5(2x 3) 2(4x 7) Finds or similar, (2)(5) but fails to finish or finishes incorrectly. 2017.1 J.18/20_MS 20/72 Page 20 of 71 exams

2017 JC Maths [HL] Paper 1 Notes: 2017.1 J.18/20_MS 21/72 Page 21 of 71 exams

2017 JC Maths [HL] Paper 1 Q.7 (Suggested maximum time: 10 minutes) (20) ** Deduct 1 mark off correct answer only if final answer is not rounded or is incorrectly rounded or for the omission of or incorrect use of units in part(s) of question asterisked - this deduction should be applied only once per section (a), (b), (c), etc. of the question. ** No deduction should be applied for the omission of or incorrect use of units involving currency. Aoife is a sales representative for a pharmaceutical company. She earns a basic pay of 2750 per month plus commission of 1 5% on her total sales. For the month of December, Aoife s gross pay, including commission, was 4,700. 7(a) (i) Calculate Aoife s commission for the month of December. (5A) Aoife s commission 4,700 2,750 1,950 Scale 5A (0, 5) Partial credit: (0 marks) Hit or miss. * No deduction applied for the omission of or incorrect use of units involving currency. (ii) Hence, calculate Aoife s total sales for the month of December. (5C) 1 5% of sales 1,950 1% of sales 1,950 1 5 1,300 Total sales (100%) 1,300 100 130,000 Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. indicates division by 1 5. High partial credit: (4 marks) Finds 1,300 or 1,300 100, but fails to finish or finishes incorrectly. Correct answer with no supporting work or correct answer with invalid supporting work. * No deduction applied for the omission of or incorrect use of units involving currency. 2017.1 J.18/20_MS 22/72 Page 22 of 71 exams

2017 JC Maths [HL] Paper 1 Q.7 (cont d.) The standard rate of tax is 20% and the higher rate of tax is 40%. The standard rate cut-off point is 33,800 per annum. 7(b) Aoife pays 1026 34 in income tax for December. Calculate Aoife s tax credits for the month. Tax @ 20% 33,800 20% 12 20 2,816 67 100 563 333333... 563 33 Tax @ 40% (4,700 2,816 67) 40% 40 1,883 33 100 753 332 753 33 Gross tax Tax @ 20% + Tax @ 40% 563 33 + 753 33 1,316 66 Net tax Gross tax Tax credit 1026 34 1,316 66 Tax credit Tax credits 1,316 66 1026 34 290 32 or 2,816 67 0 20 or 1,883 33 0 40 (10D*) Scale 10D* (0, 4, 6, 8, 10) Low partial credit: (4 marks) Any work of merit, e.g. finds monthly 33,800 SRCOP [ans. or 2,816 67] 12 and stops. Finds tax @ 20% or @ 40% correctly of some relevant figure, but fails to continue correctly. Middle partial credit: (6 marks) Finds tax @ 20% and @ 40% correctly, but fails to find or finds incorrect figure for Gross tax. High partial credit: (8 marks) Finds Gross tax correctly, but fails to find or finds incorrect figure for monthly Tax credits. * Deduct 1 mark off correct answer only if not rounded or incorrectly rounded - apply only once per each section (a), (b), (c), etc. of question. * No deduction applied for the omission of or incorrect use of units involving currency. 2017.1 J.18/20_MS 23/72 Page 23 of 71 exams

2017 JC Maths [HL] Paper 1 Q.8 (Suggested maximum time: 10 minutes) (25) 8(a) The nth term of a sequence is given by the formula T n 5n 2. (i) Write down the first three terms of the sequence. (5C) T n 5n 2 T 1 5(1) 2 5 T 2 5(2) 2 20 T 3 5(3) 2 45 Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. some correct substitution into relevant formula. One correct term. High partial credit: (4 marks) Two correct terms. (ii) Is 1000 a term in this sequence? Justify your answer. Answer no Justification T n 5n 2 5n 2 1,000 n 2 200 n 200 14 142135... N (5C) Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Correct answer, but no justification given. Some work of merit, e.g. equates 5n 2 1,000 or n 2 200 and stops. High partial credit: (4 marks) Correct answer, but incomplete or unsatisfactory justification given. 2017.1 J.18/20_MS 24/72 Page 24 of 71 exams

2017 JC Maths [HL] Paper 1 Q.8 (cont d.) 8(b) Stephen plays golf. The table below shows the average distance, in metres, he regularly hits the ball with various golf clubs. Golf Club Putter 9 Iron 8 Iron 7 Iron 6 Iron 5 Iron Average Distance (m) 75 80 85 90 95 100 (i) Is the pattern of distances shown in the table linear, quadratic or exponential? Justify your answer. Answer linear (5C) Justification as the first differences are constant, the average distance forms a linear sequence (see table) Club Avg. D (m) 1st Diff. P 75 9 80 8 85 7 90 6 95 5 5 5 5 Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Correct answer but no justification given. Some work of merit, e.g. finds one first difference. High partial credit: (4 marks) Correct answer, but incomplete or unsatisfactory justification given. 2017.1 J.18/20_MS 25/72 Page 25 of 71 exams

2017 JC Maths [HL] Paper 1 Q.8 (cont d.) 8(b) (cont d.) (ii) On the grid below, draw a graph showing the pattern of average distances for the different golf clubs. (5C) 120 110 100 90 Average Distance (m) 80 70 60 50 40 30 20 10 0 x P 9 8 7 6 5 4 3 Golf Club Scale 5D (0, 2, 4, 5) Low partial credit: (2 marks) One or two points correctly plotted, labelled or not. High partial credit: (4 marks) Between three and five points correctly plotted, labelled or not. All points correctly plotted, but not joined together with line. (iii) Using your graph, or otherwise, estimate the average distance Stephen would expect to achieve with a 3 Iron golf club. From graph Distance (3 Iron) 110 m ** Accept students answers based on fully plotted graph in part (b)(ii). Scale 5C* (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. extends graph of points plotted as far as 3 Iron (with or without x-axis labelled). Draws vertical line from 3 Iron position, but does not extend graph to intersect. High partial credit: (4 marks) Extends line correctly as far as 3 Iron and draws vertical line from 3 Iron with or without horizontal line from point of intersection to the y-axis. Answer outside tolerance (±1 grid box), but inside tolerance of ±2 grid boxes. Correct answer with no supporting work. 2017.1 J.18/20_MS 26/72 Page 26 of 71 exams (5C*) * Deduct 1 mark off correct answer only for the omission of or incorrect use of units ( m ) - apply only once to each section (a), (b), (c), etc. of question.

2017 JC Maths [HL] Paper 1 Q.9 (Suggested maximum time: 15 minutes) (30) 9(a) Solve the following inequality and graph your solution on the number line. 21 3 6x < 12, x Z. (10D) Solution 21 3 6x 6x 21 + 3 6x 24 x 4 and 3 6x < 12 6x < 12 3 < 9 9 x > 6 3 > 2 2 3 < x 4 or 21 3 6x < 12... subtract 3 21 3 3 6x 3 < 12 3 24 6x < 9 24 6x < 9... divide by 6 24 6x 9 > 6 6 6 3 4 x > 2 2 3 < x 4 Number line 3 2 1 0 1 2 3 4 5 Scale 10D (0, 4, 6, 8, 10) Low partial credit: (4 marks) Some work of merit, e.g. substitutes in value for x. Middle partial credit: (6 marks) Finds x > 2 3 or x 4 (accept answer with or without inequality sign). High partial credit: (8 marks) Solution to inequality fully correct or number line correctly graphed. 2017.1 J.18/20_MS 27/72 Page 27 of 71 exams

2017 JC Maths [HL] Paper 1 Q.9 (cont d.) 9(b) Factorise fully each of the following expressions. (i) 27x 2 45x (5B) 27x 2 45x 9x(3x 5) Scale 5B (0, 3, 5) Partial credit: (3 marks) Some work of merit, e.g. any common factor identified. (ii) 6xy 3sy 4tx + 2st 6xy 3sy 4tx + 2st 3y(2x s) 2t(2x s) (3y 2t)(2x s) or 6xy 3sy 4tx + 2st 6xy 4tx 3sy + 2st 2x(3y 2t) s(3y 2t) (2x s)(3y 2t) (5C) Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. one common factor or group indicated. High partial credit: (4 marks) Finds one correct factorisation - both inside and outside bracket, e.g. 3y(2x s). Finds two correct factorisations, but with sign errors. 2017.1 J.18/20_MS 28/72 Page 28 of 71 exams

2017 JC Maths [HL] Paper 1 Q.9 (cont d.) 9(c) A 99 Flake ice cream, more commonly known as a 99 or ninety-nine, is an ice cream cone in which a Flake bar is inserted. Mary buys 2 plain ice cream cones and 3 ninety-nines costing 10 20. Joe buys 3 plain ice cream cones and 4 ninety-nines costing 14 20. Calculate the cost of a flake. (10D) Mary: 2x + 3y 10 20 Joe.: 3x + 4y 14 20 2x + 3y 10 20 ( 3) 3x + 4y 14 20 ( 2) 6x + 9y 30 60 6x 8y 28 40 y 2 20 2x + 3y 10 20 2x + 3(2 20) 10 20 2x + 6 6 10 20 2x 10 20 6 60 3 60 x 1 80 or 3x + 4y 14 20 3x + 4(2 20) 14 20 3x + 8 80 14 20 3x 14 20 8 80 5 40 x 1 80 or 2x + 3y 10 20 ( 4) 3x + 4y 14 20 ( 3) 8x + 12y 40 80 9x 12y 42 60 x 1 80 x 1 80 2x + 3y 10 20 2(1 80) + 3y 10 20 3 60 + 3y 10 20 3y 10 20 3 60 6 60 y 2 20 Cost of flake y x 2 20 1 80 0 40 or 40c or 3x + 4y 14 20 3(1 80) + 4y 14 20 5 40 + 4y 14 20 4y 14 20 5 40 8 80 y 2 20 Scale 10D (0, 4, 6, 8, 10) Low partial credit: (4 marks) Any work of merit, e.g. multiplying any equation by appropriate constant to facilitate cancellation of x or y term. Finds one or both expressions correctly, i.e. 2x + 3y 10 20 and/or 3x + 4y 14 20. Finds either variable (x or y) correctly by trial and error, but fails to verify in both equations or verifies incorrectly. Middle partial credit: (6 marks) Finds first variable (x or y) correctly, but fails to find second variable or finds incorrectly. Finds both variables (x and y) correctly with no work shown. Finds both variables (x and y) by trial and error, but does not verify into both equations. High partial credit: (8 marks) Finds both variables (x and y) correctly, but fails to find or finds incorrect cost of flake. * No deduction applied for the omission of or incorrect use of units involving currency. 2017.1 J.18/20_MS 29/72 Page 29 of 71 exams

2017 JC Maths [HL] Paper 1 Q.10 (Suggested maximum time: 10 minutes) (20) A triangle has three sides as follows where x R. 3x + 2, 2x 5, x + 1 10(a) Calculate the length of the perimeter of the triangle in terms of x. Give your answer in its simplest form. (5B) Perimeter (3x + 2) + (2x 5) + (x + 1) 3x + 2x + x + 2 5 + 1 6x 2 Scale 5B (0, 3, 5) Partial credit: (3 marks) Some work of merit, e.g. adds two sides correctly. Finds correct one component. 10(b) Show that the triangle is isosceles when x is equal to 6. Side 1 3x + 2 3(6) + 2 18 + 2 20 Side 2 2x 5 2(6) 5 12 5 7 Side 3 x + 1 6 + 1 7 triangle is isosceles as side 2 / 2x 5 and side 3 / x + 1 are equal in length (7) Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. finds length of one or two sides (different lengths) of triangle when x 6. (5C) High partial credit: (4 marks) Finds correct length of two relevant sides of triangle, but no conclusion or incorrect conclusion given. 2017.1 J.18/20_MS 30/72 Page 30 of 71 exams

2017 JC Maths [HL] Paper 1 Q.10 (cont d.) 10(c) Mark says that x equal to 6 is the only value of x such that the triangle is isosceles. Is he correct? Justify your answer. (10C) Answer yes Justification Consider: Side 1 Side 2 3x + 2 2x 5 x 7 3x + 2 3( 7) + 2 21 + 2 19 as side < 0, it can t be a triangle or 2x 5 2( 7) 5 19 as side < 0, it can t be a triangle or x + 1 7 + 1 6 as side < 0, it can t be a triangle Consider: Side 1 Side 3 3x + 2 x + 1 2x 1 x 1 2 2x 5 2( 2 1 ) 5 1 5 6 as side < 0, it can t be a triangle Scale 10C (0, 4, 7, 10) Low partial credit: (4 marks) Some work of merit, e.g. equates correctly Side 1 Side 2 or Side 1 Side 3 and finds correct x value. Correct answer but no justification given. High partial credit: (7 marks) Equates correctly Side 1 Side 2 or Side 1 Side 3 and find correct x values and finishes correctly. Equates correctly Side 1 Side 2 and Side 1 Side 3 and finds correct x values, but fails to finish or finishes incorrectly. 2017.1 J.18/20_MS 31/72 Page 31 of 71 exams

2017 JC Maths [HL] Paper 1 Q.11 (Suggested maximum time: 15 minutes) (35) Two gyms, A and B, offer special rates to students for the month of July. The grid below shows the graphs of the total cost of membership for the number of days used, for each gym, during July. 100 90 80 B Total Cost ( ) 70 60 50 40 33 333... 30 A 20 10 0 x 5 6 666... 10 15 20 25 Number of Days 11(a) Which gym has a joining fee? Justify your answer with reference to the graph above. (5B) Answer gym A Justification Any 1: consider the equation of a line y mx + c that represents the cost of gym A - as c 20, therefore 20 is charged whether a student uses the gym or not // as the equation of the line representing gym A cuts the y-axis at 20, this indicates a cost of 20 before going to the gym or not // as the equation of the line representing gym A goes through (0, 20), this means that cost is 20 when the number of days used is zero // etc. ** Accept other appropriate answers and interpretations. Scale 5B (0, 3, 5) Partial credit: (3 marks) Correct answer but no explanation or incorrect explanation given. Valid explanation, but no conclusion stated. 2017.1 J.18/20_MS 32/72 Page 32 of 71 exams

2017 JC Maths [HL] Paper 1 Q.11 (cont d.) 11(b) How might a student decide which gym is better value for money? Justify your answer. (5C) Answer determine the point of intersection (6 666..., 33 333...) and consider the case above / below this point Justification Any 1: below this point (6 666 visits), gym B is better value for money // above this point (6 666 visits), gym A is better value for money Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. finds the point of intersection of the two graphs. Correct answer but no explanation or incorrect explanation given. High partial credit: (4 marks) Valid explanation, but no conclusion stated [e.g. above the point of intersection one gym will be more expensive than the other, etc.] Correct answer, but incomplete or unsatisfactory explanation given. 11(c) Find the slope (rate of change) of the graph for gym A. Explain what this value means. Refer to both the cost and the number of days the gym is used in July. Slope of gym A (0, 20), (25, 70) (x 1, y 1 ) (x 2, y 2 ) m A y2 y1 x2 x1 70 20 25 0 50 25 2 or m A rise run 50 25 2 (5C) Explanation after paying an initial fee of 20 for the month, the cost will be 2 per day to attend the gym Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. writes down correct relevant formula for slope. High partial credit: (4 marks) Finds correct slope, but incomplete or unsatisfactory explanation given. Correct explanation, but fails to find or finds incorrect slope. 2017.1 J.18/20_MS 33/72 Page 33 of 71 exams

2017 JC Maths [HL] Paper 1 Q.11 (cont d.) 11(d) Write down a formula to represent the total cost for the number of days gym A is used in July. State clearly the meaning of any letters you use in your formula. (5B) m A 2 Cost of gym A c 20 + 2x where c cost x number of days gym used Scale 5B (0, 3, 5) Partial credit: (3 marks) Correct expression for cost, but no explanation of terms used. 11(e) Write down a formula to represent the total cost for the number of days gym B is used in July. State clearly the meaning of any letters you use in your formula. (5C) Slope of gym B: (0, 0), (10, 50) (x 1, y 1 ) (x 2, y 2 ) m B y2 y x2 x 50 0 10 0 50 10 5 or rise m B run 50 10 5 1 1 Cost of gym B c 5x where c cost x number of days gym used Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. writes down correct relevant formula for slope. High partial credit: (4 marks) Finds correct slope, but no expression or incorrect expression for cost. Correct expression for cost, but no explanation of terms used. 2017.1 J.18/20_MS 34/72 Page 34 of 71 exams

2017 JC Maths [HL] Paper 1 Q.11 (cont d.) 11(f) Use your formulae from parts (d) and (e) to justify your answer in part (b). (10C) gym A: c 20 + 2x gym B: c 5x 5x 20 + 2x 5x 2x 20 3x 20 x 20 3 6 666... or 2x c 20 ( 1) 5x c 0 ( 1) 2x c 20 5x + c 0 3x 20 20 x 3 6 666... c 5x 20 5( ) 3 100 3 33 333... point of intersection (6 666..., 33 333...) or c 20 + 2x 20 20 + 2( ) 3 60 40 + 3 3 100 3 33 333... Scale 10C (0, 4, 7, 10) Low partial credit: (4 marks) Some work of merit, e.g. equates 5x 20 + 2x and stops or continues incorrectly [Method ]. Sets up two simultaneous equations and multiplies equation(s) by appropriate constant(s) to facilitate the cancellation of x or c term [Method ]. High partial credit: (7 marks) Finds first variable (x or c) correctly, but fails to find second variable or finds incorrectly [Method and ]. Finds both variables (x and c) correctly with no work shown. Finds both variables (x and c) by trial and error, but does not verify into both equations. 2017.1 J.18/20_MS 35/72 Page 35 of 71 exams

2017 JC Maths [HL] Paper 1 Q.12 (Suggested maximum time: 10 minutes) (20) 12(a) Six functions are defined as follows: (10C) e(x) 4 3x x 2 f (x) 4 g(x) (x 2) 2 h(x) x 2 + x 6 j(x) 2 x k(x) 5 4 x The table below shows the sketches of all six functions. Write the name of the function into the box underneath the correct sketch for each of the functions. y y y x x x f (x) j(x) h(x) y y y x x x e(x) k(x) g(x) Scale 10C (0, 4, 7, 10) Low partial credit: (4 marks) One or two functions correctly identified. High partial credit: (7 marks) Three or four functions correctly identified. 2017.1 J.18/20_MS 36/72 Page 36 of 71 exams

2017 JC Maths [HL] Paper 1 Q.12 (cont d.) 12(b) Let f be the function f : x 2x 2 + 9, where x R. (i) Find the values of x for which f(x) 9x. (5C) f(x) 9x 2x 2 + 9 9x 2x 2 9x + 9 0 (2x 3)(x 3) 0 2x 3 0 2x 3 3 x 2 or x 3 0 x 3 Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. equates 2x 2 + 9 9x or 2x 2 9x + 9 0 and stops or continues incorrectly. High partial credit: (4 marks) Correctly solves quadratic equation, but fails to find roots or only finds one root. (ii) Hence, or otherwise, find the values of t for which f ( t 1 ) t 9. (5C) Let x t 1 1 t t 3 2 2 3 or 1 t t 3 1 3 ** Accept answers from part (b)(i) if not oversimplified. Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Any work of merit, e.g. writes down x t 1 or equivalent. High partial credit: (4 marks) Correct method and finishes correctly, but one minor error. Finds only one value of t. 2017.1 J.18/20_MS 37/72 Page 37 of 71 exams

Name/version: Printed: Whom: exams Checked: To: Ret d: Pre-Junior Certificate Examination, 2017 Updated: Name/version: Whom: Mathematics Complete (y/n): Whom: 2005 Print Stamp.doc Higher Level Paper 2 Marking Scheme (300 marks) Structure of the Marking Scheme Students responses are marked according to different scales, depending on the types of response anticipated. Scales labelled A divide students responses into two categories (correct and incorrect). Scales labelled B divide responses into three categories (correct, partially correct, and incorrect), and so on. These scales and the marks that they generate are summarised in the following table: Scale label A B C D No. of categories 2 3 4 5 5 mark scale 0, 5 0, 3, 5 0, 2, 4, 5 0, 2, 3, 4, 5 10 mark scale 0, 4, 7, 10 0, 4, 6, 8, 10 15 mark scale 0, 4, 8, 12, 15 A general descriptor of each point on each scale is given below. More specific directions in relation to interpreting the scales in the context of each question are given in the scheme, where necessary. Marking scales level descriptors A-scales (two categories) incorrect response (no credit) correct response (full credit) B-scales (three categories) response of no substantial merit (no credit) partially correct response (partial credit) correct response (full credit) C-scales (four categories) response of no substantial merit (no credit) response with some merit (low partial credit) almost correct response (high partial credit) correct response (full credit) D-scales (five categories) response of no substantial merit (no credit) response with some merit (low partial credit) response about half-right (middle partial credit) almost correct response (high partial credit) correct response (full credit) In certain cases, typically involving incorrect rounding, omission of units, a misreading that does not oversimplify the work or an arithmetical error that does not oversimplify the work, a mark that is one mark below the full-credit mark may also be awarded. Such cases are flagged with an asterisk. Thus, for example, scale 10C* indicates that 9 marks may be awarded. The * for units to be applied only if the student s answer is fully correct. The * to be applied once only within each section (a), (b), (c), etc. of all questions. The * penalty is not applied to currency solutions. Unless otherwise specified, accept correct answer with or without work shown. Accept students work in one part of a question for use in subsequent parts of the question, unless this oversimplifies the work involved. 2017.1 J.18/20_MS 38/72 Page 38 of 71 exams

Summary of Marks 2017 JC Maths (Higher Level, Paper 2) Q.1 (a) 5B (0, 3, 5) Q.7 (a) (i) 5B (0, 3, 5) (b) 5C* (0, 2, 4, 5) (ii) 5B (0, 3, 5) (c) 5B* (0, 3, 5) (iii) 5B (0, 3, 5) (e) 5B* (0, 3, 5) (iv) 5C* (0, 2, 4, 5) 20 (b) (i) 10C (0, 4, 7, 10) (ii) 5C (0, 2, 4, 5) (iii) 5C (0, 2, 4, 5) Q.2 (a) 5C* (0, 2, 4, 5) 40 (b) (i) 5C* (0, 2, 4, 5) (ii) 5C* (0, 2, 4, 5) (c) (i) 5B* (0, 3, 5) Q.8 (a) (i) 10D (0, 4, 6, 8, 10) (ii) 5C* (0, 2, 4, 5) (ii) 5B (0, 3, 5) (iii) 5B (0, 3, 5) (iii) 5C (0, 2, 4, 5) 30 (b) (i) 5B* (0, 3, 5) (ii) 5B* (0, 3, 5) 30 Q.3 (a) 10C (0, 4, 7, 10) (b) (i) (ii) 10C (0, 4, 7, 10) Q.9 15D (0, 4, 8, 12, 15) (iii) 15 (c) 5C (0, 2, 4, 5) 25 Q.10 (a) 5C* (0, 2, 4, 5) (b) 5C* (0, 2, 4, 5) Q.4 (a) 10C (0, 4, 7, 10) (c) 5C* (0, 2, 4, 5) (b) 5B (0, 3, 5) (d) 5C* (0, 2, 4, 5) (c) 5C* (0, 2, 4, 5) (e) 5C* (0, 2, 4, 5) 20 25 Q.5 (a) 5B (0, 3, 5) Q.11 (a) 5C (0, 2, 4, 5) (b) (i) 5B (0, 3, 5) (b) 5C (0, 2, 4, 5) (ii) 5C* (0, 2, 4, 5) (c) 5B (0, 3, 5) (c) (i) 10D* (0, 4, 6, 8, 10) (d) 5C (0, 2, 4, 5) (ii) 5C* (0, 2, 4, 5) 20 30 Q.6 (a) 5A* (0, 5) (b) 5B* (0, 3, 5) (c) 5C* (0, 2, 4, 5) (d) 5A (0, 5) (e) 5C (0, 2, 4, 5) (f) 10C* (0, 4, 7, 10) (g) 5D (0, 2, 3, 4, 5) (h) 5B (0, 3, 5) 45 Assumptions about these marking schemes on the basis of past SEC marking schemes should be avoided. While the underlying assessment principles remain the same, the exact details of the marking of a particular type of question may vary from a similar question asked by the SEC in previous years in accordance with the contribution of that question to the overall examination in the current year. In setting these marking schemes, we have strived to determine how best to ensure the fair and accurate assessment of students work and to ensure consistency in the standard of assessment from year to year. Therefore, aspects of the structure, detail and application of the marking schemes for these examinations are subject to change from past SEC marking schemes and from one year to the next without notice. 2017.1 J.18/20_MS 39/72 Page 39 of 71 exams

exams Pre-Junior Certificate Examination, 2017 Mathematics Higher Level Paper 2 Marking Scheme (300 marks) General Instructions 1. There are 11 questions on this examination paper. Answer all questions. 2. Questions do not necessarily carry equal marks. 3. Marks will be lost if all necessary work is not clearly shown. 4. Answers should include the appropriate units of measurement, where relevant. 5. Answers should be given in simplest form, where relevant. Q.1 (Suggested maximum time: 10 minutes) (20) ** Deduct 1 mark off correct answer only if final answer is not rounded or is incorrectly rounded or for the omission of or incorrect use of units in part(s) of question asterisked - this deduction should be applied only once per section (a), (b), (c), etc. of the question. The pie chart opposite shows the sources of energy production in the EU in 2016. 25% 10% x Oil Coal Gas 2x 20% Nuclear Renewables 1(a) What type of data is displayed? Put a tick ( ) in the correct box below. (5B) Numerical Discrete Numerical Continuous Categorical Nominal Categorical Ordinal Scale 5B (0, 3, 5) Partial credit: (3 marks) Ticks Categorical Ordinal box. 2017.1 J.18/20_MS 40/72 Page 40 of 71 exams

2017 JC Maths [HL] Paper 2 Q.1 (cont d.) 1(b) What percentage of energy was produced from coal in 2016? (5C) % energy produced from coal 25 + 10 + x + 20 + 2x 100 3x + 55 100 3x 100 55 45 x 15% Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. writes down any three of 25, x, 10, 20 or 2x. Finds 25 + 10 + x + 20 + 2x 100 and stops or continues incorrectly. High partial credit: (4 marks) Finds 3x + 55 100, but fails to finish or finishes incorrectly. * No deduction applied for the omission of or incorrect use of symbol ( % ). 1(c) Calculate the measure of the angle in the pie chart that represents the percentage of energy produced from nuclear in 2016. x 15 2x 30 angle 360 30% 30 360 100 108 (5B*) Scale 5B (0, 3, 5) Partial credit: (3 marks) Some work of merit, e.g. writes down 2x 30. Writes down angle 360 30% 30 or 360, but fails to finish 100 or finishes incorrectly. * Deduct 1 mark off correct answer only for the omission of or incorrect use of units ( ) - apply only once to each section (a), (b), (c), etc. of question. 1(d) 75 tonnes of energy was produced from renewables in 2016. How much energy was produced from oil in 2016? 25% 75 tonnes 1% 75 25 3 tonnes 10% 3 10% 30 tonnes (5B*) Scale 5B* (0, 3, 5) Partial credit: (3 marks) Some work of merit, e.g. equates 25% to 75 tonnes and stops. Finds 1% 3 tonnes, but fails to finish or finishes incorrectly. * Deduct 1 mark off correct answer only for the omission of or incorrect use of units ( tonnes ) - apply only once to each section (a), (b), (c), etc. of question. 2017.1 J.18/20_MS 41/72 Page 41 of 71 exams

2017 JC Maths [HL] Paper 2 Q.2 (Suggested maximum time: 15 minutes) (30) The Great Pyramid of Giza (also known as the Pyramid of Khufu) on the banks of the River Nile in Egypt is the oldest of the Seven Wonders of the Ancient World. The pyramid consists of four equilateral triangular sides sitting on a square base. 2(a) The base of the pyramid has an area of 52,900 m 2. Find the length of one side of its base. (5C*) Area of square base l w l 2 l 2 52,900 l 52, 900 230 m Scale 5C* (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. writes down l w or l 2 and stops. Finds l 2 52,900, but fails to finish or finishes incorrectly. High partial credit: (4 marks) Finds l 52, 900, but fails to finish or finishes incorrectly. * Deduct 1 mark off correct answer only for the omission of or incorrect use of units ( m ) - apply only once to each section (a), (b), (c), etc. of question. 2(b) When it was built, the triangular faces of the pyramid were covered in polished limestone. (i) Find the height of one of the triangular faces. Give your answer correct to the nearest metre. Consider the perpendicular height of a single triangular face: Using Pythagoras theorem side 2 2 1 base 2 + h 2 (230) 2 2 1 (230) 2 + h 2 h 2 230 2 115 2 52,900 13,225 39,675 h 39, 675 199 185842... 199 m (5C*) Scale 5C* (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, writes down correct formula for Pythagoras theorem and stops. Some correct substitution into formula for Pythagoras theorem (not stated). High partial credit: (4 marks) Fully correct substitution into formula for Pythagoras theorem with some manipulation, e.g. h 2 230 2 115 2, 52,900 13,225 or 39,675, but fails to finish or finishes incorrectly. * Deduct 1 mark off correct answer only if not rounded or incorrectly rounded - apply only once per each section (a), (b), (c), etc. of question. * No deduction applied for the omission of or incorrect use of units ( m ) as units are mentioned in the question. 2017.1 J.18/20_MS 42/72 Page 42 of 71 exams

2017 JC Maths [HL] Paper 2 Q.2 (cont d.) 2(b) (ii) Find the surface area of the polished limestone that would have been required to cover the pyramid. (5C*) Surface area 4 Area of triangular side 1 4 base height 2 4 2 1 (230) 199 4 115 199 91,540 m 2 ** Accept students answers from parts (a) and (b)(i) if not oversimplified. Scale 5C* (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. writes down correct formula for area of a triangle and stops. Some correct substitution into correct area formula (not stated). High partial credit: (4 marks) Fully correct substitution into correct area formula, but fails to finish or finishes incorrectly. Finds correct area of one side of pyramid [ans. 22,885 m 2 ] and stops. * Deduct 1 mark off correct answer only for the omission of or incorrect use of units ( m 2 ) - apply only once to each section (a), (b), (c), etc. of question. 2(c) There are two smaller pyramids, Khafre and Menkaure, near the Great Pyramid. The triangular faces of all three pyramids are similar, as shown in the diagram below. (i) Great Pyramid (Khufu) Khafre Menkaure Find the percentage decrease in the length of a side of Khafre, whose sides measure 216 m compared to that of the Great Pyramid (Khufu). Give your answer correct to one decimal place. Decrease in length of side 230 216 14 m % Decrease 14 100 230 1 6 086956... 6 1% ** Accept students answers from part (a) if not oversimplified. (5B*) Scale 5B* (0, 3, 5) Partial credit: (3 marks) Some work of merit, e.g. writes down correct formula % Decrease and stops. Finds correct decrease in length of side. * Deduct 1 mark off correct answer only if not rounded or incorrectly rounded - apply only once per each section (a), (b), (c), etc. of question. * No deduction applied for the omission of or incorrect use of symbol ( % ). 2017.1 J.18/20_MS 43/72 Page 43 of 71 exams

2017 JC Maths [HL] Paper 2 Q.2 (cont d.) 2(c) (ii) The percentage decrease in the length of a side of Menkaure compared to that of Khafre is 49 5%. Calculate the length of the perimeter of the base of Menkaure. Give your answer correct to the nearest metre. (5C*) Decrease in length of side 49 5% of 216 49 5 216 100 106 92 m Length of side of Menkaure 216 106 92 109 08 m Perimeter of the base of Menkaure 109 08 4 436 32 436 m Scale 5C* (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. writes down 49 5% of 216 or equivalent and stops. Finds correct decrease in length of side. High partial credit: (4 marks) Finds correct length of side, but fails to finish or finishes incorrectly. * Deduct 1 mark off correct answer only if not rounded or incorrectly rounded - apply only once per each section (a), (b), (c), etc. of question. * No deduction applied for the omission of or incorrect use of units ( m ) as units are mentioned in the question. (iii) Does the surface area of the faces of the pyramids decrease at the same rate? Give a reason for your answer. Answer no (5B) Reason Any 1: area is two dimensional - each dimension decreases by a scale factor, k - therefore area decreases by a factor of k 2 // area is two dimensional - each dimension decreases at calculated rate, therefore area decreases at increased rate // etc. ** Accept other appropriate answers. Scale 5B (0, 3, 5) Partial credit: (3 marks) Some work of merit, e.g. reference to scale factor, k or k 2 and stops. Correct answer, but no reason given. 2017.1 J.18/20_MS 44/72 Page 44 of 71 exams

2017 JC Maths [HL] Paper 2 Notes: 2017.1 J.18/20_MS 45/72 Page 45 of 71 exams

2017 JC Maths [HL] Paper 2 Q.3 (Suggested maximum time: 10 minutes) (25) 3(a) The co-ordinate diagram shows the lines l, m, n, o and p. The table shows the equation of each line. Write the letters l, m, n, o and p into the table to match each line to its equation. (10C) y l m Equation Line x y + 3 0 m n x 2y y 5 0 p n x y 2x + 5 l p o 2x + y 9 o Scale 10C (0, 4, 7, 10) Low partial credit: (4 marks) One line correctly labelled. High partial credit: (7 marks) Two or three lines correctly labelled. 3(b) Complete the following sentences. Write one of the letters l, m, n, o and p in each box. (10C) (i) Lines p and o have negative slopes. (ii) Lines l and p are perpendicular to each other. (iii) Lines l and n intersect at the point (0, 5). Scale 10C (0, 4, 7, 10) Low partial credit: (4 marks) One correct answer. High partial credit: (7 marks) Two correct answers. 2017.1 J.18/20_MS 46/72 Page 46 of 71 exams

2017 JC Maths [HL] Paper 2 Q.3 (cont d.) 3(c) The lines m, n and o intersect at the same point. Find the co-ordinates of this point of intersection. n: y 5 0 y 5 m: x y + 3 0 n m: x (5) + 3 0 x 2 0 x 2 or n m (2, 5) n m o (2, 5) n: y 5 0 y 5 o: 2x + y 9 n o: 2x + (5) 9 2x 9 5 4 x 2 n o (2, 5) n m o (2, 5) ** Accept students answers from part (a) if not oversimplified. (5C) Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Any work of merit, e.g. multiplying equation(s) by appropriate constant to facilitate cancellation of x or y term. Finds either variable (x or y) correctly by trial and error, but fails to verify in both equations or verifies incorrectly. Finds first variable (y) correctly and stops. High partial credit: (4 marks) Finds first variable (y) correctly and substitutes value into second equation correctly, e.g. m: x (5) + 3 0 or o: 2x + (5) 9, but fails to find or finds incorrect value of second variable (x). Finds both variables (x and y) correctly, but no work shown. Finds both variables (x and y) by graphical means. Finds both variables (x and y) by trial and error, but does not verify in all three equations. 2017.1 J.18/20_MS 47/72 Page 47 of 71 exams

8cm 2017 JC Maths [HL] Paper 2 Q.4 (Suggested maximum time: 10 minutes) (20) ** Deduct 1 mark off correct answer only if final answer is not rounded or is incorrectly rounded or for the omission of or incorrect use of units in part(s) of question asterisked - this deduction should be applied only once per section (a), (b), (c), etc. of the question. 4(a) Construct a right-angled triangle ABC, where: AB 6 cm CAB 90 BC 8 cm. (10C) C A 6cm 41 B ** Allow a tolerance of ±0 2 cm or ±2. Scale 10C (0, 4, 7, 10) Low partial credit: (4 marks) One side or one angle (90 ) correctly drawn. Sketch drawn with given measurements shown. High partial credit: (7 marks) Triangle correctly drawn, but unlabelled or incorrectly labelled. Triangle correctly drawn, but outside allowable tolerances. 4(b) On your diagram, measure the angle ABC. Give your answer correct to the nearest degree. From diagram ABC 41 (± 2 ) ** Accept students triangle from part (a) if not oversimplified. (5B) Scale 5B (0, 3, 5) Partial credit: (3 marks) Wrong angle correctly measured. Triangle incorrect and unlabelled, but one angle correctly measured. * No deduction applied for the omission of or incorrect use of units ( ) as units are mentioned in the question. 2017.1 J.18/20_MS 48/72 Page 48 of 71 exams

2017 JC Maths [HL] Paper 2 Q.4 (cont d.) 4(c) Using trigonometry, verify your answer to part (b) above. (5C*) cos ABC Adj Hyp AB BC 6 8 0 75 ABC cos 1 (0 75) 41 409622... 41 ** Accept students answers from parts (a) and (b) if not oversimplified. Scale 5C* (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. writes down correct trigonometric ratio (cos). Some correct substitution into correct trigonometric ratio (cos). Finds cos ABC and stops [ans. 0 75]. Using answer from part (b), finds correct value of cos 41 [ans. 0 754709...] High partial credit: (4 marks) Correct substitution into trigonometric ratio and correctly manipulated, e.g. ABC cos 1 (0 75), but fails to finish or finishes incorrectly. * Deduct 1 mark off correct answer only if not rounded or incorrectly rounded or for the omission of or incorrect use of units ( ) - apply only once per each section (a), (b), (c), etc. of question. 2017.1 J.18/20_MS 49/72 Page 49 of 71 exams

2017 JC Maths [HL] Paper 2 Q.5 (Suggested maximum time: 15 minutes) (30) ** Deduct 1 mark off correct answer only if final answer is not rounded or is incorrectly rounded or for the omission of or incorrect use of units in part(s) of question asterisked - this deduction should be applied only once per section (a), (b), (c), etc. of the question. Landslides occur when the stability of a slope of ground is compromised. This can happen after there has been heavy rainfall. The sketch shows the incline (slope) of a section of ground on the side of a mountain prior to a landslide. A 14 m Ground line 5(a) Find the incline (slope) of the land in the sketch between the points A and B. Incline (slope) Rise Run 14 10 10 m B (5B) 1 4 Scale 5B (0, 3, 5) Partial credit: (3 marks) Some work of merit, e.g. writes down Rise Rise slope or and stops. Run Run 14 Gives or 1 4 as final answer. 10 5(b) The diagram below shows a sketch of the incline after a landslide. EDC 36. C Ground line E 23 m 36º D 2017.1 J.18/20_MS 50/72 Page 50 of 71 exams

2017 JC Maths [HL] Paper 2 Q.5 (cont d.) 5(b) (i) Find the incline (slope) of the ground shown in the sketch between the points C and D. (5B) Incline (slope) tan EDC tan36 0 726542... ** Accept students answers that are appropriately round. Scale 5B (0, 3, 5) Partial credit: (3 marks) Some work of merit, e.g. writes down tan EDC or tan 36 and stops. (ii) Hence, find CE, giving your answer correct to the nearest metre. (5C*) tan EDC Opp Adj CE DE 0 726542...... answer from part (i) 0 726542... CE 23 CE 23 0 726542... 16 710478... 17 m ** Accept students answers from part (b)(i) if not oversimplified. Scale 5C* (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. writes down correct trigonometric ratio (tan). Some correct substitution into correct trigonometric ratio (tan). CE Finds 0 726542... and stops. 23 High partial credit: (4 marks) Finds CE 23 0 726542..., but fails to finish or finishes incorrectly. * Deduct 1 mark off correct answer only if not rounded or incorrectly rounded - apply only once per each section (a), (b), (c), etc. of question. * No deduction applied for the omission of or incorrect use of units ( m ) as units are mentioned in the question. 2017.1 J.18/20_MS 51/72 Page 51 of 71 exams

2017 JC Maths [HL] Paper 2 Q.5 (cont d.) 5(c) The National Roads Authority has decided to construct a concrete retaining wall, 80 m long, to protect a section of motorway from future landslides. The cross-section of the retaining wall, of height h, is shown. (i) Find h, giving your answer correct to the nearest metre. (10D*) cos A Adj 0 5 m Hyp 1 5 0 5 3 16 1 3 16 0 316455... A cos 1 (0 316455...) 71 551284... 71 6 h m 3 16 m tan A Opp Adj tan 71 551284... h 1 h 1 tan 71 551284... 2 997599... 3 m 1 5 m A Scale 10D* (0, 4, 6, 8, 10) Low partial credit: (4 marks) Some work of merit, e.g. writes down correct trigonometric ratio (cos or tan). Some correct substitution into correct trigonometric ratio (cos or tan). 1 Finds cos A or 0 316455... 3 16 and stops. Mid partial credit: (6 marks) Finds A cos 1 1 or 3 16 cos 1 0 316455... and stops or continues incorrectly. High partial credit: (8 marks) Finds A 71 551284... correctly (accept rounded value), but fails to finish or finishes incorrectly. h Finds tan 71 551284... or h, but 1 fails to finish or finishes incorrectly. Finds h 1 tan 71 551284... or similar, but fails to finish or finishes incorrectly. * Deduct 1 mark off correct answer only if not rounded or incorrectly rounded - apply only once per each section (a), (b), (c), etc. of question. * No deduction applied for the omission of or incorrect use of units ( m ) as units are mentioned in the question. 2017.1 J.18/20_MS 52/72 Page 52 of 71 exams

2017 JC Maths [HL] Paper 2 Q.5 (cont d.) 5(c) (cont d.) (ii) Given that the wall is 80 m long, find the volume of concrete required to construct the wall. (5C*) Area of cross section of retaining wall Area of rectangle + area of triangle 1 l w + base height 2 3 0 5 + 2 1 1 5 0 5 3 3 1 + 1 3 2 2 3 3 + 2 2 6 2 3 m 2 Volume of concrete required Area of cross section 80 3 80 240 m 3 ** Accept students answers from part (c)(i) if not oversimplified. Scale 5C* (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. writes down correct area formula from Tables. Some correct substitution into correct area formula (not stated). High partial credit: (4 marks) Finds correct area of cross section, but fails to finish or finishes incorrectly. * Deduct 1 mark off correct answer only for the omission of or incorrect use of units ( m 3 ) - apply only once per each section (a), (b), (c), etc. of question. 2017.1 J.18/20_MS 53/72 Page 53 of 71 exams

2017 JC Maths [HL] Paper 2 Q.6 (Suggested maximum time: 25 minutes) (45) Transition year students conducted a survey for their Green Schools project. They handed out a questionnaire to a sample group of students in their school, which asked how they travel to school and how long it takes. The length of time (in minutes) taken to travel to school by the students surveyed is displayed in the back-to-back stem-and-leaf plot shown. Junior School Senior School 8 6 0 3 5 7 8 9 8 6 5 4 3 2 1 1 4 5 5 5 7 9 8 6 4 3 1 2 2 3 4 5 8 9 3 3 4 6 7 4 0 5 9 Key: 2 2 means 22 minutes 6(a) What is the mode of the Senior School data? (5A*) Mode 15 minutes Scale 5A* (0, 5) Hit or miss. * Deduct 1 mark off correct answer only for the omission of or incorrect use of units ( mins. ) - apply only once per each section (a), (b), (c), etc. of question. 6(b) Find the median time taken to travel to school by Junior School students. (5B*) Median 6, 8, 12, 13, 14, 15, 16, 18, 21, 23, 24, 26, 28 Median 16 minutes Scale 5B* (0, 3, 5) Partial credit: (3 marks) Indication of middle time. * Accept correct answers with or without work shown. * Deduct 1 mark off correct answer only for the omission of or incorrect use of units ( mins. ) - apply only once per each section (a), (b), (c), etc. of question. 2017.1 J.18/20_MS 54/72 Page 54 of 71 exams

2017 JC Maths [HL] Paper 2 Q.6 (cont d.) 6(c) Find the interquartile range of the Junior School data. (5C*) Graphical means 6, 8, 12, 13, 14, 15, 16, 18, 21, 23, 24, 26, 28 12 + 13 Lower Quartile, Q 1 2 12 5 minutes 23 + 24 Upper Quartile, Q 3 2 23 5 minutes Interquartile range Q 3 Q 1 23 5 12 5 11 minutes... Answer 1 or Numerical means 6, 8, 12, 13, 14, 15, 16, 18, 21, 23, 24, 26, 28 n number of results in dataset Lower quartile, Q 1 n k 4 13 4 3 25 4 Q 1 4th value in list 13 Upper quartile, Q 3 k 3n 4 3(13) 4 9 75 10 Q 3 10th value in list 23 Interquartile range Q 3 Q 1 23 10 10 minutes... Answer 2 Scale 5C* (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. writes down definition of interquartile range and stops. Some relevant work towards finding upper and/or lower quartile by either method. High partial credit: (4 marks) Finds correct value of upper or lower quartile by either method, but fails to finish or finishes incorrectly. Finds correct value of interquartile range, but no work shown. * Deduct 1 mark off correct answer only for the omission of or incorrect use of units ( mins. ) - apply only once per each section (a), (b), (c), etc. of question. 2017.1 J.18/20_MS 55/72 Page 55 of 71 exams

2017 JC Maths [HL] Paper 2 Q.6 (cont d.) 6(d) What time in the stem-and-leaf-plot could best be described as an outlier? (5A) Outlier 59 minutes Scale 5A (0, 5) Hit or miss. 6(e) Fill in the grouped frequency table below using the data from the stem-and-leaf plot. (5C) Time (minutes) 0 10 10 20 20 30 30 40 40 50 50 60 Frequency 7 13 11 4 1 1 Note: 20 30 means at least 20 minutes but less than 30 minutes, etc. Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) One or two correct entries. High partial credit: (4 marks) Three, four or five correct entries. 6(f) Use mid-interval values of the data in the grouped frequency table to estimate the mean time taken to travel to school by the students in the sample. Give your answer correct to the nearest minute. Mean (5 7) + (15 13) + (25 11) + (35 4) + (45 7 + 13 + 11 + 4 + 1 + 1 35 + 195 + 275 + 140 + 45 + 55 745 37 20 135135... 20 minutes 37 (10C*) 1) + (55 1) Scale 10C* (0, 4, 7, 10) Low partial credit: (4 marks) Some work of merit, e.g. shows indication of division by 37. Finds one correct mid-interval value. High partial credit: (7 marks) Finds correct numerator, i.e. 745 or (5 7) + (15 13) + (25 11) + (35 4) +(45 1) + (55 1). Finds incorrect mean using consistent incorrect mid-interval values. 745 Finds, but fails to finish or finishes 37 incorrectly. * Deduct 1 mark off correct answer only if not rounded or incorrectly rounded - apply only once per each section (a), (b), (c), etc. of question. * No deduction applied for the omission of or incorrect use of units ( minutes ) as units are mentioned in the question. 2017.1 J.18/20_MS 56/72 Page 56 of 71 exams

2017 JC Maths [HL] Paper 2 Q.6 (cont d.) 6(g) Display the data from the grouped frequency table on a histogram. (5D) 14 12 10 Frequency 8 6 4 2 0 0 10 20 30 40 50 60 Time (minutes) Scale 5D (0, 2, 3, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. draws correctly labelled axes (with no bars inserted). One or two bars correct (each of consistent width). Mid partial credit: (3 marks) Three or four bars correct (each of consistent width). Bar chart correctly drawn, but both axes not numerated or labelled. High partial credit: (4 marks) Five bars correct (each of consistent width). All six bars correct (each of consistent width), but both axes not labelled. Bar chart correctly drawn. 6(h) The students felt that the wording of Q1 in the survey could have been improved, because some students didn t take the survey seriously when filling in their answers as shown. Green Schools Project Survey Q1: How did you travel to school this morning? by Jet Pack Q2: How long did it take you to get to school this morning? 25 minutes Rewrite Q1 of the survey in a more suitable form. (5B) Any 1: How do you travel to school? Walk Bus Cycle Car Other // How did you travel to school this morning? (Tick one) Walk Bus Cycle Car Other // etc. ** Accept other appropriate answers. Scale 5B (0, 3, 5) Partial credit: (3 marks) Question written in an unsuitable form, but with deficiencies. 2017.1 J.18/20_MS 57/72 Page 57 of 71 exams

2017 JC Maths [HL] Paper 2 Q.7 (Suggested maximum time: 20 minutes) (40) 7(a) An electric appliance store sells three different brands of television, all in various screen sizes and features, as shown. Televisions Brand Features Screen Size Zony Smart TV 24 inches Hamsung Full HD 32 inches Fillips 40 inches 49 inches 55 inches 65 inches (i) Calculate how many different televisions can be purchased from this store. (5B) #options 3 2 6 36 Scale 5B (0, 3, 5) Partial credit: (3 marks) Some work of merit, e.g. mention of the Fundamental Principle of Counting. Writes a list of possible options. Finds 3 2 6, but fails to finish or finishes incorrectly. Uses addition instead of multiplication, i.e. # options 3 + 2 + 6 11. (ii) A customer purchased a television from the store. Assuming all brands and types of televisions are equally likely to be purchased, find the probability that this person purchased a Zony 32 inch smart TV. (5B) #Zony, 32 inch, Smart TV 1 #All TV types 36 1 P(Zony, 32 inch, Smart TV) 36 or 0 027777... Scale 5B (0, 3, 5) Partial credit: (3 marks) Correct numerator or denominator. (iii) A customer is chosen at random from those who bought a Fillips TV. Find the probability that this person purchased a smart TV. #All Fillips TV 1 2 6 12 #Fillips, Smart TV 1 6 6 P(Smart TV) 6 12 1 2 or 0 5 (5B) Scale 5B (0, 3, 5) Partial credit: (3 marks) Some work of merit, e.g. identifies 6 or 12 and stops or continues. 2017.1 J.18/20_MS 58/72 Page 58 of 71 exams

2017 JC Maths [HL] Paper 2 Q.7 (cont d.) 7(a) (iv) Television screen sizes are measured according to the length of their diagonal. A television on special offer has a screen 24 inches in height and 32 inches in width. Diagonal 24 inches Write down the screen size of the television. Give a reason for your answer. Answer 40 inches 32 inches (5C*) Reason according to Pythagoras theorem: d 2 32 2 + 24 2 1,024 + 576 1,600 d 1, 600 40 inches Scale 5C* (0, 2, 4, 5) Low partial credit: (2 marks) Correct answer, but no reason or incorrect reason given. High partial credit: (4 marks) Correct answer, but reason given insufficient or incomplete. * Deduct 1 mark off correct answer only for the omission of or incorrect use of units ( inches ) - apply only once per each section (a), (b), (c), etc. of question. 2017.1 J.18/20_MS 59/72 Page 59 of 71 exams

2017 JC Maths [HL] Paper 2 Q.7 (cont d.) 7(b) The English-language edition of Scrabble contains 100 tiles, 98 of which have a single letter printed on them. There are 42 vowels and 56 consonants. To make a game more difficult, the two blank tiles are removed. The board game starts with each player picking tiles from a bag containing the remaining 98 tiles. When a tile is picked from the bag, it is not replaced. (i) The first player picks two tiles from the bag. Complete the tree diagram below to show all possible outcomes and fill in the boxes in the diagram which represent the probability that vowels or consonants will be picked. (10C) Letters 42 Vowel 98 56 98 Consonant 41 97 56 97 42 97 55 97 Vowel Consonant Vowel Consonant Scale 10C (0, 4, 7, 10) Low partial credit: (4 marks) Between one and four outcomes or probabilities correct. High partial credit: (7 marks) Between five and eight outcomes or probabilities. (ii) Find the probability that the tiles selected are either both consonants or both vowels. (5C) P(consonant, consonant) 56 55 98 97 3,080 9,506 220 679 P(vowel, vowel) 42 41 98 97 1,722 9,506 123 679 or 0 324005... P(consonant, consonant or vowel, vowel) 220 123 + 679 679 343 49 or 679 97 or 0 505154... 2017.1 J.18/20_MS 60/72 Page 60 of 71 exams

2017 JC Maths [HL] Paper 2 Q.7 (cont d.) 7(b) (ii) (cont d.) ** Accept students answers from part (b)(i) if not oversimplified. Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. identifies at least one correct pair of probabilities 56 55 42 41 and or and, but does not 98 97 98 97 attempt to multiply them. Uses addition and finishes correctly 56 55 773 [ans. + or equivalent]. 98 97 679 Mid partial credit: (3 marks) Finds one probability correctly, 3,080 220 i.e. P(c, c) or 9,506 679 1,722 123 or P(v, v) or, but fails 9,506 679 to finish or finishes incorrectly. High partial credit: (4 marks) Finds both probability correctly, i.e. P(c, c) and P(v, v), but fails to finish or finishes incorrectly. (iii) Find the probability that at least one of the tiles selected is a vowel. (5C) P(at least 1 vowel) 1 P(consonant, consonant) 220 1 679 Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) High partial credit: (4 marks) 459 or 0 675994... 679 220 Some work of merit, e.g. identifies 679 and stops or continues. 220 Finds 1, but fails to finish 679 or finishes incorrectly. 2017.1 J.18/20_MS 61/72 Page 61 of 71 exams

2017 JC Maths [HL] Paper 2 Q.8 (Suggested maximum time: 10 minutes) (30) ** Deduct 1 mark off correct answer only if final answer is not rounded or is incorrectly rounded or for the omission of or incorrect use of units in part(s) of question asterisked - this deduction should be applied only once per section (a), (b), (c), etc. of the question. 8(a) ABCD is a parallelogram with the side AB extended to E to form an external angle. D 29 54 C (i) Find the measure of each of the angles marked x, y and z. Give a reason for your answer in each case. E z A y x B (10D) Angle: Reason: x 113 x + 29 + 40 180... the sum of the three angles in a triangle is equal to 180 x 180 (29 + 54 ) 180 83 97 y 29 alternate angles as [ AB ] is parallel to [ DC ] z 126 z + 54 180... opposite angles in z 180 54 a parallelogram are equal 126 and EB is a straight line / straight angle 180 or z x + y... corresponding angles as 97 + 29 [ AD ] is parallel to [ BC ] 126 Scale 10D (0, 4, 6, 8, 10) Low partial credit: (4 marks) Some work of merit, e.g. identifies one correct angle or reason. Mid partial credit: (6 marks) Finds two correct angles, but no reason(s) or incorrect reason(s) given. Gives two correct reasons, but angle(s) not evaluated or evaluated incorrectly. High partial credit: (8 marks) Finds two correct angles and correct reasons given. Finds three correct angles, but no reason(s) or incorrect reason(s) given. (ii) What type of triangle does DBC represent? Give a reason for your answer. (5B) Answer scalene (triangle) Reason all three internal angles are unequal and hence the triangle has three unequal sided // etc. ** Accept other appropriate material. Scale 5B (0, 3, 5) Partial credit: (3 marks) Correct answer, but no reason or incorrect reason given. 2017.1 J.18/20_MS 62/72 Page 62 of 71 exams

2017 JC Maths [HL] Paper 2 Q.8 (cont d.) 8(a) (iii) State whether triangles DBC and ABD are congruent. Give a reason for your answer. (5C) Answer triangles DBC and ABD are congruent Reason SSS: DB DB, AD BC, AB DC, // or SAS: DB DB, ABD CDB, AB DC or AD BC, BDA DBC, DB DB or AB DC, DAB BCD, AD BC // or ASA: ABD CDB, DB DB, BDA DBC or DAB BCD, AD BC, BDA DBC or DAB BCD, AB DC, ABD CDB // etc. ** Accept other appropriate answers. Scale 5C (0, 2, 4, 5) Low partial credit: (2 marks) Correct answer, but no reason or incorrect reason given. High partial credit: (4 marks) Correct answer, but reason given insufficient or incomplete. S 8(b) P, Q, R and S are points on a circle with centre O. ROP 115, as shown. P O 115 (i) Find RSP. Q R (5B*) RSP 1 ROP 2 1 (115 ) 2 57 5 Scale 5B* (0, 3, 5) Partial credit: (3 marks) Some work of merit, e.g. writes down that the angle at the centre is equal to twice the angle at the circumference. Writes down ROP 2 RSP or similar. Finds RSP 2 115 230. * Deduct 1 mark off correct answer only for the omission of or incorrect use of units ( ) - apply only once per each section (a), (b), (c), etc. of question. 2017.1 J.18/20_MS 63/72 Page 63 of 71 exams

2017 JC Maths [HL] Paper 2 Q.8 (cont d.) 8(b) (cont d.) (ii) Find PQR. (5B*) PQR 1 [360 ROP ] 2 1 (360 115 ) 2 1 (245 ) 2 122 5 or PQR + RSP 180 PQR 180 RSP 180 57 5 122 5 ** Accept students answers for RSP from part (b)(i) if not oversimplified. Scale 5B* (0, 3, 5) Low partial credit: (3 marks) Some work of merit, e.g. writes down that in a cyclic quadrilateral opposite angles sum to 180. Finds 245 and stops [method ]. Finds PQR 180 115 65 [method ]. * Deduct 1 mark off correct answer only for the omission of or incorrect use of units ( ) - apply only once per each section (a), (b), (c), etc. of question. 2017.1 J.18/20_MS 64/72 Page 64 of 71 exams

2017 JC Maths [HL] Paper 2 Q.9 (Suggested maximum time: 5 minutes) (15) Prove that the angle at the centre of a circle standing on a given arc is twice the angle at any point of the circle standing on the same arc. (15D) Given Circle, with centre O, containing points A, B and C. A To prove COB 2 CAB O Construction Join A to O and extend to D. Proof B D C DOB OAB + OBA... exterior angle equal to the sum of 2 opposite interior angles OA OB... both radius of the circle OAB OBA... base angles of isosceles triangle are equal DOB OAB + OAB 2 OAB similarly COD 2 CAO COB DOB + COD 2 OAB + 2 CAO 2( OAB + CAO ) 2 CAB Some steps may be indicated on the diagram. Scale 15D (0, 4, 8, 12, 15) Low partial credit: (4 marks) Some work of merit, e.g. draws correct diagram. Middle partial credit: (8 marks) Diagram, Given, To Prove and Construction only; or More than one step missing in proof. High partial credit: (12 marks) One step missing in proof; or fully correct but with no reason given; or gets as far as step or equivalent. 2017.1 J.18/20_MS 65/72 Page 65 of 71 exams

2017 JC Maths [HL] Paper 2 Q.10 (Suggested maximum time: 10 minutes) (25) ** Deduct 1 mark off correct answer only if final answer is not rounded or is incorrectly rounded or for the omission of or incorrect use of units in part(s) of question asterisked - this deduction should be applied only once per section (a), (b), (c), etc. of the question. The diagram below is a scale drawing of a dome which is in the shape of a hemisphere. The figure of a man, who is 1 8 m tall, standing beside the dome, allows the scale of the drawing to be estimated. 10(a) Estimate the radius of the dome. (5C*) Scale From the diagram: Height of man 2 grid units 2 grid units 1 8 m 1 grid unit 1 8 2 0 9 m Radius From the diagram: Height of dome 11 grid units 11 0 9 m 9 9 m Radius 9 9 m Scale 5C* (0, 2, 4, 5) Low partial credit: (2 marks) Some work of merit, e.g. equates height of man 2 grid units or height of dome 11 grid units. Writes down height of dome radius of hemisphere. Finds correct scale. High partial credit: (4 marks) Uses correct scale to determine height / radius of dome, but error(s) in finding radius. Finds correct diameter of dome [ans. 19 8m]. * Deduct 1 mark off correct answer only for the omission of or incorrect use of units ( m ) - apply only once per each section (a), (b), (c), etc. of question. 2017.1 J.18/20_MS 66/72 Page 66 of 71 exams