Mathematics (Project Maths Phase 2) Higher Level. Marking Scheme. Pre-Junior Certificate Examination, Paper 1 Pg. 2. Paper 2 Pg. 29 J.

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1 J.18/0 Pre-Junior Certificate Examination, 014 Mathematics (Project Maths Phase ) Higher Level Marking Scheme Paper 1 Pg. Paper Pg. 9 Page 1 of 56

2 exams Pre-Junior Certificate Examination, 014 Mathematics (Project Maths Phase ) Higher Level Paper 1 Marking Scheme (300 marks) Structure of the Marking Scheme Student responses are marked according to different scales, depending on the types of response anticipated. Scales labelled A divide student responses into two categories (correct and incorrect). Scales labelled B divide responses into three categories (correct, partially correct, and incorrect), and so on. The scales and the marks that they generate are summarised in this table: Scale label A B C D No of categories mark scale 0, 5 0, 3, 5 0,, 3, 5 0,, 3, 4, 5 10 mark scale 0, 10 0, 5, 10 0, 3, 7, 10 0, 5, 8, 9, mark scale 0, 15 0, 10, 15 0, 10, 1,15 0, 10, 1, 14, 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 (four categories) response of no substantial merit (no credit) response with some merit (low partial credit) response abut half-right (partial credit) almost correct response (high partial credit) correct response (full credit) In certain cases, typically involving incorrect rounding or omission of units, 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. However, it is important to note that this reduction only applies under the following criteria: The * for units to be applied only if the student s answer is fully correct. The * to be applied once only per question. The * penalty is not applied to currency solutions. 014 J.18/0_MS /56 Page of 56 exams

3 Summary of Marks (Paper 1) Q.1 (a) (i) 5C (0,, 3, 5) Q.8 (i) 10C (0, 3, 7, 10) (ii) 5C (0,, 3, 5) (ii) 5C* (0,, 3, 5) (b) (i) 5D (0,, 3, 4, 5) (iii) 5C* (0,, 3, 5) (ii) 5C (0,, 3, 5) 0 0 Q.9 (a) 10C (0, 3, 7, 10) Q. (a) (i)+(ii) 10C (0, 3, 7, 10) (b) 10C* (0, 3, 7, 10) (b) 10C (0, 3, 7, 10) 0 0 Q.10 (i) 10C (0, 3, 7, 10) Q.3 (i) 10C (0, 3, 7, 10) (ii) 5C (0,, 3, 5) (ii) 5A (0, 5) (iii) 5B* (0, 3, 5) (iii) 5B (0, 3, 5) 0 0 Q.11 (i) 5B (0, 3, 5) Q.4 (i) 10C (0, 3, 7, 10) (ii) 5B (0, 3, 5) (ii) 10C (0, 3, 7, 10) (iii) 5B (0, 3, 5) 0 15 Q.5 10C (0, 3, 7, 10) Q.1 (i) 5B (0, 3, 5) 10C (0, 3, 7, 10) (ii) 10B (0, 5, 10) 0 15 Q.6 (i) 10C (0, 3, 7, 10) Q.13 (i) 5, Att. (ii) 5B (0, 3, 5) (ii) 5, Att. (iii) 5A (0, 5) (iii) 15, Att. 5 (iv) 5B (0, 3, 5) 15, Att. 5 (v) 5B (0, 3, 5) (iv) 5, Att. (vi) 5B (0, 3, 5) (v) 5, Att. (vii) 5A (0, 5) Q.7 (i) 5B (0, 3, 5) (ii) 10C (0, 3, 7, 10) (iii) 5C (0,, 3, 5) 0 General Instructions 1. There are 13 questions on this examination paper. Answer all questions.. 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. 013 J.18/0_MS 3/56 Page 3 of 56 exams

4 exams Pre-Junior Certificate Examination, 014 Mathematics (Project Maths Phase ) Higher Level Paper 1 Marking Scheme (300 marks) Question 1 (suggested maximum time: 10 minutes) (0) 1(a) (i) What is a prime number? (5C) Answer Any 1: a natural number greater than 1 that has no positive divisors other than 1 and itself // a natural number greater than 1 that has only two factors, itself and 1 Scale 5C (0,, 3, 5) Low partial credit: ( marks) Writes an example of a prime number (with a maximum of one error) in an attempt to list primes. High partial credit: (3 marks) Definition of a composite number. Lists three or more correct prime numbers. (ii) Determine which of the following numbers are prime. (5C) Give a reason for your answers. 17, 117, 17, 317. Answer 17, 317 are primes Reason factors of 17: 1, 17 factors of 317: 1, 317 Answer 117, 17 are not primes Reason factors of 117: 1, 3, 9, 13, 39, 117 factors of 17: 1, 7, 31, 17 Scale 5C (0,, 3, 5) Low partial credit: ( marks) One number correctly identified as a prime or not. High partial credit: (3 marks) Four numbers correctly identified but no reasons given. Three numbers correctly identified with correct reasons. 014 J.18/0_MS 4/56 Page 4 of 56 exams

5 1(b) (i) By plotting the following numbers on the number line below, write the numbers in ascending order: π 5, 19,,, (5D) π Number line π Order: ( ), 3 19 ( ), 3 5 ( ), 3 19 (4 75) (5) 5 Scale 5D (0,, 3, 4, 5) Low partial credit: ( marks) One or two numbers in the correct position on the number line. Middle partial credit: (3 marks) Three or four numbers in the correct position on the number line. High partial credit: (4 marks) All numbers in the correct position on the number line but not written out in ascending order. (ii) From the list of numbers in part (i) above, identify one: (5C) Natural Number (N) Rational Number (Q) or or Irrational Number (R \ Q) 4π 19 or 3 Prime Number Scale 5C (0,, 3, 5) Low partial credit: ( marks) One or two correct entries. High partial credit: (3 marks) Three correct entries (one per set). 013 J.18/0_MS 5/56 Page 5 of 56 exams

6 Question (suggested maximum time: 10 minutes) (0) (a) x < 14 < y. (10C) (i) Find, correct to one decimal place, the maximum value of x. x < 14 x < max. value of x 3 7 (ii) Find, correct to one decimal place, the minimum value of y. y > 14 y > min. value of y 3 8 Scale 10C (0, 3, 7, 10) Low partial credit: (3 marks) 14 evaluated correctly. High partial credit: (7 marks) Maximum value of x or minimum value of y correct. (b) A right-angled triangle has a base length and a perpendicular height 5 7 as shown in the diagram Show that the total length of the three sides of the triangle is a natural number. (10C) Hypotenuse Hypotenuse 64 8 Length of 3 sides N Scale 10C (0, 3, 7, 10) Low partial credit: (3 marks) Any correct relevant step. Some correct use of Pythagoras theorem. High partial credit: (7 marks) Calculates hypotenuse correctly and stops. Error in use of theorem of Pythagoras and finishes correctly. 014 J.18/0_MS 6/56 Page 6 of 56 exams

7 Question 3 (suggested maximum time: 10 minutes) (0) 60 students attend a language summer camp to study one or more of the following languages: German, Spanish and Chinese. A student is chosen at random from the group. The probability that the student studies German is 3 1. The probability that the student studies Spanish only is 4 1. The probability that the student studies German and Spanish is 5 1. The probability that the student studies Chinese and Spanish is The probability that the student studies all three languages is The probability that the student studies German and Chinese but not Spanish is. 0 (i) Represent the above information on the Venn diagram. (10C) U [ 60 ] S [31] G [ 31 ] [5] [16] [6] [3] [6] [15] [106] [4] [6039] [1] C [34] Scale 10C (0, 3, 7, 10) Low partial credit: (3 marks) One or two correct elements. High partial credit: (7 marks) Between three and six correct elements. (ii) Which language is the most popular among students? (5A) Chinese ** Accept student s answer consistent with part (i). Scale 5A (0, 5) Hit or miss. (iii) Is it reasonable to say that students use this summer camp to study more than one new language? Justify your answer. Answer no (5B) 19 Reason less than one-third ( ) studied more than one language 60 ** Accept student s answer consistent with part (i). Scale 5B (0, 3, 5) Partial credit: (3 marks) Correct answer but no reason given. 013 J.18/0_MS 7/56 Page 7 of 56 exams

8 Question 4 (suggested maximum time: 10 minutes) (0) A clothing company sells a particular shirt in Spain for 59. This price includes VAT charged at 18%. The same shirt is available to buy in Ireland for VAT in Ireland is charged at 3%. (i) Calculate the price of the shirt in each country excluding VAT. (10C) Price in Spain (ex. VAT) 50 Price in Ireland (ex. VAT) Scale 10C (0, 3, 7, 10) Low partial credit: (3 marks) Any work of merit. High partial credit: (7 marks) Spanish price or Irish price correct. (ii) Given that the company s mark-up in Spain is 5%, calculate the mark-up on the same shirt in Ireland. (10C) Cost excluding mark-up (Spain) Mark-up in Ireland % mark-up Profit Cost Price % ** Accept student s answers from part (i). Scale 10C (0, 3, 7, 10) Low partial credit: (3 marks) Any work of merit, e.g. correct formula for % mark-up written down. High partial credit: (7 marks) Incorrect formula for % mark-up used and finishes. 014 J.18/0_MS 8/56 Page 8 of 56 exams

9 Question 5 (suggested maximum time: 10 minutes) (0) On 1 January 01, Mary invested 6,000 in a special savings account which paid 7% interest on the first day on condition that the money remains in the account for 3 years. On the same day, her sister Joan invested 6,000 in another special savings account which guaranteed an annual interest rate of 5% for 3 years, compounded annually. DIRT tax is deducted each year from the interest earned. The table below shows the DIRT tax rates applicable for the years 01 to 014, inclusive. Year 01 30% % DIRT Tax Rate % (includes 3% PRSI) Calculate how much each sister will receive at the end of 014, correct to two decimal places, after DIRT tax is deducted. Mary (10C) 7 Gross Interest 6, DIRT Net Interest Gross interest DIRT Final sum 6, ,94 Scale 10C (0, 3, 7, 10) Low partial credit: (3 marks) Any work of merit. Errors calculating Interest and DIRT. High partial credit: (7 marks) Calculates Net interest and stops. One error calculating Interest or DIRT. Joan (10C) Year 1: 5 Gross Interest 6, DIRT Net Interest Year : Amount 6, ,105 5 Gross Interest 6, DIRT J.18/0_MS 9/56 Page 9 of 56 exams

10 Question 5 (cont d.) Year : (cont d.) Net Interest Year 3: Amount 6, , Gross Interest 6, DIRT Net Interest Final sum 6, ,9 61 Scale 10C (0, 3, 7, 10) Low partial credit: (3 marks) Any work of merit. Errors calculating Interest and DIRT. High partial credit: (7 marks) Calculates Net interest and stops. One error calculating Interest or DIRT. 014 J.18/0_MS 10/56 Page 10 of 56 exams

11 Question 6 (suggested maximum time: 10 minutes) (40) Peter and his family plan to move home to Ireland from the UK next year. He is concerned about the cost of water charges being introduced in Ireland. The table below shows the proposed pricing structure for water in Ireland and the pricing structure of Peter s current water provider in the UK. Both pricing structures include an annual standing charge, even if no water is used, and the cost of different amounts of water used (in m 3 ). Water Used (m 3 ) IRL Cost ( ) UK Cost ( ) (i) Draw a graph to show the relationships between the amount of water used and the cost in both Ireland and the UK. (10C) Cost () Ireland UK Water Used (m ) Scale 10C (0, 3, 7, 10) Low partial credit: (3 marks) Marks one or two points only or draws a histogram correctly from table. High partial credit: (7 marks) Marks between three and eight points or histogram correctly from table. Points of each line not joined up. 013 J.18/0_MS 11/56 Page 11 of 56 exams

12 6(ii) Use your graph to estimate the annual standing charge in both Ireland and the UK. (5B) Annual standing charge Ireland 5 UK 100 Scale 5B (0, 3, 5) Partial credit: (3 marks) One correct answer. (iii) What level of water usage would result in the same annual charge in both countries? (5A) Same charge 37 5 m 3 (tolerance ± 5 m 3 ) Scale 5A (0, 5) Hit or miss. (iv) Write down two formulae to represent the annual water charges in both Ireland and the UK. State clearly the meaning of any letters used in the formulae. (5B) Ireland C x 5 C 5 + 5x... C cost, x water usage in m 3 UK C x 5 C x... C cost, x water usage in m 3 Scale 5B (0, 3, 5) Partial credit: (3 marks) One correct answer. Any work of merit. (v) Use the formulae from part (iv) to confirm that your answer in part (iii) is correct. (5B) 5 + 5x x 5x 3x x 75 x 37 5 m 3 Ireland: 5 + 5x 5 + 5(37 5) UK: x (37 5) ** Accept student s formulae from part (iv). Scale 5B (0, 3, 5) Partial credit: (3 marks) Any work of merit. Finds x 37 5 m J.18/0_MS 1/56 Page 1 of 56 exams

13 6(vi) Peter decides that he will install a rainwater harvester in his new home in Ireland. Based on his UK usage, he estimates that his household water usage will be 45 m 3 per quarter in Ireland. The rainwater harvester company claims that Peter will save one-third of his current water usage. The cost of installing the rainwater harvester is How long will it take before Peter benefits financially from installing the rainwater harvester, based on current market rates? (5B) or Cost of water used per quarter Company claims savings of one-third Estimated saving per quarter Return on installation cost 1,500 quarters 75 0 quarters 5 years Annual estimated water usage m 3 Annual cost of water used Company claims savings of one-third Estimated annual saving Return on installation cost 1, years Scale 5B (0, 3, 5) Partial credit: (3 marks) Any work of merit. Quarterly or annual savings calculated. (vii) Under what circumstances would Peter benefit earlier? (5A) Reason Any 1: water charges increases // water usage increases Scale 5A (0, 5) Accept any reasonable explanation. 013 J.18/0_MS 13/56 Page 13 of 56 exams

14 Question 7 (suggested maximum time: 10 minutes) (0) A series of shapes is made using square blocks. (i) Draw the next two shapes in the series. (5B) Scale 5B (0, 3, 5) Partial credit: (3 marks) One shape correctly drawn. (ii) How many blocks are there in the 4nd shape? (10C) a 1 d T T T n a + (n 1)d 1 + (n 1) n 3 3n T 4 3(4) blocks Scale 10C (0, 3, 7, 10) Low partial credit: (3 marks) Any work of merit, e.g. first term or common difference identified, formula for nth term written down. Sixth or subsequent terms identified. High partial credit: (7 marks) Correct formula identified. 4nd shape calculated with minor errors. 014 J.18/0_MS 14/56 Page 14 of 56 exams

15 (iii) Where in the series is the shape that consists of blocks? (5C) T n 3n 3n 3n + 4 n 8 8th shape consists of blocks ** Accept student s formula from part (ii). Scale 5C (0,, 3, 5) Low partial credit: ( marks) Any work of merit, e.g. formula for nth term equal to. High partial credit: (3 marks) Correct substitution but minor errors in calculations. 013 J.18/0_MS 15/56 Page 15 of 56 exams

16 Question 8 (suggested maximum time: 10 minutes) (0) A charity cycle involves two cycling clubs from Dublin and Galway travelling between Dublin and Galway and returning home over two days. On the first day of the cycle, the Dublin club cycles from Dublin to Galway and the Galway club cycles from Galway to Dublin. The two clubs, having started their cycles at the same time, arrange to meet for lunch in Athlone. The average speed of the Galway club is 1 km per hour. The average speed of the Dublin club is 31 km per hour and they travel 40 km more than the Galway club. Both clubs arrive in Athlone at the same time. (i) Calculate the distance between Dublin and Galway. (10C) Let d distance between Galway and Athlone Galway club Time taken distance speed d 1 Dublin club Time taken d 1 distance speed d d d 1(d + 40) 1d d 840 d 84 km Distance travelled by Galway club 84 km Distance travelled by Dublin club km Distance from Dublin to Galway km Scale 10C (0, 3, 7, 10) Low partial credit: (3 marks) Any work of merit, e.g. formula for average speed or time taken written down with some correct substitution. High partial credit: (7 marks) Distance travelled by Dublin or Galway club calculated. Correct method with minor errors. 014 J.18/0_MS 16/56 Page 16 of 56 exams

17 8(ii) After lunch, both clubs continue their cycle to their destination. How long does it take the Galway club to reach Dublin, assuming that they continue at the same speed as before lunch? Give your answer correct to the nearest minute. Galway club Time taken distance speed hours 5 hours, minutes 5 hours, 54 minutes (5C*) ** Accept student s answer from part (i). Scale 5C* (0,, 3, 5) Low partial credit: ( marks) Any work of merit, e.g. formula for time taken written down with some correct substitution. High partial credit: (3 marks) Error converting hours to minutes. A correct method with one error. * Deduct 1 mark off correct answer only if incorrectly rounded, no rounding, omission of or incorrect units - apply only once throughout the question. (iii) On the second day of the charity cycle, both clubs intend to complete the return leg. They plan to meet up again in Athlone on route to their respective destinations. Both clubs aim to cycle at the same speeds as they did on the first day. How much earlier does the Galway club need to leave Dublin in order to arrive in Athlone at the same time as the Dublin club? Give your answer correct to the nearest minute. Galway club Time taken 5 hours, 54 minutes Dublin club Time taken distance speed hours hours, minutes hours, 43 minutes Time difference 5 hours, 54 minutes hours, 43 minutes 3 hours, 11 minutes Galway club needs to leave Dublin 3 hours, 11 minutes earlier (5C*) ** Accept student s formula from part (i) and (ii). Scale 5C* (0,, 3, 5) Low partial credit: ( marks) Any work of merit, e.g. formula for time taken written down with some correct substitution. High partial credit: (3 marks) Calculates time taken by both clubs but fails to calculate time difference. A correct method with one error. * Deduct 1 mark off correct answer only if incorrectly rounded, no rounding, omission of or incorrect units - apply only once throughout the question. 013 J.18/0_MS 17/56 Page 17 of 56 exams

18 Question 9 (suggested maximum time: 10 minutes) (0) 9(a) Solve the equation: x 1 3x 4 8. (10C) x 1 3x (x 1) 5(3x 0 4 x 15x x ) ( 11x + 18)(15) (8)(0) ( 11x + 18)(3) (8)(4) 33x x x x 33 3 Scale 10C (0, 3, 7, 10) Low partial credit: (3 marks) Any work of merit, e.g. some correct use of any correct common denominator. High partial credit: (7 marks) Correct method with minor errors. 9(b) Solve for x: x 4x 7 0, giving your answer correct to two decimal places. (10C*) x x 014 J.18/0_MS 18/56 Page 18 of 56 exams b ± b a ( 4) ± 4ac ( 4) 4()( 7) 4 ± ± ± x , x Scale 10C* (0, 3, 7, 10) Low partial credit: (3 marks) Any work of merit, e.g. some correct substitution into b formula. () High partial credit: (7 marks) Correct method with minor errors. Find only one value of x (3 1 or 1 1). * Deduct 1 mark off correct answer only if incorrectly rounded, no rounding, - apply only once throughout the question.

19 Question 10 (suggested maximum time: 10 minutes) (0) A rectangular allotment is divided into four different sized rectangular plots according to the needs of four individuals, as shown in the diagram. x m x m 5m (i) Find, in terms of x, the total area of the allotment. (10C) Total length x + 5 Total width 3 + x Total area l w (x + 5)(3 + x) x + 11x + 15 or Area of plot A l w x 3 6x Area of plot B Area of plot C x x x Area of plot D x 5 5x Area of allotment Area of (plot A + plot B + plot C + plot D) Area of allotment 6x x + 5x x + 11x + 15 Scale 10C (0, 3, 7, 10) Low partial credit: (3 marks) Any work of merit, e.g. any correct area calculated. 3m High partial credit: (7 marks) Correct method with one error. Incorrect area of one plot. 013 J.18/0_MS 19/56 Page 19 of 56 exams

20 10(ii) If the total area of the allotment is 136 m, calculate the value of x. (5C) or Area of allotment 136 x + 11x x + 11x 11 0 (x 11)( x + 11) 0 x 11 0 x 11 x 11 as x > 0 x 11 or 5 5 x + 11x 11 0 or x x 11 x x x as x > 0 x b ± 11 ± b a 4ac 11 4()( 11) () 11 ± ± 1, ± , x or 5 5 Scale 5C (0,, 3, 5) Low partial credit: ( marks) One or both factors correctly identified. Any work of merit, e.g. some correct substitution into b formula. High partial credit: (3 marks) Correct method with minor errors. One of 5 5 or 11 unless 11 has been correctly eliminated (then full marks). 014 J.18/0_MS 0/56 Page 0 of 56 exams

21 10(iii) The allotment was re-divided into four identical rectangular plots. Calculate the dimensions of each plot. (5B*) Total length x ( ) m Plot length 1 (16) 8 m Total width 3 + x m Plot width 1 (8 5) 4 5 m or Total length x ( ) m Plot length 1 (16) 4 4 m Plot width 8 5 m or Plot length x ( ) m Total width 3 + x m Plot width 1 (8 5) 4 15 m Scale 5B* (0, 3, 5) Partial credit: (3 marks) Any work of merit. One correct dimension. * Deduct 1 mark off correct answer only if incorrectly rounded, no rounding, omission of or incorrect units - apply only once throughout the question. 013 J.18/0_MS 1/56 Page 1 of 56 exams

22 Question 11 (suggested maximum time: 10 minutes) (15) Factorise fully each of the following expressions: (i) 3x 9x (5B) 3x 9x 3x(x 3) Scale 5B (0, 3, 5) Partial credit: (3 marks) Any common factor identified. (ii) an 5a 5b + bn an 5a 5b + bn a(n 5) + b( 5 + n) (n 5)(a + b) or an 5a 5b + bn an + bn 5a 5b n(a + b) 5(a + b) (a + b)(n 5) Scale 5B (0, 3, 5) Partial credit: (3 marks) Any two terms correctly factorised. (5B) (iii) 4x + x 3. (5B) 4x + x 3 (8x + 3)(x 1) Scale 5B (0, 3, 5) Partial credit: (3 marks) 4x and/or 3correctly factorised. 014 J.18/0_MS /56 Page of 56 exams

23 Question 1 (suggested maximum time: 10 minutes) (15) In physics, the mass-energy equivalence is the concept that the mass of an object or system is a measure of its energy content. Albert Einstein proposed the equation E mc to describe the equivalence in (i) Write c in terms of E and m, where c > 0. (5B) E mc mc E c E m c E m as c > 0 Scale 5B (0, 3, 5) Partial credit: (3 marks) One correct manipulation. (ii) Calculate the value of c, if m 50 kg and E J. (10B) c E m or 300,000,000 ** Accept student s formula from part (i). Scale 10B (0, 5, 10) Partial credit: (5 marks) Some correct substitution. 013 J.18/0_MS 3/56 Page 3 of 56 exams

24 Question 13 (suggested maximum time: 10 minutes) (50) General Instructions 1. Penalties of three types are applied to students work, as follows: Blunders - mathematical errors / sign errors / omissions B ( 3) Slips - numerical slips S ( 1) Misreadings (provided task is not oversimplified) M ( 1) Frequently occurring errors to which these penalties must be applied are listed in this scheme. They are labelled B1, B, B3,..., S1, S, S3,..., M1, M, M3,..., etc. Note that these lists are not exhaustive.. When awarding attempt marks, e.g. Att. (3), it is essential to note that: any correct relevant step in a part of a question merits at least the attempt mark for that part if deductions result in a mark which is lower than the attempt mark, then the attempt mark must be awarded a mark between zero and the attempt mark is never awarded. 3. Worthless work must be awarded zero marks. Some examples of such work are listed in this scheme and they are labelled as W1, W,..., etc. 4. The phrase hit or miss means that partial marks are not awarded - the student receives all of the relevant marks or none. 5. The phrase and stops means that no more work is shown by the student. 6. Special notes relating to the marking of a particular part of a question are indicated by double asterisks. These notes immediately follow the relevant solution. 7. The sample solutions for each question are not intended to be exhaustive lists - there may be other correct solutions. 8. Unless otherwise indicated in the scheme, accept the best of two or more attempts - even when attempts have been cancelled. 9. The same error in the same section of a question is penalised once only. 10. Particular cases, verification and answers derived from diagrams (unless requested) qualify for attempt marks at most. 11. A serious blunder, omission or misreading merits the attempt mark at most. 1. Do not penalise the use of a comma for a decimal point, e.g may be written as 6, J.18/0_MS 4/56 Page 4 of 56 exams

25 Question 13 (cont d.) A long rectangular piece of sheet metal of width 18 cm is to be fabricated into a gutter by turning up sides of equal height x cm, perpendicular to the base. x cm A 18cm (i) Show that the formula for the area of the cross-section, A, of the fabricated gutter, in cm, is: A 18x x. (5, Att. ) Area l w... m (18 x) x... m 18 x x... 5m Blunders ( 3) B1: Mathematical error. B: Fails to finish. Misreadings ( 1) M1: Misreads a digit, providing it does not oversimplify the question. Slips ( 1) S1: Invalid conclusion. Attempts () A1: Area formula. A: Any relevant step. Worthless (0) W1: Incorrect answer, no work shown. W: Copies diagram. W3: Work of no merit. (ii) Explain why the formula for A is valid only for 0 < x < 9. (5, Att. ) x 0 no sides x < 0 negative length of sides impossible as length > 0 x > 0 x ; no base length x > 9 Impossible as nd side < 9; must be equal x < 9 0 < x < 9 Blunders ( 3) B1: Incorrect statement. Attempts () A1: Any correct statement. Worthless (0) W1: Incorrect answer, no work shown. 013 J.18/0_MS 5/56 Page 5 of 56 exams

26 (iii) Let f be the function f : x 18x x. Draw the graph of f for 0 x 9, x R. Table (15, Att. 5) x x x f (x) Graph (15, Att. 5) fx () ** Table is worth 15 marks, graph is worth 15 marks. ** Middle lines of table do not have to be shown. ** Student may choose not to use a table. ** Points need not be listed, but marked on position on graph. ** Graph constitutes work in this question. ** Accept reversed co-ordinates if (i) axes are not labelled or (ii) axes are reversed to compensate. ** Graph incorrect - examine work and mark accordingly. ** Tolerance ±1 box on scale. ** Error(s) in each row/column attract a maximum deduction of 3. ** Accept student s values from table when plotting points. Blunders ( 3) B1: Error in calculating 18x - apply once if consistent. B: Error in calculating x - apply once if consistent but note A. B3: Error in calculating last line of table - apply once if consistent. B4: Each incorrect point without work [S may apply]. B5: Point plotted incorrectly - apply once if consistent. B6: Incomplete domain. B7: Axes scaled incorrectly - apply once only. B8: Axes reversed. B9: No curve between (4, 40) and (5, 40) on graph. Slips ( 1) S1: Numerical slips (max. of 3). S: Each incorrect point plotted [Tolerance ±1 box on scale]. Misreadings ( 1) M1: Misreads 18x as 18x all the way across in the row headed 18x. M: Misreads x as (x) all the way across in the row headed x. Attempts (5, 5) A1: Some correct substitution. A: Error leading to a linear graph. A3: Draws axes, with some indication of scaling. A4: Some effort to plot a point. 014 J.18/0_MS 6/56 Page 6 of 56 exams

27 (iv) Use your graph from part (iii) to estimate the maximum possible area of the cross-section of the gutter. Maximum area 41 m (tolerance ± m ) (5, Att. ) ** Accept answer consistent with student s curve (within tolerance of ±1 box on scale). Blunders ( 3) B1: Fails to use graph. B: Value not consistent with student s graph. B3: States x value (width) instead of y value (area) for maximum area. Slips ( 1) S1: Answer given as maximum point instead of maximum value. Attempts () A1: Finds from function only. A: Reads x value (width) from table. A3: Some indication on graph but no value given. Worthless (0) W1: Answer inconsistent with student s graph. (v) Hence, determine the maximum volume of the gutter if it is 15 m long. (5, Att. ) Maximum volume Maximum area m m 3 ** Accept student s answer from part (iv). Blunders ( 3) B1: Error in volume formula. B: Incorrect substitution. B3: Mathematical error. Slips ( 1) S1: Slip in multiplication. Attempts () A1: Some relevant step. 013 J.18/0_MS 7/56 Page 7 of 56 exams

28 Notes: 014 J.18/0_MS 8/56 Page 8 of 56 exams

29 exams Pre-Junior Certificate Examination, 014 Mathematics (Project Maths Phase ) Higher Level Paper Marking Scheme (300 marks) Structure of the Marking Scheme Student responses are marked according to different scales, depending on the types of response anticipated. Scales labelled A divide student responses into two categories (correct and incorrect). Scales labelled B divide responses into three categories (correct, partially correct, and incorrect), and so on. The scales and the marks that they generate are summarised in this table: Scale label A B C D No of categories mark scale 0, 5 0, 3, 5 0,, 3, 5 0,, 3, 4, 5 10 mark scale 0, 10 0, 5, 10 0, 3, 7, 10 0, 5, 8, 9, mark scale 0, 15 0, 10, 15 0, 10, 1,15 0, 10, 1, 14, 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 (four categories) response of no substantial merit (no credit) response with some merit (low partial credit) response abut half-right (partial credit) almost correct response (high partial credit) correct response (full credit) In certain cases, typically involving incorrect rounding or omission of units, 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. However, it is important to note that this reduction only applies under the following criteria: The * for units to be applied only if the student s answer is fully correct. The * to be applied once only per question. The * penalty is not applied to currency solutions. 014 J.18/0_MS 9/56 Page 9 of 56 exams

30 Summary of Marks (Paper ) Q.1 (i) 5B* (0, 3, 5) Q.10 (i) 10D (0, 5, 8, 9, 10) (ii) 5C* (0,,3,5) (ii) 5B (0, 3, 5) (iii) 5B* (0, 3, 5) (iii) 5B (0, 3, 5) (iv) 5B (0, 3, 5) (iv) 5B (0, 3, 5) 0 (v) 5B (0, 3, 5) 30 Q. (i) 5C* (0,,3,5) (ii) 10D (0, 5, 8, 9, 10) Q.11 10C (0, 3, 7, 10) Q.3 (i) 5C (0,,3,5) (ii) 5C (0,,3,5) Q.1 (i) 10C (0, 3, 7, 10) (iii) 5B (0, 3, 5) (ii) 5C (0,,3,5) Q.4 (i) 5B (0, 3, 5) (ii) 5B (0, 3, 5) Q.13 10D (0, 5, 8, 9, 10) (iii) 5B (0, 3, 5) 5B (0, 3, 5) (iv) 5B (0, 3, 5) 5B (0, 3, 5) (v) 5C (0,,3,5) 0 5 Q.5 (i) 5B (0, 3, 5) Q.14 10C* (0, 3, 7, 10) (ii) 5C (0,,3,5) 10C* (0, 3, 7, 10) (iii) 5C (0,,3,5) 0 15 Q.15 (i) 10D* (0, 5, 8, 9, 10) Q.6 (i) 10D (0, 5, 8, 9, 10) (ii) 10D* (0, 5, 8, 9, 10) (ii) 5A (0, 5) 0 (iii) 5B* (0, 3, 5) (iv) 5C* (0,,3,5) 5 Q.16 (i) 5B (0, 3, 5) (ii) 5A (0, 5) (iii) 5C (0,,3,5) Q.7 (i) 5B (0, 3, 5) (iv) 5C* (0,,3,5) (ii) 5B (0, 3, 5) 0 10 Q.8 5B (0, 3, 5) 5B (0, 3, 5) 10C (0, 3, 7, 10) 0 Q.9 (i) 5B (0, 3, 5) (ii) 5A (0, 5) (iii) 5C (0,,3,5) (iv) 5B (0, 3, 5) J.18/0_MS 30/56 Page 30 of 56 exams

31 exams Pre-Junior Certificate Examination, 014 Mathematics (Project Maths Phase ) Higher Level Paper Marking Scheme (300 marks) General Instructions 1. There are 16 questions on this examination paper. Answer all questions.. 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. 014 J.18/0_MS 31/56 Page 31 of 56 exams

32 Question 1 (suggested maximum time: 10 minutes) (0) The design of a company s logo incorporates two identical semi-circles which fit exactly in a rectangle, as shown. 1(i) From the diagram, measure the radius of the semi-circles. (5B*) Radius 4 cm ** Actual size as per examination paper. ** Allow tolerance of ±0 1 cm. Scale 5B* (0, 3, 5) Partial credit: (3 marks) Diameter given. Radius outside tolerance of ±0 1 cm but within ±0 3 cm. * Deduct 1 mark off correct answer only for omission of or incorrect units - apply only once throughout the question. 1(ii) Find the area enclosed by the two semi-circles, in terms of π. (5C*) Area of semicircles πr π( 4) 5 76π cm ** Accept student s radius from part (i). Scale 5C* (0,, 3, 5) Low partial credit: ( marks) Correct formula from Tables. High partial credit: (3 marks) Writes down π( 4). * Deduct 1 mark off correct answer only if not in terms of π or for omission of or incorrect units - apply only once throughout the question. 014 J.18/0_MS 3/56 Page 3 of 56 exams

33 1(iii) Find the area of the shaded region. Give your answer correct to one decimal place. (5B*) Area of shaded region (L W) area of semicircles ((3 4) ( 4)) 5 76π (7 4 8) 5 76( ) cm ** Accept student s radius from part (ii). Scale 5B* (0, 3, 5) Low partial credit: ( marks) Any work of relevance, e.g. (L W) πr or (L W) area of semicircles or or * Deduct 1 mark off correct answer only if incorrectly rounded, no rounding, omission of or incorrect units - apply only once throughout the question. 1(iv) Calculate the ratio of the area enclosed by the two semi-circles to the area of the rectangle, giving your answer in the form π : n, n N. (5B) Area of semicircles : Area of rectangle 5 76π: π : 5 76 π : 6 ** π Accept student s answer in the form. 6 Scale 5B (0, 3, 5) Partial credit: (1 mark) Any work of relevance, e.g. 5 76π : 34 56, or 16 5 : or 1 : J.18/0_MS 33/56 Page 33 of 56 exams

34 Question (suggested maximum time: 10 minutes) (15) A vessel is in the shape of a cylinder on top of a cone, as shown. The diameter of the cylinder is 10 cm and the overall height of the vessel is 30 cm. 10 cm h cm 30 cm (i) Find the internal volume of the vessel. Give your answer in terms of π and h, the height of the cylinder. V vessel πr h πr (30 h) π(5) h π(5) (30 h) 5π(h (30 h)) 5π(h h ) (5C*) 5π( 3 h + 10) cm 3 Scale 5C* (0,, 3, 5) Low partial credit: ( marks) Any work of relevance, e.g. correct formulae from Tables. High partial credit: (3 marks) Correct substitution into relevant formulae. * Deduct 1 mark off correct answer only if not in terms of π or for omission of or incorrect units - apply deduction only once throughout the question. 014 J.18/0_MS 34/56 Page 34 of 56 exams

35 (ii) Water is poured into the vessel. Under what conditions are the volumes of water in the cone and cylinder equal? V water in cylinder V water in cone πr 1 h w πr h cone 3 1 hcone h w 3 height of the water in the cylinder must be a third of the height of the cone (10D) Assume max. height of water h Volume of cylinder Volume of cone πr h 1 πr (30 h) 3 1 h (30 h) h 3 4 h 3 10 h cm height of the water in the cylinder 7 5 cm Scale 10D (0, 5, 8, 9, 10) Low partial credit: (5 marks) One condition found but no explanation given. Middle partial credit: (8 marks) One condition found with explanation. Two conditions found but no explanation given. High partial credit: (9 marks) Two conditions found with only one explanation given. * Deduct 1 mark off correct answer only for omission of or incorrect units - apply only once throughout the question. 014 J.18/0_MS 35/56 Page 35 of 56 exams

36 Question 3 (suggested maximum time: 10 minutes) (15) Rob tested two different brands of batteries, A and B, to see which had the longer life-span. He placed batteries from each brand in a torch and recorded how long each lasted. The results are shown below in the back-to-back stem-and-leaf plot. Brand A Brand B Key: 65 1 means 651 minutes 3(i) Find the median life-span of each brand of battery. (5C) Median life-span: Brand A 694 minutes Brand B 718 minutes Scale 5C (0,, 3, 5) Low partial credit: ( marks) 7th time identified. High partial credit: (3 marks) One median identified only. 3(ii) Find the interquartile range of each brand of battery. (5C) Interquartile range: Brand A minutes Brand B minutes Scale 5C (0,, 3, 5) Low partial credit: ( marks) One upper or lower quartile found. 4th or 1th time identified. High partial credit: (3 marks) One interquartile range calculated. 3(iii) Based on the above data, would you conclude that one brand is more reliable than the other? Give a reason for your answer. Answer Brand B is more reliable (5B) Reason Any 1: more batteries from brand B lasted over 700 minutes // mean life-span of brand B (707 7 minutes) > brand A (699 4 minutes) Scale 5B (0, 3, 5) Partial credit: (3 marks) Correct answer but no reason or incorrect reason given. 014 J.18/0_MS 36/56 Page 36 of 56 exams

37 Question 4 (suggested maximum time: 10 minutes) (5) A group of 96 students from a school in Galway stopped at a food court in a shopping centre on the way home from visiting the Young Scientist Exhibition. When they got back on the bus, their teachers conducted a quick survey to determine which food outlet they had chosen to eat in at the food court. The results are recorded in the table below. Burger Joint Sandwich Bar Wok Station Boys Girls 11 x 3 4(i) Assuming that all students ate at one of the food outlets, find the value of x. (5B) x x 96 x Scale 5B (0, 3, 5) Partial credit: (3 marks) Any work of relevance, e.g (ii) If one student is chosen at random, what is the probability that the student chosen is a girl who ate at the Wok Station? P(girl, Wok Station) Scale 5B (0, 3, 5) Partial credit: (3 mark) Favourable outcomes correct. Total outcomes correct. (5B) 4(iii) If one student is chosen at random, what is the probability that the student chosen is a boy? P(boy) (5B) Scale 5B (0, 3, 5) Partial credit: (3 mark) Favourable outcomes correct. Total outcomes correct Answer J.18/0_MS 37/56 Page 37 of 56 exams

38 4(iv) If the student chosen at random is a boy, what is the probability that he did not eat at the Sandwich Bar? P(not sandwich bar boy) (5B) or P(not sandwich bar boy) 1 P(sandwich bar boy) Scale 5B (0, 3, 5) Partial credit: (3 mark) Favourable outcomes correct. Total outcomes correct. 16 Answer. 46 4(v) Place each of your answers to parts (ii), (iii) and (iv) at its correct position on the probability scale below. (5C) (ii) 0 33 (iii) 0 48 (iv) Scale 5C (0,, 3, 5) Low partial credit: ( marks) One probability correctly positioned. High partial credit: (3 marks) Two probabilities correctly positioned. 014 J.18/0_MS 38/56 Page 38 of 56 exams

39 Question 5 (suggested maximum time: 5 minutes) (15) Students in a school have been asked to select the subjects that they would like to study for their Leaving Certificate next year. Each student can only select one subject from each of the following options lines. Option Subjects A Biology Business Geography Accounting B French Art C German History Chemistry 5(i) How many different subject selections are possible? (5B) Scale 5B (0, 3, 5) Partial credit: (3 marks) Any work of relevance, e.g (ii) Sophie does not want to study any of the languages offered. From how many different subject selections can she pick? (5C) Scale 5C (0,, 3, 5) Low partial credit: ( marks) Any work of relevance, e.g. mention of 7 subjects (4 + +1). High partial credit: (3 marks) (iii) The school has decided to add Music to one of the option lines. On which option line should Music be placed, in order to maximise the possible subject selections from which students can pick? Explain your answer. (5C) Answer line B Explanation Line A maximum possible selections (4 + 1) 3 30 Line B maximum possible selections 4 ( + 1) 3 36 Line C maximum possible selections 4 (3 + 1) 3 Scale 5C (0,, 3, 5) Low partial credit: ( marks) Correct answer but no reason or incorrect reason given. High partial credit: (3 marks) 30, 36 and 3 calculated but no answer given. 014 J.18/0_MS 39/56 Page 39 of 56 exams

40 Question 6 (suggested maximum time: 10 minutes) (5) Third-year students were asked to choose their favourite sport offered in their school. The results are recorded in the table below. Sport Soccer Tennis Rugby Basketball Number of Students (i) Display the data on a pie-chart, showing clearly how the size of each angle is calculated. (10D) Total number of students Angle per student Soccer 360 or Tennis 360 or Rugby 360 or Basketball 360 or Pie Chart ** Any suitable pie chart. Scale 10D (0, 5, 8, 9, 10) Low partial credit: (5 marks) Any work of relevance, e.g. angle per student (4 ), indicates that angles are fraction of 360. Middle partial credit: (8 marks) Angles correctly calculated. High partial credit: (9 marks) Angles correctly calculated and one incorrect on diagram. 014 J.18/0_MS 40/56 Page 40 of 56 exams

41 6(ii) Why is this an appropriate method to display the above data? (5A) Answer data is categorical Scale 5A (0, 5) Hit or miss. 6(iii) What percentage of students chose Basketball as their favourite sport in school? Give your answer correct to the nearest whole number. (5B*) Basketball % Scale 5B (0, 3, 5) Partial credit: (3 marks) 15 1 Any work of relevance, e.g. or * Deduct 1 mark off correct answer only if incorrectly rounded, no rounding, for omission of or incorrect units - apply only once throughout the question. 6(iv) It was later discovered that the surveys from one class of 30 students had been mislaid. When these surveys were accounted for, the number of students who chose Rugby increased by 9. Calculate the measure of the angle that would now represent the students who chose Rugby. Rugby (5C*) Scale 5C (0,, 3, 5) Low partial credit: ( marks) Any work of relevance, e.g. new angle per student (3 ), indicates that answer is a fraction of 360. High partial credit: (3 marks) 34 but not completed or incorrect. 10 * Deduct 1 mark off correct answer only for omission of or incorrect units - apply only once throughout the question. 014 J.18/0_MS 41/56 Page 41 of 56 exams

42 Question 7 (suggested maximum time: 10 minutes) (10) PQRS is a parallelogram. X is the image of R under axial symmetry in the line QS. Y is the point of intersection of QX and PS. P Y Y X S Q R 7(i) Prove that triangle QSP and triangle QSX are congruent. (5B) X is the image of R under axial symmetry in the line QS Q is the image of Q under axial symmetry in the line QS S is the image of S under axial symmetry in the line QS ΔQXS is the image of ΔQRS under axial symmetry in the line QS In triangles QPS and QXS, QP SR... opposite sides in a parallelogram SX QS QS... common side PS QR... opposite sides in a parallelogram QX ΔQPS and ΔQXS are congruent (SSS) Scale 5B (0, 3, 5) Partial credit: (3 marks) Proof that ΔQXS is the image of ΔQRS under axial symmetry in the line QS. 7(ii) Show that PY YX. (5B) ΔQPS and ΔQXS are congruent QP SX and QPS QXS PYQ XYS... vertically opposite angles PQY XSY... third angle in a triangle ΔQPY and ΔYXS are congruent (ASA) PY XY Scale 5B (0, 3, 5) Partial credit: (3 marks) Two pairs of equal angles correctly identified. Sides and one pair of equal angles correctly identified. 014 J.18/0_MS 4/56 Page 4 of 56 exams

43 Question 8 (suggested maximum time: 10 minutes) (0) Prove that in a right-angled triangle the square of the hypotenuse is the sum of the squares of the other two sides. (5B, 5B, 10C) Given ΔABC where BAC 90. To prove BC AB + AC Construction Draw AD BC. Proof CAB BDA... both 90 ABC ABD... common angle triangles ABC and ABD are similar. BC AB AB BD AB BC. BD... corresponding sides are proportional Likewise, triangles ABC and ADC are similar. BC AC AC DC AC BC. DC... corresponding sides are proportional AB + AC BC. BD + BC. DC BC.( BD + DC ) BC. BC BC Some steps may be indicated on the diagram. or Diagram Given Right-angled triangle with length of sides a, b, c, where c is the hypotenuse To prove a + b c Construction Construct a square ABCD of side a + b Construct the point E on [AB] such that AE b (and hence EB a). Similarly construct points F, G and H on the other sides as shown. Join E, F, G and H to divide the square ABCD into a quadrilateral and four triangles. Label the angles 1,, 3 and 4 as shown. Proof Each of the four inscribed triangle is congruent to the original triangle c a b D a G b 1 C a b a H 4 F 3 ba b A b E a B... SAS Each side of the inner quadrilateral has length c angle sum of triangle corresponding parts in congruent triangles straight angle The inscribed quadrilateral is a square 014 J.18/0_MS 43/56 Page 43 of 56 exams

44 Q.8 (cont d.) Area of large square 4(area of one triangle) + area of inscribed square (a + b) 4 (area of one triangle) + c (a + b) 4(½ ab) + c a + ab + b ab + c a + b c Diagram/Construction Some steps may be indicated on the diagram. Scale 5B (0, 3, 5) Partial credit: (3 marks) Right-angled triangle drawn. Given/To prove Scale 5B (0, 3, 5) Partial credit: (3 marks) Given correct. To prove correct. Proof Scale 10C (0, 3, 7, 10) Low partial credit: (3 marks) Any correct step. High partial credit: (7 marks) One step missing or incomplete. 014 J.18/0_MS 44/56 Page 44 of 56 exams

45 Question 9 (suggested maximum time: 10 minutes) (0) Lisa has a set of eight coloured plastic strips as shown below. 9 cm 1 cm 0 cm 4cm 4cm 9cm 6cm 6cm 9(i) Is it possible for Lisa to make an equilateral triangle using any three of the strips shown above? Give a reason for your answer. Answer no Reason cannot combine the strips to make three equal sides (5B) Scale 5B (0, 3, 5) Partial credit: (3 marks) Correct answer but no reason or incorrect reason given. 9(ii) What strips can Lisa use to make a parallelogram? (5A) Answer 4 cm, 4 cm, 6 cm and 6 cm Scale 5A (0, 5) Hit or miss. 9(iii) How many different isosceles triangles can Lisa form using the strips shown above? Explain your answer. Answer 3 Explanation using strips 4 cm, 4 cm and 6 cm using strips 6 cm, 6 cm and 4 cm using strips 6 cm, 6 cm and 9 cm (5C) Scale 5C (0,, 3, 5) Low partial credit: ( marks) Correct answer but no explanation or incorrect explanation given. High partial credit: (3 marks) Correct answer with one part of explanation missing. 9(iv) If Lisa wanted to form a right-angled triangle, which three strips could she use? Explain your answer. Answer 0 cm, 1 cm and 9 cm Explanation Using Pythagoras: (5B) Scale 5B (0, 3, 5) Partial credit: (3 marks) Correct answer but no reason or incorrect reason given. 014 J.18/0_MS 45/56 Page 45 of 56 exams

46 Question 10 (suggested maximum time: 10 minutes) (30) The table below gives the equations of five lines. Line Equation Slope a y x + 3 b x + y 3 c 3x + y t d y x + e y 5x (i) Write down the slope of each line in the table above. (10D) Line Equation Slope a y x + 3 m a b x + y 3 m b c 3x + y t m c 3 d y x + 1 m d e y 5x + 6 m e 5 Scale 10D (0, 5, 8, 9, 10) Low partial credit: (5 marks) One or two correct answers. Middle partial credit: (8 marks) Three correct answers. High partial credit: (9 marks) Four correct answers. 10(ii) Which line has the greatest slope? Give a reason for your answer. (5B) Answer line e or y 5x + 6 Reason m e or 5 is the biggest Scale 5B (0, 3, 5) Partial credit: (3 marks) Correct answer but no reason or incorrect reason given. 10(iii) Which lines are perpendicular? Give a reason for your answer. (5B) Answer lines a and d or y x + 3 and y x + Reason m a m d 1 1 Scale 5B (0, 3, 5) Partial credit: (3 marks) Correct answer but no reason or incorrect reason given. m a m d J.18/0_MS 46/56 Page 46 of 56 exams

47 10(iv) The line c contains the point (1, 1). Find the value of t. (5B) c (1, 1) 3x + y t 3(1) + ( 1) t 3 1 t t Scale 5B (0, 3, 5) Partial credit: (3 marks) (1, 1) correctly substituted. 10(v) Which lines makes equal intercepts of the y-axis? Give a reason for your answer. (5B) Line Equation y-axis intercept a y x + 3 y 3 b x + y 3 y 3 c 3x + y t y d y x + y 1 e y 5x + 6 y 6 Answer lines a and b or y x + 3 and x + y 3 Reason both lines intersects the y-axis at y 3 Scale 5B (0, 3, 5) Partial credit: (3 marks) Correct answer but no reason or incorrect reason given. 014 J.18/0_MS 47/56 Page 47 of 56 exams

48 Question 11 (suggested maximum time: 5 minutes) (10) Using only a compass and a straight-edge, divide the line segment below into four equal parts. Show all construction work. (10C) ** Allow tolerance of ±0 cm. ** All construction lines and arcs must be shown. Scale 10C (0, 3, 7, 10) Low partial credit: (3 marks) Any correct construction line or arc. High partial credit: (7 marks) Accurate construction but without construction lines and/or arcs. 014 J.18/0_MS 48/56 Page 48 of 56 exams

49 Question 1 (suggested maximum time: 5 minutes) (15) 1(i) Construct a right-angled triangle containing an angle A such that cos A 7. (10C) 7 A Scale 10C (0, 3, 7, 10) Low partial credit: (3 marks) cos 1 correctly found and/or triangle 7 drawn. Rough diagram drawn with correct measurements shown. High partial credit: (7 marks) Correct diagram fully drawn but angle A not labelled. One side incorrect on diagram but fully labelled. 1(ii) Find tan A, giving your answer in surd form. (5C) cos A O + A H O + 7 O O 45 O 45 Adjacent Hypoteneuse J.18/0_MS 49/56 Page 49 of 56 exams tan A Opposite Adjacent Scale 5C (0,, 3, 5) Low partial credit: ( marks) Any work of relevance, e.g. Pythagoras Opp stated with some substitution, tan. Adj High partial credit: (3 marks) O 45 or O 45. Tan A

50 Question 13 (suggested maximum time: 5 minutes) (0) 13(i) In the table below, show the values of cos and sin of the angles listed. Give your answers correct to three significant figures. (10D) cos sin Scale 10D (0, 5, 8, 9, 10) Low partial credit: (5 marks) One correct answer. Middle partial credit: (8 marks) Two correct answers. High partial credit: (9 marks) Three correct answers. 13(ii) What can you conclude from the above results? Give a reason for your answer. (5B) Conclusion Any 1: cos30 sin // cos40 sin // cos50 sin // cos60 sin Reason Any 1: cosθ sin (90 θ) // sinθ cos (90 θ) // or similar Scale 5B (0, 3, 5) Partial credit: (3 marks) Correct answer but no reason or incorrect reason given. 13(iii) Brid notices during an examination that her calculator is faulty. The sin function is giving her an error message. Assuming all other functions are working correctly, explain how she might use her calculator to calculate the value of sin 4. (5B) Use calculator to calculate cos 48 as: sin4 cos (90 4 ) cos48 Scale 5B (0, 3, 5) Partial credit: (3 marks) Mention of 90 4 or J.18/0_MS 50/56 Page 50 of 56 exams

51 Question 14 (suggested maximum time: 10 minutes) (0) The Costa Concordia disaster was the partial sinking of the Italian cruise ship when it ran aground at Isola del Giglio during the night of 13 January 01. When planning to raise the ship, engineers calculated various angles and distances. Using the information shown in the diagram, find the height of the top of the smoke stack, marked A, above the level of the water. (10C*, 10C*) A 34 5m 76 Let H common hypotenuse of both triangles cos A Adjacent Hypotenuse cos 76 5 H H 5 cos m Let h distance between the top of the smoke stack marked A and the water level sin A Opposite Hypotenuse sin 34 h h (sin 34 ) ( ) m H (common hypotenuse of both triangles) Scale 10C* (0, 3, 7, 10) Low partial credit: (3 marks) Trig. value with some substitution, Adj e.g. cos 76. Hyp High partial credit: (7 marks) Trig. value with full substitution, 5 5 i.e. H or. cos (10C*) * Deduct 1 mark off correct answer only if incorrectly rounded, no rounding, for omission of or incorrect units - apply only once throughout the question. h (distance between the top of the smoke stack marked A and the water level) Scale 10C* (0, 3, 7, 10) Low partial credit: (3 marks) Trig. value with some substitution, Opp e.g. sin 34. Hyp (10C*) High partial credit: (7 marks) Trig. value with full substitution, i.e. h (sin 34 ) or ( ). * Deduct 1 mark off correct answer only if incorrectly rounded, no rounding, for omission of or incorrect units - apply only once throughout the question. 014 J.18/0_MS 51/56 Page 51 of 56 exams