JUNIOR CERTIFICATE EXAMINATION

Transcription

1 JUNIOR CERTIFICATE EXAMINATION 01 MARKING SCHEME MATHEMATICS ORDINARY LEVEL PAPER 1 Page 1

2 GENERAL GUIDELINES FOR EXAMINERS 1. Penalties of three types are applied to candidates work as follows: Blunders - mathematical errors/omissions (-3) Slips- numerical errors (-1) Misreadings (provided task is not oversimplified) (-1). Frequently occurring errors to which these penalties must be applied are listed in the scheme. They are labelled: B1, B, B3,, S1, S,, M1, M, etc. These lists are not exhaustive.. When awarding attempt marks, e.g. Att(3), 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 is awarded zero marks. Some examples of such work are listed in the scheme and they are labelled as W1, W, etc. 4. The phrase hit or miss means that partial marks are not awarded the candidate receives all of the relevant marks or none. 5. The phrase and stops means that no more work is shown by the candidate. 6. Special notes relating to the marking of a particular part of a question are indicated by an asterisk. These notes immediately follow the box containing 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, verifications and answers derived from diagrams (unless requested) qualify for attempt marks at most. 11. A serious blunder, omission or misreading results in the attempt mark at most. 1. Do not penalise the use of a comma for a decimal point, e.g may be written as 5,50. Page

3 BONUS MARKS FOR ANSWERING THROUGH IRISH Bonus marks are applied separately to each paper as follows: If the mark achieved is 5 or less, the bonus is 5% of the mark obtained, rounded down. e.g. 198 marks 5% 9.9 bonus 9 marks. If the mark awarded is above 5, the following table applies: Bunmharc (Marks obtained) 6 Marc Bónais (Bonus Mark) 11 Bunmharc (Marks obtained) Marc Bónais (Bonus Mark) Page 3

4 QUESTION 1 Part (a) 10 marks Att (3) Part (b) 0 (5, 5, 5, 5) marks Att (,,, ) Part (c) 0 (10,5,5) marks Att (3,,) (a) 10 marks Att 3 1. (a) S =,,,, P =,, Fill the elements of S and P into the following diagram. S P (a) 10 marks Att 3 S r s u p t w P Slips 1 S1 Each element incorrectly filled into diagram S Each element omitted from diagram but see W1 S3 Each unlisted element used but see W1 (some relevant element must be present to use S3) Misreading (-1) M1 Interchanging S and P totally Attempts (3 marks) A1 Totally incorrect filling of the Venn diagram using given elements A Correct number of dots in each set without labels W1 No filling in of the Venn diagram or use of unlisted elements only but see S3 Page 4

5 (b) 0 marks (5,5,5,5) Att,,, (b) A = {1,, 3, 6} is the set of the divisors of 6. B = {1,, 4, 8} is the set of the divisors of 8. C = {1,, 4, 5, 10, 0} is the set of the divisors of 0. A B C List the elements of: (i) B C (ii) A \ B C (iii) B C (iv) the common divisors of 6, 8 and 0. (i) 5 marks Att B C= {1,,4,5,8,10,0} B1 Any incorrect set of the elements of B and C other than the misreading below Misreading (-1) M1 B C = {1,, 4} Attempts ( marks) A1 3or 6 appear in the answer (ii) 5 marks Att A \ B C {3,6} B1 Any incorrect set of the elements of A, B and C Misreading (-1) M1 (A\B) C = {3,5, 6,10,0,1,,4 } Page 5

6 (iii) 5 marks Att B C= {1,, 4} B1 Any incorrect set of the elements of A, B and C other than the misreading below Misreading (-1) M1 B C giving {1,, 4, 5, 8, 10, 0} Attempts ( marks) A1 1, or 4 appear in the answer (iv) 5 marks Att the common divisors of 6, 8 and 0 = { 1, } S1 Each missing or incorrect element to a max of 3 Attempts ( marks) A1 Any correct divisors of 6,8 or 0 appears, but see S1 A Ans. 10 W1 Elements listed that are not divisors of 6, 8 or 0 (c) 0 ( 10,5,5) Marks Att 3,, (c) In a survey, 60 households were asked if they had a cat (C) or a dog (D). 0 said they had a cat. 5 said they had a dog. 1 said they had both a cat and a dog. (i) Represent this information in the Venn diagram below. C D (ii) How many households had only a cat or a dog? (iii) What percentage of households had neither a cat nor a dog? Page 6

7 (c)(i) 10 marks Att 3 C D B1 Each incorrect or omitted entry (unless consistent error) in Venn diagram subject to S1 below. S1 Numerical errors where work is clearly shown Misreading (-1) M1 Interchanges cats and dogs Attempts (3 marks) A1 Any one correct/relevant entry c(ii) 5 marks Att = 1 *A correct answer written in the space provided takes precedence over an incorrect Venn diagram. *Accept candidates work from previous part c (i). B1 Any incorrect use of the given numbers or the numbers from the candidates incorrect Venn diagram [Subject to S1]. S1 Numerical errors where work is clearly shown S Fails to add their correct relevant figures Page 7

8 c(iii) 5 marks Att % 60 *A correct answer written in the space provided takes precedence over an incorrect Venn diagram. *Accept candidates work from previous parts (c) (i), (c) (ii). B1 No work shown B Mishandles the percentage B3 Any incorrect use of the given numbers or numbers from the previous work [Subject to Second *above] B4 Fails to find the percentage Misreading (-1) M1 100 or similar and continues S1 Numerical errors where work is clearly shown, to a max of 3 Attempts ( marks) A1 Any one correct/relevant step A 100 appears W1 Incorrect answer with no work shown Page 8

9 QUESTION Part (a) 10 marks Att 3 Part (b) 0 (10,5, 5) marks Att (3,, ) Part (c) 0 (5, 10, 5) marks Att (, 3,) (a) 10 marks Att 3 (a) 3 packets of soup cost What would be the cost of 5 packets of the same soup? (a) 10 marks Att 3 1costs. = 1.17 Or 3:5 Or 3:5 = 3.51 : x. 5 cost =5.85 = 1.17 =. Ans = x = = x =. = 5.85 *Correct answer without work *Special Case *Stops at 1.17 or *Stops at (=17.55) 7marks marks 4marks (no use of 5, B(-3) and B4 or B5) 4 marks ( no use of 3 and possible slips) B1 Divisor 3but see above B incorrect multiplier B3 5:3 = 3.5 : x and continues B4 Error in decimal point (apply once) B5 Fails to finish S1 Numerical errors where work is clearly shown, to a max of -3 Attempts (3 marks) A1 Indicates or 3:5 or 3.51: x only and stops A 1.17 or or or ( only) appear with no work shown A3 only appears A4 (3.51 3) or (3.51 5) and stops A is multiplied or divided by any wrong number correctly W1 Incorrect answer without work but see A1 and A W = 6.54 or similar, and stops Page 9

10 (b) 0 marks (10, 5, 5 ) Att (3,, ). (b) (i) By rounding each of these numbers to the nearest whole number, 4 31 estimate the value of is approximately equal to: = = (ii) Using a calculator, or otherwise, find the exact value of (iii) Find the difference between the exact value in (ii) and the estimated value in (i). b(i) 10 marks Att is approximately equal to: = = * 16 4 and stops or 16 4 = 1 7 marks *No penalty if the intermediate step between approximations and final answer not shown i.e. 4 not shown. 10 marks. 1 *Special Case: =.05 in this part 3 marks. Page 10

11 B1 Correct answer without work B Error(s) in rounding off to the nearest whole number (once only if consistent) B3 Decimal error in calculation of approximate value B4 An arithmetical operation other than indicated. B5 4 =.5 or (4 4 16) = 10 (breaking order) or similar and continues S1 Numerical errors to a max of -3. Attempts (3 marks) A1 Only one or two approximations made to the given numbers and stops W1 Wrong answer without work but note Special Case above b(ii) 5 marks Att B1 Decimal error or early rounding off B Fails to finish B3 Treats as ( ) 3.78 = B4 Treats as: ( ) 15.6 = B5 Treats as: 4.31 ( ) = B6 Treats as: ( ) = S1 Numerical errors to a max of 3 Attempts ( marks) A1 Any correct relevant step and stops. A Any of the following (see above): , , , or = or = (minimum 4 decimal places) with or without work W1 Wrong answer without work but see A Page 11

12 b(iii)` 5 marks Att = 0 05 or *Allow candidate s figures B1 Fails to finish B Decimal error (once only if consistent) B3 Finds the sum of (i) and (ii) S1 Numerical errors to a max of -3 Attempts ( marks) A1 Any relevant step i.e. transfers answers from (i) and/or (ii) W1 Incorrect answer without work (c)` 0 (5,5,5,5) marks Att,,, (c) (i) Using a calculator, or otherwise, multiply Then express your answer in the form a 10 n, where 1 a <10 and 7 a n (ii) Write in the form a 3, where n N. a 7 11 Hence or otherwise evaluate (iii) It takes three workers four days to build a wall. How long would it take two workers to build the same wall? n N. c(i) 5 marks Att = = * 3.51 or (without work) 4 marks B1 Decimal error B An arithmetic operation other than that indicated e.g = S1 Numerical errors to a max of -3 S Rounds off to S3 Incorrect format, where a 1or a 10 S4 Finds and stops and n Z Attempts ( marks) A1 Any relevant step and stops Page 1

13 c (ii) 5marks Att a a 7 3 a 7 3 a 4 or a a a a a a a a a a a a 7 3 = a 4 * a a a a and stops 4marks * a 7-3 and stops 4marks B1 Each error in calculation involving indices B Each incorrect number of a s in the extended form B3 Each incorrect elimination of the a s in extended form S1 Numerical errors to a max of -3 Attempts ( marks) A1 Some correct manipulation of indices A 4 only written down W1 Writes a only or incorrect answer with no work shown other than A c(ii)hence 5 marks Att *Accept candidate s answer from above unless it oversimplifies the question B1 Each error in calculation involving indices B Each incorrect number of 11 s in the extended form B3 Fails to finish B4 Each incorrect elimination of the 11 s in extended form S1 Numerical errors to a max of -3 Attempts ( marks) A1 Some correct manipulation of indices A 11 = 11 or similar and stops A3 Candidate transfers their answers from above W1 Incorrect answer with no work shown Page 13

14 c(iii) 5 marks Att 1 man takes 3 4 days = 1days 1 men take 6 days * 4 8 Special case: 3 3 marks 4 ( ) * Stops at B1 Incorrect answer without work B1 Divisor and continues B Incorrect multiplier ( 3) or fails to multiply, or fails to multiply but see 1st * S1 Numerical errors where work is clearly shown to a max of -3 Attempts ( marks) A1 Mentions one man or man days A 1 or only appear (no work shown) 4 A3 4 or 3 and stops A4 4 is multiplied or divided by any wrong number, correctly Worthless(0) W1 Incorrect answer without work but see A above W = 7 or similar W3 hours only with no mention of 3 or 4 or (96 on its own) Page 14

15 QUESTION 3 Part (a) 10 marks Att 3 Part (b) 0 (10,10) marks Att (3,3) Part (c) 0 ( 10, 10) marks Att (3,3) (a) 10 marks Att 3 3. (a) The cost of a holiday came to 400. This was made up of the cost of travel, accommodation and spending money. 3 of the cost was for travel and accommodation. 5 How much spending money was there? (a) 10 marks Att = = travel + acc => spend. =60% => = 40% = = * No penalty for omitting symbol B1 Correct answer without work 3 B 400 (method 1) 5 B (method ) B4 Calculates the travel and accommodation and stops (method 1 ) B5 Operation other than subtraction in final step or omits final step. (method 1) B6 Finds 60% 0f 400 and stops (same as B5) S1 Numerical errors (to max -3) Attempts (3 marks) A1 Any attempt at getting 5 3 of 400 or 5 of 400 A Writes down 5 or 40% Page 15

16 (b) 0 (10,10) Marks Att (3,3) (b) (i) Amanda borrows She agrees to pay it back at 90 per month for a year. How much interest will she pay? (ii) A computer is ordered online. It is advertised for 550 plus VAT at 3%. There s a delivery charge of What is the total cost to be paid? (b) (i) 10 marks Att 3 Amanda borrows She agrees to pay it back at 90 per month for a year. How much interest will she pay? b(i) 10 marks Att = 1080 Int: = 80 * No penalty for omitting symbol B1 Correct answer without work B 90 1=1080 and stops B = 7.5 and continues correctly B4 Multiplies 90 by some whole number other than 1 and continues B5 Fails to finish S1 Numerical errors (to max -3) Attempts (3 marks) A1 Oversimplification A Multiplies 90 by some number other than 1 and stops Page 16

17 (b) (ii) 10 marks att 3 100% = % = 1% = VAT = 550 = Total Cost = = % = 13 = = Total Cost = = 684 Total Cost = = 684 * No penalty for omitting symbol B1 Correct answer without work B Decimal error B3 Inverts as and continues (giving answers or ) B4 Mishandles 3 % eg Note: (550 must be used} B5 550 taken as 13% and finds his 100% and continues B6 No addition of VAT (as per candidates work) to the bill B7 No addition of the delivery charge B8 Subtraction of VAT ( as per candidates work) from the bill B9 No addition of 550 S1 Numerical errors to a max of -3 Misreadings (-1) M1 Reads as 3% or 500 Attempts (3 marks) A1 and stops or A 100% = 550 and stops A3 100 and stops or A % and stops A and stops and stops and stops W1 Incorrect answer without work W = 573 and stops or continues Page 17

18 Part (c) 0(10,10) marks Att (3,3) (i) A work of art is priced at After VAT is added it costs Calculate the amount of VAT and the rate of VAT. (ii) Ronan was given a bicycle which was in need of repair. For the repairs, he spent 60 on spare parts and 1 on paint. When it was repaired he sold it for 95. Calculate the profit he made as a percentage of his costs. Give the percentage to the nearest whole number. (c) (i) 10 marks Att3 A work of art is priced at After VAT is added it costs Calculate the amount of VAT and the rate of VAT. (c) (i) 10 marks Att = 891 = VAT % * No penalty for omitting symbol * = 891 = 13.5% 10 marks *Stops after marks ( and stops still only 4 marks) B1 Correct answer without work. B Decimal error eg 1.35% B3 Inverts as and continues ( to get %) B , by 6600 and continues correctly B5 Mishandles the finding of the rate of vat B6 100 to get 11.89% = 1% B7 Rounds off to 14% without showing 13.5% B8 Fails to finish S1 Numerical errors (apart from decimal errors) max of -3 Attempts (3marks) A1 Some use of 100 A Some attempt at subtraction Page 18

19 (c) (ii) 10 marks Att3 Ronan was given a bicycle which was in need of repair. For the repairs, he spent 60 on spare parts and 1 on paint. When it was repaired he sold it for 95. Calculate the profit he made as a percentage of his costs. Give the percentage to the nearest whole number. (c) (ii) 10 marks Att = = 3 Profit = 3 % = % (13% accept) = = 3% * No penalty for omitting symbol *Answer 3 4 marks * 100 and stops 6 marks B1 Correct answer without work B Adds 95 to 7 and continues B3 Calculates profit as percentage of selling price. ie. B4 Divisor not equal to 7 B5 Mishandles the calculation of profit as a percentage B6 Fails to multiply by % 4% S1 Numerical errors to a max of -3 S Fails to round off to the nearest whole number Attempts (3 marks) A1 Some indication of subtraction A Some use of 100 A (= 7) Worthless (0 marks) W1 Incorrect answer without work = 0 marks. Page 19

20 QUESTION 4 Part (a) 15(10,5) marks Att 3, Part (b) 15 (5,5,5) marks Att (,,) Part (c) 0 ( 5,5, 10) marks Att (,,3) (a) 10,5 marks Att 3, (a) If a = 4 and b = 5, find the value of: (i) a + b (ii) ab 3 (a)(i) 10 marks Att 3 (i) a + b = (4) + 5 = = 13 *8 +5 (only) 9 marks *One substitution coupled with an implied substitution leading to correct answer e.g. = a +5 = marks. B1 Correct answer without work B Leaves (4) in the answer B3 Breaks order i.e. (4+5) = 18 B4 Treats (4) as 6 or 4 S1 Numerical errors to a max of 3 S Values of a and b interchanged. Misreadings (-1) M1 Incorrect numerical substitution for either a or b, but not both, and continues (See W1) or a + b calculated out Attempts (3 marks) A1 Incomplete substitution and stops e g a + 5 W1 Incorrect substitution for both a and b Page 0

21 (a)(ii) 5 marks Att (ii) ab 3 = = 0 3 = 17 *0 3 (only) 4 marks *One substitution coupled with an implied substitution leading to correct answer e g 4b 3 = 17 or 5a 3 =17 5 marks B1 Correct answer without work B Leaves 4(5) in the answer B3 Breaks order i.e. 4(5-3) = 8 B4 Treats 4(5) as 9 or 45 S1 Numerical errors to a max of -3 Misreadings (-1) M1 Incorrect numerical substitution for either a or b, but not both, and continues (See W1) Attempts ( marks) A1 Incomplete substitution and stops e g 4b -3 W1 Incorrect substitution for both a and b. Page 1

22 (b) 15 (5,5,5) Marks Att,, (b) f(x) = x 1. (i) Draw a graph of f(x) in the domain 1 x 1, x R. (ii) Use your graph to estimate the value of x when f(x) = 0. (b)(i) 5 marks Att f(x) = x 1 f(-1) = (-1) 1= - -1 = -3 (-1,-3) f(0) = (0 ) 1 = 0 1 = -1 (0,-1) f(1) = (1) 1= 1 = 1 (1,1) OR f(x) = x 1 f(-1) = (-1) 1 = -3 (-1,-3) f(0) = (0 ) 1 = -1 (0,-1) f(1) = (1) 1 = 1 (1,1) OR x x f (x) * Error(s) in each row/column calculation attracts a maximum deduction of 3marks * points correct (full marks) _ (need not be in domain) B1 + x taken as all the way. [In the row headed + x by candidate] B -1 calculated as - x all the way. [In the row headed -1 by candidate] B3 Adds in top row when evaluating f (x) in Box B4 Omits -1 row B5 Omits + x row B6 Takes x as + x and applies it in his calculations B7 Each incorrect image without work i.e. calculation through the function method S1 Numerical errors to a max of -3 in any row / column Misreadings (-1) M1 Misreads -1 as +1 and places +1 in the table or function. M Misreads x as x and places x in the table or function Attempts ( marks) A1 Any effort at calculating point(s) A Only one point calculated and stops Page

23 (b)(ii) 5 marks Att * Answers need not be written in table.. *Accept candidate s value from (i) but see B1 and S4 (see S) *Tolerance ± 0.5 ( ± 1Box on grid) *Correct graph but no table award full marks i.e. (5 + 5) *Only one correct point graphed correctly but no table Att + Att *Accept reversed co-ordinates if (i) if axes not labelled or (ii) if axes are reversed to compensate (see B1 below) B1 Full domain not covered B Scale error (once) B3 Reversed co-ordinates plotted against non-reversed axes (once only) {See 6 th * above} S1 All points not joined or joined in incorrect order S Each incorrectly plotted point S3 Each point { points needed } from table not graphed [ See nd * above ] S4 Not a straight line if not already penalised in b(i) or b(ii) but see nd * Attempts ( marks) A1 Graduated axes (need not be labelled) A Some effort to plot a point { See nd * above} A3 Random straight line with or without axes A4 One correct point, with /without work Page 3

24 b(iii) 5 marks Att Answer to be written here: x= 0 5 when f(x) = 0 * Allow candidate s figures B1 Fails to finish but draws some relevant line S1 Numerical errors to a max of -3 S Correct answer indicated and/or written on graph only Attempts ( marks) A1 Some correct indication on graph A Attempts at algebraic evaluation or calculator A3 Finds answer -1 i.e. find x = 0 (where crosses y-axis) W1 Wrong answer without work (c) 0(5,5,10) marks Att,,3 (c) (i) Conor spent y on a book. He then spent (4y + 6) on a football jersey. In total, he spent 61. Write an equation in y to represent this information. (ii) Solve your equation from (i) to find the value of y. (iii) Solve the equation: x 5x 14 = 0. c(i) 5 marks Att y + 4y + 6 = 61 5y + 6 =61 B1 Incorrect expression for the cost of a book and football jersey other than misreading below S1 No 61 included in answer Misreadings (-1) M1 Answer given as y + 4y 6 = 61 or similar Attempts ( marks) A1 Any effort at forming an expression (y included) W1 Cost of book given as a constant Page 4

25 c(ii) 5 marks Att 5y + 6 =61 5y = y = 55 y = 11 * Accept candidates answer from previous work. Blunders(-3) B1 Correct answer without work B Error in forming equation B3 Distribution error B4 Transposition error B5 Stops at 5y = 55 or fails to solve equation B6 Error in collecting like term Misreadings (-1) M1 Transfers information in (i) incorrectly if not oversimplied S1 Numerical errors to a max of -3 Attempts ( marks) A1 Answer from part c (i) written down and stops. A Any effort at forming an expression A3 Any effort at solving their equation A4 Successful Trial and Error Worthless (0 marks ) W1 Incorrect answer with no work c(iii) 10 marks Att 3 x - 5x 14 = 0 x - 5x 14 = x - 7x + x 14 = 0 1 x(x -7) + (x 7) = 0 x (x + )(x 7) = (x + ) = 0 or (x 7) = 0 x x = - or x = 7 and 7 (x + ) = 0 or (x 7) = 0 x = - or x = 7 * correct solutions by Trial and Error 10 MARKS * 1 correct solution by Trial and Error 3 MARKS (Attempt) Page 5

26 Factor Method B1 Correct answers without work B Incorrect two term linear factors of x -5x -14 formed from correct (but inapplicable) factors of x and/or ±14,e.g. (x+14)(x-1) B3 No roots given, or two incorrect roots (once only) B4 Incorrect factors of x and/or ±14 B5 Correct cross method but factors not shown and stops [Note: B3 applies also] B6 x(x 7) + (x 7) or similar and stops [Note: B3 applies also]. B7 Error(s) in transposition S1 Numerical errors to a max of -3 S One root only from factors Attempts (3 marks) A1 Some effort at factorization e.g. (x )( ) or the cross with at least one x written in A States one correct root without work Worthless (0 marks) W1 x 5x = 14 or similar and stops W Incorrect Trial and error W3 Oversimplification, resulting in a linear equation Formula Method B1 Error in a,b,c substitution (apply once only) B Sign error in substituted formula (apply once only) B3 Error in square root or square root ignored B4 Stops at B5 Incorrect quadratic formula and continues S1 Numerical errors to a max of -3 S p Roots left in the form q S3 One root only Attempts (3 marks) A1 Correct formula and stops A One correct substitution and stops Page 6

27 QUESTION 5 Part (a) 10 marks Att 3 Part (b) 0 (5,5.10) marks Att (,,3) Part (c) 0 ( 10, 10) marks Att (3,3) (a) 10 marks Att 3 (a) Simplify fully (x + 1) + 5(x + 3). (a) 10 marks Att 3 (x + 1) + 5(x + 3) = x + +10x +15 = 1x + 17 * Stops after correct removal of brackets 7 Marks * Gathering of terms at most one blunder B1 Correct answer without work B Error(s) in distribution (each time) B3 Combining unlike terms after removal of brackets and continues B4 Fails to group like terms B5 Fails to finish S1 Numerical errors to a max of -3 Misreadings (-1) M1 (x+) and continues Attempts (3 marks) A1 Any one term correctly multiplied A Combines unlike terms at the start and finishes correctly W1 Combining unlike terms before attempting multiplication and stops e.g. 5(5x) = 5x Page 7

28 (b) 0 (5,5,10) marks Att,,3 (i) Factorise 5xy + 3y. (ii) Factorise ax + ay + 3x + 6y. (iii) Solve for x and y: x + 5y = 19 3x y = 3 b((i) 5 marks Att 5xy + 3y = y(5x + 3 ) B1 Removes factor incorrectly Attempts ( marks) A1 Indication of common factor e g underlines y s and stops b(ii) 5 marks Att ax + ay + 3x + 6y = a(x + y) + 3(x + y) or x(a+3) + y(a + 3) = (a + 3)(x + y) = (a+3)(x+ y) *Accept also (with or without brackets) for 5 marks any of the following (a+3) and (x+y) [The word and is written down.] (a+3) or (x+y) [The word or is written down.] (a+3), (x+y) [A comma is used] B1 Correct answer without work B Stops after first line of correct factorization e.g. a(x + y) + 3(x + y) or equivalent i,e. x(a + 3) + y(a + 3) B3 Error(s) in factorising any pair of terms B4 Correct first line of factorisation but ends as (a+3).xy or equivalent S1 (a+3) (x+y) Attempts ( marks) A1 Pairing off, or indication of common factors and stops A Correctly factorises any pair and stops Page 8

29 b (iii) 10 marks Att 3 x + 5y = 19 OR 6x +15y = 57 Or 3x -3 = y 3x y = 3 X5-6x+y = -6 x +5(3x-3) = 19 x + 5y = 19 17y = 51 x +15x 15 = 19 15x 5y = 15 y = 3 17x = 34 17x = x= x + 15 = 19 17x = 34 () + 5y = 19 x = 4 x = 4 + 5y = 19 x = 4 + 5y = 19 5y = 15 y = 3 y = 3 *Apply only one blunder deduction (B or B3) to any error(s) in establishing the first equation; in terms of x only or the first equation in terms of y only *Finding the second variable is subject to a maximum deduction of -3 B1 Correct answers without work (stated or substituted) B Error or errors in establishing the first equation in terms of x only (17x = 34) or the first equation in terms of y only (17y = 51) through elimination by cancellation (but see S1) B3 Error or errors in establishing the first equation in terms of x only (17x = 34) or the first equation in terms of y only (17y = 51) through elimination by substitution (but see S1) B4 Errors in transposition when finding the first variable B5 Errors in transposition when finding the second variable B6 Incorrect substitution when finding second variable B7 Finds one variable only S1 Numerical errors to a max of -3 Attempt (3 marks) A1 Attempt at transposition and stops A Multiplies either equation by some number and stops A3 Incorrect value of x or y substituted correctly to find his correct nd variable A4 One correct answer without work (stated and substituted) Worthless (0 marks) W1 Incorrect values for x or y substituted into the equations Page 9

30 (c) 0 Marks (10,10) Att 3,3 (i) Write as a single fraction 3x 8 x. (ii) Solve the equation 3(x 7) 5(x 1) = 0. Verify your answer. c (i) 10 marks Att3 x 3x 4x 3x 7x * * * or + etc 7 marks and stops 4 Marks x 3x = 8 0 Marks or B1 Correct answer without work B Incorrect common denominator and continues B3 Incorrect numerator from candidate's denominator B4 Omitted or incorrect denominator S1 Numerical errors to a max of -3 Attempts (3 marks) A1 Any correct step. A Any correct common denominator found Worthless (0 marks) W1 ( and stops W Incorrect answer, with no work Page 30

31 c(ii) 10 marks Att 3 Solve 3(x 7) 5(x 1) = 0 6x -1-5x + 5 = 0 x - 16 = 0 x = 16 Verify 3(x 7) 5(x 1) 3((16) 7) 5(16 1 3( 3-7) 5(15) 3(5) 75 = 0 *Stops after correct removal of brackets 4 Marks *If changes -5 to +5 at the start: blunder (-3) *States x=16 (no work) and verifies correctly 7 Marks *States x=16 (no work) with no verification 4 Marks *Verifies correctly x=16 (not stated) Att 3 B1 Correct answer without work B Error(s) in distribution (each time) B3 Combining unlike terms (each time) and continues B4 Fails to group like terms B5 Error(s) in transposition (each time) B6 Fails to finish B7 Fails to verify S1 Numerical errors to a max of -3 S Incorrect or no conclusion from their work Misreadings (-1) M1 3(x+7) or similar and continues but see nd* above Attempts (3 marks) A1 Any one term correctly multiplied A Any correct step W1 combining unlike terms before attempting multiplication and stops e.g. 3(14x) = 4x W Invented answer verified but see * above W3 Incorrect answer with no work Page 31

32 QUESTION 6 Part (a) 10(5,5) marks Att, Part (b) 30 (15,15 ) marks Att (5,5) Part (c) 10 (5,5) marks Att (,) (a) 10(5,5) marks Att, 6. (a) P = {(1, a), (, a), (3, b), (4, c)}. Write out the domain and range of P. Domain = Range = (a) Domain 5 marks Att Domain = {1,, 3,4} S1 Each incorrect element omitted / included other than the misreading below. Misreadings (-1) M1 Correct range { a, b, c } or { a, a, b, c } given. W1 No element of the domain appears. (a) Range 5 marks Att *Accept { a, a, b, c } for full marks. S1 Each incorrect element omitted / included other then the misreading below Misreadings (-1) M1 Correct domain {1,, 3, 4} given W1 No element of the range appears. Range = { a, b, c } Page 3

33 Part (b) 30(15, 15) marks Att (5, 5) Draw the graph of the function f : x 5 x x in the domain x 4, where x R. Table 15marks Att 5 f ( x) 5 x x f ( ) 5 ( ) ( ) = f ( 1) 5 ( 1) ( 1) = 5 1 f ( 0) 5 (0) (0) = f ( 1) 5 (1) (1) = 5 1 f ( ) 5 () () = f ( 3) 5 (3) (3) = f ( 4) 5 (4) (4) = = 3 (, 3) = ( 1,) = 5 (0, 5) = 6 (1, 6) = 5 (, 5) = (3, ) = 3 (4, -3). OR A f ( ) = 5 ( ) - ( ) = -3 f ( 1) = 5 ( 1) - ( 1) = f (0) = 5 (0) - ( 0) = 5 f (1) = 5 (1) - ( 1) = 6 f () = 5 () - ( ) = 5 f (3) = 5 (3) - ( 3) = f (4) = 5 (4) - ( 4) = -3 B x x x f (x) *Error(s) in each row/column calculation attracts a maximum deduction of 3 marks B1 Correct answer, without work i.e. 7 correct couples only and no graph B + x taken as all the way. [In the row headed + x by candidate] B3 5 calculated as 5 x all the way. [In the row headed 5 by candidate] B4 Adds in top row when evaluating f (x) in B. B5 Omits 5 row B6 Omits + x row B7 Omits a value in the domain (each time). B8 Each incorrect image without work i.e. calculation through the function method (A) B9 Misreads x as x and places x in the table or function. S1 Numerical errors to a max of -3 in any row / column Page 33

34 Misreadings (-1) M1 Misreads x as x and places x in the table or function. M Misreads 5 as -5 and places -5 in the table or function. Attempts (5 marks) A1 Omits x row from table or treats x as x or x. A Any effort at calculating point(s). A3 Only one point calculated and stops. Graph 15 marks Att * Only one correct point graphed correctly Att 5 + Att 5 * Correct graph but no table full marks i.e. ( ) marks. * Accept reversed co-ordinates if (i) if axes not labelled or (ii) if axes are reversed to compensate (see B1 below) Blunders ( -3) B1 Reversed co-ordinates plotted against non-reversed axes (once only) {See 3rd * above}. B Scale error (once only) B3 Points not joined or joined in incorrect order (once only) S1 Each point of candidate graphed incorrectly {Tolerance 0. 5 } S Each point (7 points needed ) from table not graphed [ See nd * above ] Attempts (5 marks) A1 Graduated axes (need not be labelled) A Some effort to plot a point { See 1st * above} Page 34

35 Part (c) 10 (5, 5) marks Att, (c) (i) Draw the axis of symmetry of the graph you have drawn in 6(b). (ii) Use your graph to estimate the value of 5 x x when 1 5 x. (c) (i) 5 marks Att (c) (i) Draw the axis of symmetry of the graph you have drawn in 6(b). Axis of Symmetry: x = 1 f(1.5) = * Accept any vertical line (parallel to candidate s y-axis) within tolerance of Blunders ( -3) B1 Any vertical line ( parallel to the candidate s y-axis) outside of the tolerance. B Marks x 1 on the x-axis and stops. B3 States x 1 but no line is indicated on the graph. Attempts ( marks) A1 Any attempt at axial symmetry of f (x). A y-axis indicated as the axis of symmetry (See B1). Page 35

36 (c) (ii) 5 marks Att (c) (ii) Use your graph to estimate the value of 5 x x when x 1. 5 Work to be shown on the graph and answer to be written here *Correct answer (clearly consistent with candidate s graph) inside the tolerance without graphical indication marks. B1 Correct answer without work B Answer on the diagram but outside of tolerance ( 0. 5 ) B3 Fails to write down the answer, when indicated correctly on graph S1 Correct answer indicated and/or written on graph only Attempts ( marks) A1 Attempts at algebraic evaluation or calculator A Marks 1.5 in any way on either axis and stops W1 Answer outside of tolerance without graphical indication. Page 36

37 Coimisiún na Scrúduithe Stáit State Examination Commission Scrúdu an Teastais Shóisearaigh JUNIOR CERTIFICATE EXAMINATION 011 MARKING SCHEME MATHEMATICS ORDINARY LEVEL PAPER 1 Page 1

38 GENERAL GUIDELINES FOR EXAMINERS 1. Penalties of three types are applied to candidates work as follows: Blunders - mathematical errors/omissions (-3) Slips- numerical errors (-1) Misreadings (provided task is not oversimplified) (-1). Frequently occurring errors to which these penalties must be applied are listed in the scheme. They are labelled: B1, B, B3,, S1, S,, M1, M, etc. These lists are not exhaustive.. When awarding attempt marks, e.g. Att(3), 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 is awarded zero marks. Some examples of such work are listed in the scheme and they are labelled as W1, W, etc. 4. The phrase hit or miss means that partial marks are not awarded the candidate receives all of the relevant marks or none. 5. The phrase and stops means that no more work is shown by the candidate. 6. Special notes relating to the marking of a particular part of a question are indicated by an asterisk. These notes immediately follow the box containing 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, verifications and answers derived from diagrams (unless requested) qualify for attempt marks at most. 11. A serious blunder, omission or misreading results in the attempt mark at most. 1. Do not penalise the use of a comma for a decimal point, e.g may be written as 5,50. Page

39 BONUS MARKS FOR ANSWERING THROUGH IRISH Bonus marks are applied separately to each paper as follows: If the mark achieved is 5 or less, the bonus is 5% of the mark obtained, rounded down. (e.g. 198 marks 5% = 9.9 bonus = 9 marks.) If the mark awarded is above 5, the following table applies: Bunmharc (Marks obtained) 6 Marc Bónais (Bonus Mark) 11 Bunmharc (Marks obtained) Marc Bónais (Bonus Mark) Page 3

40 QUESTION 1 Part (a) 15 (10,5) marks Att (3,) Part (b) 0 (5,5,5,5) marks Att (,,,) Part (c) 15 (5,5,5) marks Att (,,) (a) 10,5 marks Att 3, (a) S = (i) Write down a subset of S that has one element. (ii) Write down a subset of S that has three elements. (a) (i) 10 marks Att 3 {w} or {x} or {y} or {z} * No penalty for the omission of brackets. * No penalty for use of Venn Diagram to show subsets. B1 Any incorrect set of elements of S other than the misreading as below. Misreadings (-1) M1 Subset of S with two or three elements. e.g. S = {w, x}. Attempts (3 marks) A1 Draws a single bracket & stops. A { } Null set or set itself Worthless(0) W1 No relevant element listed without brackets but see A1 above (a) (ii) 5 marks Att {w,x, y} or {w, x, z } or {w,y, z} or {x, y,z} * No penalty for omission of brackets. * No penalty for use of Venn Diagram to show subsets. B1 Any incorrect set of elements of S other than the misreading as below. Misreadings (-1) M1 Correct subsets of S with one or two elements e.g. S = {w,x }.etc Attempts () A1 Draws a single bracket & stops. A { } Null set or set itself Worthless(0) W1 No relevant element listed without brackets but see A1 above Page 4

41 (b) 5,5,5,5 marks Att,,, U is the universal set. P = {, 6, 9} Q = {1, 3, 5, 6} R = {3, 4, 6, 7, 9} List the elements of: U P (i) R \ Q (ii) P, the complement of set P (iii) Q P R (iv) Q R \ P (i) 5 marks Att R \ Q = {4,7,9} B1 Any incorrect set of elements of Q and R other than the misreading below. Misreadings (-1) M1 Q \ R = {1,5} Attempts ( marks) A1 4 or 7 or 9 appear in the answer. A P Q / R ={ } Worthless(0) W1 {8} (ii) 5 marks Att P, the complement of set P = {1,3,4,5,7,8} B1 Any incorrect set of the elements of P and Q and R other than the misreading below Misreadings (-1) M1 P Q R giving { 1,,3,4,5,6,7,9 } (all needed) M R = {,1,5,8} M3 Q giving {,4,7,8,9 } M4 {,6,9} Attempts ( marks) A1 At least one correct entry appears in the answer Page 5

42 (iii) 5 marks Att Q P R {1,3,5,6,9 } B1 Any incorrect set of elements of Q, P or R other than the misreadings below. Misreadings (-1) M1 Q \ P R 1,5,3. M Q P R {6} M3 Q P R {1,,3,4,5,6,7,9 } M4 Q P R {3,6 } Attempts ( marks) A1 1,3,5,6, or 9 appear in the answer. but see Misreadings above Worthless(0) W1 Answer {8}. (iv) 5 marks Att Q R \ P= {3} B1 Any incorrect set of elements of P and Q and R other than the misreading as below. B (Q R) = {6,3} and stops Misreadings (-1) M1 Q \ R P 1,3,5. M Q \ ( R P) {1,5 } M3 Q ( R / P) {1,3,4,5,7 } Attempts ( marks) A1 6 or 3 appear in the answer. Worthless(0) W1 Answer {8}. (c) 5,5,10 marks Att,,3 (i) List all the divisors of 18 and 4. (ii) Write down the highest common factor of 18 and 4. (iii) {5, 7, 9, 11, 13, 15} is the set of odd numbers between 4 and 16. Which of these numbers are not prime numbers? Give a reason for your answer. (i) 5 marks Att Divisors of 18: = 1,,3, 6, 9, 18 Divisors of 4: = 1,, 3, 4, 6, 8, 1, 4 S1 Each missing or incorrect element to a max of 3 Attempts ( marks) A1 Any one correct element identified Worthless(0) W1 Elements listed that are not divisors of 18 or 4 Page 6

43 (ii) 5 marks Att Highest common factor = 6 *Accept candidate s answer from c(i) B1 A common factor that is not the highest S1 Answer written as 3 Misreadings (-1) M1 Writes down LCM = 7 Attempts ( marks) A1 Any common factor listed Worthless(0) W1 Incorrect answer without work but see M1 or * above (iii) 5 marks Att Not prime numbers: 9 and 15 Reason: Each has more than factors B1 Each incorrect or omitted entry S1 No or incorrect reason given Misreadings (-1) M1 Gives prime numbers only Attempts ( marks) A1 Any one relevant entry between 4 and 16 inclusive A Correct reason as to why numbers are not primes Worthless(0) W1 Incorrect answer with no work shown Page 7

44 QUESTION Part (a) 10 marks Att 3 Part (b) 0(10,5,5) marks Att (3,,) Part (c) 0(5,5,5,5) marks Att (,,,) (a) 10 marks Att 3 5 is divided between Fiona and Orla in the ratio 9:4. How much does each receive? (a) 10 marks Att 3 9+4= 13 OR 9+4 =13 OR 9x : 4x 13x = = 4 x = = =16 = 36 4x=16 Fiona: 36 Orla: 16 or 5 36 =16 9x = 36 * Correct answer without work 7 marks * Incorrect answer without work 0 marks, except for answers given in A4 below * = 13 and = / 5.78 or 5.8 merits 4 marks B1 Divisor 13 and continues B Incorrect multiplier or fails to multiply (each time) B3 Adds instead of subtracts i.e = 98 B4 Fails to find second amount B6 Error in transposition S1 Numerical errors where work is clearly shown to a max of 3 Attempts (3 marks) A1 Divisor 13 e.g, and/ or and stops A Indicates 13 parts or 9 parts or 4 parts or and stops A3 Indicates multiplication of 5by 9 and/or 4 and stops A4 Both answers added together equal 5 (no work shown) A5 Finds 9% of 5 (4.68) and 4% of 5 (.08) A6 One correct answer without work Worthless(0) W = 61 or similar W Incorrect answer without work. (subject to A4) Page 8

45 (b) 10,5,5 marks Att 3,, (i) By rounding each of these numbers to the nearest whole number, estimate the value of (ii) Using a calculator, or otherwise, find the exact value of (iii) Find the difference between the exact value in (ii) and the estimated value in (i). (i) 10 marks Att * * and stops 7 marks. and stops 7 marks.(-3) = * No penalty if the intermediate step between approximations and correct final answer is not shown i.e. not shown * Special Case:.. = 5.15 in this part Attempt 3 marks. (or or 5 ). B1 Error(s) in rounding off to the nearest whole number (once only if consistent) B Decimal error in calculation of final value B3 An arithmetic operation other than indicated e.g. 14 (4 ) = 7 (breaking order) B4 Error(s) in the manipulation of the denominator e.g. or B5 Incorrect cancellation S1 Numerical errors to a max of 3 Attempts (3 marks) A1 Only one approximation made to the given numbers and stops A Ans. 5 with no preceding rounding off W1 Incorrect answer without work but note Special Case * above Page 9

46 (ii) 5 marks Att B1 Decimal error or early rounding off B Leaves as.. B3 Treats as /1.96 = B4 Treats as = B5 Treats as = S1 Numerical errors to a max of 3 Attempts ( marks) A1 Any correct relevant calculation and stops. A Any of the following; (see above) , or merits marks (minimum 4 decimal places) (with or without work) W1 Incorrect answer without work but see A (iii) 5 marks Att = 0 15 * Allow candidate s previous answers B1 Correct answer without work B Decimal error (once only if consistent) B3 Finds the sum of b(i) and (ii) Attempts ( marks) A1 Any relevant step i.e. transfers answers from b(i) and/or b(ii) W1 Incorrect answer without work Page 10

47 (c) 5,5,5,5 marks Att,,, (i) Write (a 3 ) in the form a n, n N (ii) Using your answer from (i) or otherwise evaluate (5 3 ). Before going on holidays to the USA Seán changed 500 into dollars. The exchange rate was 1 = US$1. (iii) How many dollars did Seán get? (iv) When Seán came home he changed US$50 back into euro ( ). The exchange rate was the same. How much, in euro, did Seán receive? Give your answer to the nearest cent. (i) 5 marks Att (a 3 ) = a 3 = a 6 or (a 3 ) = a 3 a 3 = a 6 or a a a a a = a 6 * a a a a a and stops 4 marks * a 3 and stops 4 marks * 6 only written down marks B1 a 3 = a a a and stops B each error in calculation involving indices e.g. (a 3 ) = a 5 B3 Each incorrect number of a s in the extended form S1 Numerical errors to a max of 3 Attempts ( marks) A1 (a 3 ) = a 3+ and stops A Some correct manipulation of indices W1 Writes a only (ii) 5 marks Att (5 3 ) = 5 6 = 1565 or 5 3 = =1565 * Accept candidate s answer fom c(i) unless it oversimplifies the question B1 Correct answer, without work B Each error in calculation involving indices B3 Each incorrect number of 5 s in the extended form B4 Fails to finish S1 Numerical errors to a max of 3 Attempts ( marks) A1 Some correct manipulation of indices A 5 = 5 and stops A3 5 3 = 15 and stops A4 Candidate transfers answer from c(i) Worthless(0) W1 Incorrect answer with no work shown Page 11

48 (iii) 5 marks Att = $610 * No penalty for omission of or $ signs B1 Correct answer, without work B Incorrect operator i.e. divides by 1. correctly : B3 Decimal error B4 Fails to finish i.e and stops S1 Numerical errors to a max of 3 Attempts ( marks) A1 Some correct manipulation of 500 and/ or 1. Worthless(0) W1 Incorrect answer with no work shown (iv) 5 marks Att B1 Correct answer, without work B Multiplies by 1. `i.e = 61 B3 Incorrect ratio i.e.. or B4 Decimal error B5 Fails to finish i.e. leaves answer as. S1 Numerical errors to a max of 3 S Fails to round off or rounds off incorrectly Attempts ( marks) A1 Some manipulation of 50 and/ or 1. A If answer is 41 or 40.9 with no work shown but see W1 Worthless(0) W1 Incorrect answer with no work shown but see A Page 1

49 QUESTION 3 Part (a) 15 marks Att 5 Part (b) 15 (5,5,5) marks Att (,,) Part (c) 0 (5,5, 5,5) marks Att (,,,) (a) 10 marks Att 3 Three books were bought. They cost 8 75, 9 50 and respectively. If a 50 note was used to pay for the books, how much change was given? Part (a) 15 marks Att = ( ) = 1.0 or Change = = 1.0 Change = 1.0 *Accept 10 or 1.. *No penalty for the omission of the sign *Final subtraction step subject to maximum deduction of 3. B1 Correct answer without work 1 marks B Fails to find the change. B3 Operation other than addition when finding the total cost. B4 Operation other than subtraction when finding the change. B5 Each missing addition. B6 Decimal error eg..1 (Note 1 st * above). Slips ( 1) S1 Numerical errors to a max of 3 Attempts (3 marks) A1 Any attempt at addition or subtraction of the given numbers and stops W1 Incorrect answer without work is 0 marks. W Multiplication or division of the given numbers. Page 13

50 (b) 5,5,5 marks Att,, (i) A washing machine costs 30 plus VAT at 1 0%. Calculate the total cost of the washing machine after the VAT is added. (ii) A popular breakfast cereal comes in two sizes of packet, Regular (360 g) and Large (900 g). A standard portion of cereal is 30 g. How many portions are there in each size of packet? (iii) A Regular box costs 0 96 and a Large box costs 5. Using the number of portions per box, or otherwise, find which size is better value? (i) 5 marks Att 100% = 30 1% = x 1.1 1% = 30 Total Bill = VAT = 11% = = 67. = 3. x 11 Total Bill = Total Bill = Total Bill = * % = marks. * 30 x 1% = 67. and stops marks. * % and stops or 30 1% and stops marks. * 67.0 without work and stops merits marks. B1 Correct answer without work B Decimal error. B3 Inverts as 100 or 100 and continues (giving answers or ) B4 Mishandles 11% or 1% eg or or similar. (Note: 30 must be used) B5 30 taken as 11% or 1%. B6 No addition of VAT (as per candidates work). B7 Subtraction of VAT (as per candidates work). S1 Numerical errors to a max of 3. Attempts (3 marks) A1 11 or 1 or 30 and stops A 100% = 30 and stops. A and stops. A or similar and stops. 11 W1 Incorrect answer without work W = 341 and stops or continues. Page 14

51 (ii) 5 marks Att Regular: Number of portions = 360/30 =1 Large: Number of portions = 900/30 = 30 B1 Correct answers without work B Multiplication instead of division when finding the number of portions (once only) B3 Finds only one answer B4 Decimal error S1 Numerical errors to a max of 3 Attempts ( marks) A1 Any attempt at division and stops A or any correct step W1 Incorrect answer without work Page 15

52 b (iii) 5 marks Att Method 1 Method Regular: 96 1 = 8c per portion Regular: 360g = 96cent Large: 900g = 5 c Large: 5 30 = 7.5c per portion 1g = 96 1g = 5 Large box is better value g = 0.67cent 1g = 0.5cent Large box is better value. Method 3 Method 4 Regular: 96cent = 360g Large: 5cent = 900g Regular: 10 boxes = 3600g = 10 x 0.96 = cent = cent = Large: 4 boxes = 3600g = 4 x.5 = Large box is better value. 1 cent = 3.75g 1 cent = 4g Large box is better value * Candidate must indicate in some way that the Large box is better value. See S. * Accept candidate s previous answer B1 Operation other than division in unitary methods 1,, and 3 B Operation other than multiplication in common denominator method 4 B3 Finds unit cost or weight for one size box only B4 Decimal error S1 Numerical errors to a max of 3 S Fails to highlight or indicate Large box as better value Misreading (-1) M1 Transposes costs or weight for each box (eg. Regular box costs.5 or similar) and continues. Attempt ( marks) A1 States Larger box without any relevant supporting work. A Some attempt at division or multiplication using either 0.96 or.5. A3 Some attempt at division using 1 or 30 or 360 or 900 A4 1 and 30 or 360 and 900 both multiplied as alternative in method 4 W1 Incorrect answer without work W Adds given figures Page 16

53 (c) 5,5,5,5 marks Att,,, Geraldine s annual wage is She pays income tax at the rate of 0% on the first of her wage and income tax at the rate of 41% on the remainder of her wage. Geraldine has an annual tax credit of (i) Calculate the tax on the first of her wage, at the rate of 0%. (ii) How much of Geraldine s wage is taxed at the rate of 41%? (iii) Calculate the amount of tax payable at the rate of 41%. (iv) Calculate the tax due. (i) 5 marks Att 100% = % = 330 0% = 6600 Tax = 6600 Tax = Tax = 6600 Tax = Tax = % = Tax = 6600 * No penalty for omitting symbol B1 Correct answer without work. B Mishandles 0% eg = or = 1650 B3 Uses instead of B4 Decimal error. S1 Numerical error to a max of 3. Attempts ( marks) A1 Some use of 100 in attempt to find percentage eg. 0% = or 0. or and stops A Writes and stops W1 Incorrect answer without work W and stops or continues Page 17

54 3(c) (ii) How much of Geraldine s wage is taxed at the rate of 41%? (c) (ii) 5 marks Att = 7000 taxed at 41% * No penalty for omitting symbol B1 Correct answer without work. B Operation other than subtraction used with or B or 3500 is used in a subtraction with or S1 Numerical error to a max of 3. Attempts ( marks) A1 Some subtraction involving or W1 Incorrect answer without work. (c) (iii) 5 marks Att 100% = % = 70 41% = 870 Tax = 870 Tax = Tax = 870 Tax = Tax = 870 * No penalty for omitting symbol *Accept use of candidate s answer from (ii) above. B1 Correct answer without work. B Mishandles 41% eg = or similar. Note: (No penalty if already penalised in (c) (i)... consistent error.) B3 Does not use 7000 but see nd * above. B4 Decimal error. S1 Numerical error to a max of 3. Attempts ( marks) A1 Some correct use of 100 in attempt to find percentage eg. 41% = or 0.41 and stop A Some correct use of 7000 A3 Uses or instead of W1 Incorrect answer without work W = 7041 and stops or continues Page 18

55 (iv) 5 marks Att 6, = = 5970 Total Tax 9470 Tax Credit 3500 Tax Due 5970 * No penalty for omitting symbol * Accept use of candidate s answer from (i) and (iii) above. * If all 3 boxes are correctly filled in award full marks B1 Correct answer without work. B Subtracts to find gross tax. e.g = B3 Misuse or no use of Tax Credit B4 Decimal error B5 Total tax incorrectly calculated S1 Numerical error to a max of 3. Attempts ( marks) A1 Answer from c (i) or (iii) written in this part. W1 Incorrect answer without work. Page 19

56 QUESTION 4 Part (a) 15(10,5) marks Att (3,) Part (b) 15(5,10) marks Att (, 3) Part (c) 0(5,5,10) marks Att (,,3) (a) 10,5 marks Att 3, If a = 4, find the value of: (i) 3a + 5 (ii) 3a 0 (i) 10 marks Att 3 (i) 3a + 5 3(4) + 5 = = 17 * marks B1 Correct answer, without work B Leaves 3(4) in the answer B3 Incorrect substitution and continues B4 Breaks order i.e. 3(4+5) = 3(9) = 7 B5 Treats 3(4) as 7 or 34 S1 Numerical errors to a max of 3 S Treats as 3a 5 S3 Fails to finish Misreadings (-1) M1 Uses 5a + 3 Attempts ( marks) A1 Any number substituted for a and stops e.g. 3(6) A Any correct step A 3 Treats as 15a = 15(4) = 60 or 8a =8(4) = 3 W1 Incorrect answer with no work Page 0

57 (a)(ii) 5 marks Att 3a 0 3(4) 0 = 3(16) 0 = 48 0 = 8 * marks B1 Correct answer without work B Leaves 4 in the answer B3 Incorrect substitution and continues B4 Breaks order e.g. 3(16 0) = 3( 4 ) = 1. B5 Treats 3(16) as B6 Incorrect squaring eg. 4 = 8 B7 Treats as a 0 i.e omits the 3 S1 Numerical errors to a max of 3 S Fails to finish but see * above Misreadings (-1) M1 Treats as 3a +0 Attempts ( marks) A1 Any substitution for a and stops A3 Any correct step W1 Incorrect answer, with no work Page 1

58 (b) 5,10 marks Att,3 (i) x 5x Write as a single fraction. 3 6 (ii) Multiply (x 5) by (3x 4) and write your answer in its simplest form. (i) 5 marks Att x 5x 3 6 x 5x 6 7x 6 * = * * + 0 Marks, but allow 5Marks and stops 5 Marks or or B1 Correct answer without work B Incorrect common denominator and continues B3 Incorrect numerator from candidate's denominator or etc for full marks B4 Omitting denominator S1 Drops denominator S Numerical errors to a max of 3 Attempts ( marks) A1 Any correct step. A Any correct common denominator found W1 ( and stops W Incorrect answer, with no work Page

59 b(ii) 10 marks Att 3 (x 5) (3x 4) = x(3x 4) 5(3x 4) = 6x 8x 15x + 0 = 6x 3x + 0 * If 6x 8x 15x + 0 is correct (minimum 7 MARKS) B1 Correct answer without work B Error in distribution each time B3 Errors in multiplication of powers B4 Errors in collecting like terms B5 Mathematical (sign ) errors eg 5 4 = 0 B6 (x 5) written as (x + 5 ) and continues and/or (3x 4) written as (3x +4) --- oversimplifies S1 Numerical errors to a max of 3 Misreadings (-1) M1 (5x )(4x 3) etc and continues Attempts (3 marks) A1 One term correctly multiplied and stops e.g. 6x A x(3x 4) or 5(3x 4) and stops A3 x(3x 4) 5(3x 4) and stops W1 Incorrect answer with no work Page 3

60 (c) 5,10,10 marks Att,3,3 (i) The cost of a DVD is x. The cost of a CD is 3 less. What is the cost of a CD in terms of x? (ii) The total cost of 3 DVDs and CDs is 54. Write an equation in x to represent this information. Solve your equation to find the cost of a DVD. c(i) 5 marks Att CD : x 3 * Algebraic work required to earn marks B1 Incorrect expression for the cost of a CD other than misreading below Misreadings (-1) M1 Answer given as 3 + x or 3 x Attempts ( marks) W1 Cost of CD given as a constant or x. Page 4

61 (ii) 5 marks Att Equation : 3x + (x 3) =54 3x + x 6 = 54 5x = 60 x = 1 Cost of a DVD = 1 * Accept candidates answer from previous work. B1 Error in forming equation. B Distribution error B3 Transposition error B4 Stops at 5x = 60 or fails to solve equation B5 Error in collecting like terms Misreading (-1) M1 x + 3(x 3) = 54 or similar S1 Numerical errors to a max of 3 Attempts ( marks) A1 Answer from part c (i) written down and stops. A Any effort at forming an expression. A3 Writes x = 1 A4 Any effort at solving their equation A5 Successful Trial and Error W1 Incorrect answer with no work. Page 5

62 (iii) 10 marks Att 3 Solve for x and y: x + 3y = 1 3x + y = 11 (iii) 10 marks Att 3 5x + 3y = 1 ( ) OR 5x + 3y = 1 ( 3) OR x = 3x + y = 11 ( 3) 3x+y =11 ( 5) 3( ) +y =11 10 x 6y = 4 15x +9y = y +10y = 55 9x + 6y = 33 15x 10y= 55 y =55 36 x = 9 y = 19 x = 9 y =19 y = 19 5( 9) + 3y = 1 5x +3(19) =1 x = y = 1 5x +57 =1 3y = 57 5x =1 57 x = x = x = 9 y= 19 5x = 45 x = 9 x = 9 * Apply only one blunder deduction (B or B3) to any error(s) in establishing the first equation; in terms of x only or the first equation in terms of y only. * Finding the second variable is subject to a maximum deduction of (3). B1 Correct answers without work (stated or substituted) B Error or errors in establishing the first equation in terms of x only ( x = 9) or the first equation in terms of y only ( y = 19) through elimination by cancellation (but see S1) B3 Error or errors in establishing the first equation in terms of x only (x = 9) or the first equation in terms of y only ( y = 19) through elimination by substitution (but see S1) B4 Errors in transposition when finding the first variable B5 Errors in transposition when finding the second variable B6 Incorrect substitution when finding second variable B7 Finds one variable only S1 Numerical errors to a max of 3 Attempt (3 marks) A1 Attempt at transposition and stops A Multiplies either equation by some number and stops A3 Incorrect value of x or y substituted correctly to find candidate s correct nd variable W1 Incorrect values for x or y substituted into the equations Page 6

63 QUESTION 5 Part (a) 10 marks Att 3 Part (b) 0(5,5,5,5) marks Att (,,,) Part (c) 0(10,10) marks Att (3,3) (a) 10 marks Att 3 (a) Write in its simplest form ( x + 5) + 7( x + 3). (a) 10 marks Att 3 ( x + 5) + 7( x + 3) = x x +1 = 16x + 31 *Stops after correct removal of brackets 7 Marks B1 Correct answer without work B Error(s) in distribution (each time) B3 Combining unlike terms after removal of brackets and continues B4 Fails to group like terms B5 Fails to finish S1 Numerical errors to a max of 3 Misreadings (-1) M1 (x + ) and continues. Attempts (3 marks) A1 Any one term correctly multiplied A Combines unlike terms at the start and finishes correctly W1 Combining unlike terms before attempting multiplication and stops e.g. (5x) = 10x Page 7

64 (b) 5,5,5,5 marks Att,,, Factorise: (i) 4xy 8y (ii) xy xz + 3y 3z (iii) x² + 7x +1 (iv) x² 64 (i) 5 marks Att 4xy 8y = 4y(x ) * y(4x 8) or y(x 4) or (xy 4y) or 4(xy y) merit 4 Marks B1 Removes factor incorrectly Attempts ( marks) A1 Indication of common factor e.g. underlines y s and stops A Lists factors of 4 and factors of 8 (ii) 5 marks Att xy xz + 3y 3z = x(y z) + 3(y z) or y(x + 3) z(x + 3) = ( y z )(x + 3) = ( y z )(x + 3) * Accept also (with or without brackets) for 5 marks any of the following (y z) and (x+3) [The word and is written down.] (y z) or (x+3) [The word or is written down.] (y z), (x+3) [A comma is used] B1 Correct answer without work B Stops after first line of correct factorisation. e.g. x(y z) +3 (y z) or equivalent. B3 Error(s) in factorising any pair of terms B4 Correct first line of factorisation but ends as (x+3). yz or equivalent S1 (y z) (x+3) Attempts ( marks) A1 Pairing off, or indication of common factors and stops A Correctly factorises any pair and stops Page 8

65 (iii) 5 marks Att x + 7x x + 4x + 3x + 1 x x(x + 4) +3(x + ) (x + 3)(x +4) x (x + 3)(x +4) 3 and 4 (x + 3)(x + 4) Factor Method B1 Incorrect two term linear factors of x +7x+1 formed from correct (but inapplicable) factors of x and/or ±1. e.g. (x+1)(x 1) B Incorrect factors of x and/or ±1 B3 Correct cross method but factors not shown and stops B4 x(x+3)+4(x+3)or similar and stops S1 Numerical errors to a max of 3 Attempts ( marks) A1 Some effort at factorization e.g. (x )( ) or the cross with at least one x written in A States one correct factor without work Worthless (0 marks) W1 x + 7x = 1 or similar and stops W Incorrect Trial and error W3 Oversimplification, resulting in a linear equation W4 Combines x with numbers and continues or stops Formula Method B1 Error in a,b,c substitution (apply once only) B Sign error in substituted formula (apply once only) B3 Error in square root or square root ignored B4 7 1 Stops at B5 Incorrect quadratic formula and continues B6 No factors from roots or incorrect factors S1 Numerical errors to a max of 3 S3 One factor only Attempts ( marks) A1 Correct formula and stops Worthless (0 marks) W1 Combines x with numbers and continues or stops Page 9

66 (iv) 5 marks Att x² 64 = x² (8) = (x+ 8)(x 8) * Accept also (with or without brackets) for 5 marks any of the following x + 8 and (x 8) [The word and is written down.] (x+8) or (x 8) [The word or is written down.] (x+8), (x 8) [A comma is used] * Quadratic equation formula method is subject to slips and blunders. * (x 64)(x+ 64) merits 5 marks * x ± 8 merits 4 marks B1 Incorrect two term linear factors of x 64 formed from correct (but inapplicable) factors of x and 64 e.g ( x 64)( x 1) B Incorrect factors of 64 B3 Incorrect factors of x B4 (8 x)(8 + x). B5 (x 64)(x + 64) B6 Answer left as roots. (x = ±8) S1 x 8(x+8) Attempts ( marks) A1 Some effort at factorization e.g. (x )( ) or the cross with at least one x written in A 8 appears A3 x 64 =x.x 8.8 only A4 Mention of the difference of two squares.e.g. x 64 A5 Correct quadratic equation formula quoted and stops A6 64 W1 Combines xs to numbers and continues or stops Page 30

67 (c) 10,10 marks Att 3,3 (i) Solve the equation 5(3x + 1) (5x + 35) = 0. Verify your answer. (ii) Solve x + 3x 10 = 0. (i) 10 marks Att 3 5(3x + 1) (5x + 35) = 0 15x x 70 = 0 5x 65 = 0 5x = 65 x = 13 Verify 5(3x + 1) (5x + 35) x = 13 5(3(13) + 1) (5(13) + 35) 5(39 + 1) ( ) 5(40) (100) = 0 * If changes to + at the start: Blunder( 3) * States x =13 (no work) and verifies correctly 7 Marks * States x =13 (no work) with no verification 4 Marks * Verifies correctly x = 13 (not stated) Att 3 B1 Correct answer without work B Error(s) in distribution (each time) B3 Combining unlike terms (each time) and continues B4 Fails to group like terms B5 Error(s) in transposition (each time) B6 Fails to finish B7 Fails to verify or verifies incorrectly S1 Numerical errors to a max of 3 Misreadings (-1) M1 5(3x 1) or similar and continues but see * above Attempts (3 marks) A1 Any one term correctly multiplied A Any correct step W1 combining unlike terms before attempting multiplication and stops e.g. 5(4x) = 0x W Invented answer verified but see * above W3 Incorrect answer with no work Page 31

68 (ii) 10 marks Att 3 x + 3x 10 = 0 (x + 5)(x -) = x + 5x x 10 = 0 (x + 5) = 0 or (x ) = 0 1 x(x + 5) x 10 = 0 x + 5 x(x + 5) (x + 5) = (x + 5)(x ) = 0 (x + 5) = 0 or (x ) = 0 x and x = -5 or x = x = -5 or x = * correct solutions by Trial and Error 10 Marks * 1 correct solution by Trial and Error 3 Marks (Attempt) Factor Method B1 Correct answers without work B Incorrect two term linear factors of x +3x 10 formed from correct (but inapplicable) factors of x and/or ±10,e.g. (x + 10)(x 1) B3 No roots given. (once only) B4 Incorrect factors of x and/or ±10 B5 Correct cross method but factors not shown and stops [Note: B3 applies also]. B6 x(x+5) (x+5) or similar and stops [Note: B3 applies also]. B7 Error(s) in transposition S1 Numerical errors to a max of 3 S One root only from factors Attempts (3 marks) A1 Some effort at factorization e.g. (x )( ) or the cross with at least one x written in A States one correct root without work W1 x +3x = 10 or similar and stops W Incorrect Trial and error W3 Oversimplification, resulting in a linear equation Formula Method B1 Error in a,b,c substitution (apply once only) B Sign error in substituted formula (apply once only) B3 Error in square root or square root ignored B4 Stops at B5 Incorrect quadratic formula and continues S1 Numerical errors to a max of 3 S p Roots left in the form q S3 One root only Attempts (3 marks) A1 Correct formula and stops A One correct substitution and stops Page 3

69 QUESTION 6 Part (a) 15(5,10) marks Att (,3) Part (b) 0 (10,10) marks Att (3,3) Part (c) 15 (5,5,5) marks Att (,,) (a) 5, 10 marks Att,3 (a) f (x) = x 7. Find: (i) f(4) (ii) f( 3) (a) 5 marks Att (i) f (x) = x 7 f (4) = (4) 7 = 8 7 = 1 B1 Correct answer without work. B Mathematical error. e.g. 4 4, B3 Leaves (4) in the answer. B4 Combines x' s to numbers and continues e.g. B5 Mathematical error e.g B6 Breaks order i.e. (4 7) ( 3) 6. S1 Numerical errors to a max of 3. S Leaves x in the answer e.g. 1 x x 7 5x 5(4) 0. Misreadings (-1) M1 Correctly substitutes in any number other than 4 and continues. Attempts (marks) A1 Treats as equation and continues or stops.. i.e x 7 4. A Substitutes for x and stops. i.e. (4). W1 Combines x' s to numbers and stops. W Ignores x giving 7 5. W3 4[ f ( x)] 8x 8. W4 Replaces coefficient i.e. x 4x. W5 Incorrect answer without work. Page 33

70 (a) (ii) 10 marks Att3 (a) (ii) f ( 3) ( 3) 7 = 6 7 = 13 B1 Correct answer without work. [Do not penalise if already penalised in part (a) (i) or work is shown in part (a) (i).] B Mathematical error. i.e. 6 7 = 13 B3 Leaves ( 3) in the answer. B4 Combines x s to numbers and continues e.g. x 7 5x B5 Breaks order i.e. ( 3 7) = ( 10) = 0 5( 3) 15. S1 Numerical errors to a max of 3. S Leaves x in the answer e.g. 13 x Misreadings (-1) M1 Substitutes in any negative number other than 3 and continues. Attempts (3marks) A1 Treats as equation and continues or stops. i.e. x 7 3. A Substitutes in any positive number A3 Substitutes for x and stops. i.e. ( 3). W1 Ignores x giving 7 = 5 W 3f(x) = 6x + 1 W3 Combines x s to numbers and stops. W4 Replaces coefficient i.e. x 3x. W5 Incorrect answer without work. Page 34

71 (b) 10 (Table), 10 (Graph) marks Att 3,3 Draw the graph of the function g : x x² 4x + 1 in the domain 1 x 3, where x R. (b) 10 marks ( table ) Att 3 g : x x² 4x + 1 g(x) =x² 4x + 1 g( 1) =( 1) ² 4( 1) + 1 = = 7 ( 1,7) g(0) =(0) ² 4(0) + 1 = = 1 (0, 1) g(1) =(1) ² 4(1) + 1 = 4 +1 = 1 (1, 1) g() =() ² 4() + 1 = = 1 (, 1) g(3) =(3) ² 4(3) + 1 = = 1 (3, 7) Table 10 marks Att 3 A f ( 1) = ( 1) -4(-1) +1 = 7 f (0) = ( 0) -4(0) +1 = 1 f (1) = ( 1) -4(1) +1 = -1 f () = ( ) -4() +1 = 1 f (3) = ( 3) -4(3) +1 = 7 B x x x f (x) * Error(s) in each row/column calculation attracts a maximum deduction of 3marks B1 Correct answer, without work i.e. 5 correct couples only and no graph B Takes x as x and places x in the table or function. B3 Errors in evaluating x, e.g. ( 1) ( ) 4, once only if consistent. B4 4 x taken as 4 all the way [In the row headed -4 x by candidate] B5 +1 calculated as +1 x all the way. [In the row headed +1 by candidate] B6 Adds in top row when evaluating f (x) in table method (B). B7 Omits +1 row B8 Omits 4 x row B9 Omits a value in the domain (each time). B10 Each incorrect image, without work, or, calculation through the function method (A). Page 35

72 S1 Numerical errors to a max of 3 in any row / column S Fails to find a value of Range each time in table to a max of 3 Misreadings (-1) M1 Misreads 4x as 4x and places 4x in the table or function. M Misreads +1 as 1 and places 1 in the table or function. Attempts (3marks) A1 Omits x row or treats x as x or x, (i.e. evaluates a linear function) A Any effort at calculating point(s) in the Domain A3 Only one point calculated and stops. (b) 10 marks (graph) Att * Accept candidates values from previous work ( 5 co-ordinates needed ) but see S * Only one correct point graphed correctly Att 3 + Att 3 * Correct graph but no table full marks i.e. ( ) marks. * Accept reversed co-ordinates if (i) if axes not labelled or (ii) if axes are reversed to compensate (see B1 below) Page 36

73 Blunders ( -3) B1 Reversed co-ordinates plotted against non-reversed axes (once only) {See 4 th * above}. B Scale error (once only) B3 Points not joined or joined in incorrect order (once only). S1 Each point of candidate graphed incorrectly. {Tolerance 0. 5 } S Each point { 5 points needed } from table not graphed [ See nd * above ] Attempts (3 marks) A1 Graduated axes (need not be labelled) A Some effort at plotting a point { See nd * above} (c) 5,5,5 marks Att,, (c) (i) Given that y = x 1, complete the table below. x y On the grid below the graph of the line y = 3 x is drawn. Using your answers from (i), draw the graph of y = x 1 on the same grid. 4 3 y = 3 x Page 37

74 (iii) Use the graphs drawn in 6(c) (ii) to write down the co-ordinates of the point of intersection of the two lines y = 3 x and y = x 1. Answer to be written here. (c) 5 marks Att (i) Given that y = x 1, complete the table below. x y * Accept candidate s values without work S1 Each y value omitted or incorrect. Misreadings (-1) M1 Treats y x 1 as y x 1. (consistent error) Attempts () marks) A1 Any one correct y value. A Any effort at calculating points. A3 Treats as y = x and continues W1 Copies x values into y row. W All y values incorrect with no work shown but (See M1 and A3 above) Page 38

75 (ii) 5 marks Att On the grid below the graph of the line y = 3 x is drawn. Using your answers from (i), draw the graph of y = x 1 on the same grid. 4 3 y = 3 x * Accept candidates values from previous work Blunders ( -3) B1 Reversed co-ordinates plotted. B Points not joined or joined in incorrect order. S1 Each point of candidate graphed incorrectly. {See B1}. S Each point from table not graphed. Attempts ( marks) A1 Any one correct point plotted. A Any incorrect straight line drawn W1 No correct point plotted. {See B1 above}. Page 39

76 (iii) 5 marks Att (,1) * Accept correct answer based on candidate s graph from c(ii), otherwise, attempt marks at most. B1 Answer beyond tolerance ( 0. 5 ). B Answer given with co-ordinates reversed, i.e. ( y, x). S1 Correct answer written on graph but not presented in the answer box. Attempts ( marks) A1 Algebraic evaluation. (fully correct) A Point of intersection clearly indicated correctly on graph, but not written down. W1 Answer outside of tolerance without graphical indication. W Incorrect answer from candidate s graph. Page 40

77 JUNIOR CERTIFICATE EXAMINATION 010 MARKING SCHEME MATHEMATICS ORDINARY LEVEL PAPER 1

78 GENERAL GUIDELINES FOR EXAMINERS 1. Penalties of three types are applied to candidates work as follows: Blunders - mathematical errors/omissions (-3) Slips- numerical errors (-1) Misreadings (provided task is not oversimplified) (-1). Frequently occurring errors to which these penalties must be applied are listed in the scheme. They are labelled: B1, B, B3,, S1, S,, M1, M, etc. These lists are not exhaustive.. When awarding attempt marks, e.g. Att(3), 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 is awarded zero marks. Some examples of such work are listed in the scheme and they are labelled as W1, W, etc. 4. The phrase hit or miss means that partial marks are not awarded the candidate receives all of the relevant marks or none. 5. The phrase and stops means that no more work is shown by the candidate. 6. Special notes relating to the marking of a particular part of a question are indicated by an asterisk. These notes immediately follow the box containing 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, verifications and answers derived from diagrams (unless requested) qualify for attempt marks at most. 11. A serious blunder, omission or misreading results in the attempt mark at most. 1. Do not penalise the use of a comma for a decimal point, e.g may be written as 5,50. Page 1

79 QUESTION 1 Part (a) 10(5, 5) marks Att(, ) Part (b) 5(10, 5, 5, 5) marks Att(3,,, ) Part (c) 15(5, 5, 5) marks Att(,, ) Part (a) 10(5, 5) marks Att (, ) (i) Using the Venn diagram below, shade in the region that represents A B. (ii) Using the Venn diagram below, shade in the region that represents A \ B A B (a)(i) 5 marks Att A B A B B1 Any incorrect indication other than the misreading below Misreadings (-1) M1 A B indicated. Worthless(0) W1 No filling in of the Venn Diagram (a)(ii) 5 marks Att A/B A B B1 Any incorrect indication other than misreading below Misreadings (-1) M1 B/A indicated Worthless(0) W1 No filling in of the Venn diagram Page

80 Part (b) 5(10, 5, 5, 5) Att(3,,, ) U is the universal set P P = {, 4, 6, 8, 10, 1} Q = {3, 6, 9, 11, 1} R = {4, 8, 11, 1} 1 9 Q 5 R (b)(i) 10 marks Att 3 (i) List the elements of P Q R. (b)(i) 10 marks Att 3 P Q R = {1} B1 Any incorrect set of the elements of P and Q and R other than the misreading below Misreadings (-1) M1 P Q R giving {,3,4,6,8,9,10,11,1} (all needed) Attempts (3 marks) A1 1 or 5 appear in the answer (b)(ii) 5 marks Att (ii) List the elements of R', the complement of the set R. (ii) R' = {1,, 3, 5, 6, 9, 10} B1 Any incorrect set of elements of R' other than the misreadings below. Misreadings (-1) M1 R\Q giving {4, 8}, R\P giving {11}, R\(P Q) giving { } M P' = {3, 9, 11, 5} or Q' = {1,, 4, 5, 8, 10} Attempts ( marks) A1 1,,3,5,6,9, or 10 appear in the answer. A R or any proper subset of R. Page 3

81 . (b) (iii) 5 marks Att (iii) List the elements of P\ ( Q R). (iii) P\ ( Q R) = {, 4, 6, 8, 10} B1 Any incorrect set of elements of P and Q and R other than the misreadings below Misreadings (-1) M1 P\(Q R) giving {,10} or (Q R)\ P giving {11} Attempts ( marks) A1 1 or 5 appear in the answer (b) (iv) 5 marks Att (iv) Write down # (Q R). (iv) #(Q R) = 7 B1 Any incorrect cardinal number (Q R) 11 other than the misreadings as below Misreadings (-1) M1 Q R giving {3,4,6,8,9,11,1} M #(Q R)' = 4 M3 #(Q R) = Attempts ( marks) A1 Some understanding of notation e.g. Cardinal numbers or number of elements W1 Any number greater than 11 Page 4

82 (c) 15(5, 5, 5) marks Att(,, ) In a survey, a group of 7 students were asked if they played basketball or tennis 37 of these students said they played basketball (B) 30 of these students said they played tennis (T) 8 of these students said they played basketball but not tennis (c)(i) 5 marks Att (i) Represent this information in the Venn diagram below. (c)(i) 5 marks Att B T B1 Each incorrect or omitted entry but see S1 and M1 below S1 Numerical errors, where work is clearly shown, to a max of 3 Misreadings (-1) M1 Interchanges Basketball and Tennis Attempts ( marks) A1 Any one relevant entry A #B = 37 or #T = 30 or #U = 7 (c)(ii) 5 marks Att (ii) How many students played neither basketball nor tennis? (c)(ii) 5 marks Att c(ii) = =14 * Any correct answer written here in the space provided takes precedence over an incorrect Venn diagram (Subject to S1) * Accept candidate s work from previous part c(i) B1 Any incorrect use of the given numbers or the numbers from an incorrect Venn diagram (Subject to S1) S1 Numerical errors where work is clearly shown, to a max of 3 Attempts ( marks) A1 Any one relevant sum where work is clearly shown W1 Incorrect answer with no work shown Page 5

83 (c)(iii) 5 marks Att (iii) What percentage of the students surveyed played both basketball and tennis? (c)(iii) 5 marks Att = 1.5% or = 1.5% 7 8 * Any correct answer written here in the space provided takes precedence over an incorrect Venn diagram (Subject to S1) * Accept candidate s work from previous part c(i) B1 Any incorrect use of the given numbers or the numbers from an incorrect Venn diagram (Subject to S1) B Correct answer without work S1 Numerical errors, where work is clearly shown, to a max of 3 Attempts ( marks) A1 Mention of 9 or candidate s work from c(i) A Some use of 100 W1 Incorrect answer with no work shown Page 6

84 QUESTION Part (a) 10 marks Att 3 Part (b) 5(10, 5, 10)marks Att(3,, 3) Part (c) 15(5, 5, 5)marks Att(,,) Part (a) 10 marks Att 3 There is 100 in a prize fund. The first prize is 10 7 of the fund. Find the value of the first prize. (a) 10 marks Att 3 Method (1) Method () Method (3) Method (4) = parts= parts= = =10 1 part= = parts=10 3= =10 7= parts= parts= = 840 = 840 = 840 * Correct answer without work 7 marks * 100 Special Case 10 = marks 7 * Stops at 100 or 10 4 marks 10 * Stops at 100 7[= 8400] 4 marks * Incorrect answer without work 0 marks, except for answers given in A below B1 Divisor 10 and continues but see nd * B Incorrect multiplier i.e. 7 and continues but see nd * B3 Gets 10 3 of 100 only B4 Decimal error (once only) B5 Fails to finish S1 Numerical errors where work is clearly shown to a max of 3 Attempts (3 marks) A Indicates 0.7, or 7: 10, or, or, or, or.3 only, and stops A 10 or 8400 or 360 or only, appears (no work shown) 100 A3 or and stops A4 100 is multiplied or divided by any wrong number correctly Worthless(0) W = 107 or similar Page 7

85 Part (b) 5(10, 5, 10) marks Att (3,, 3) (i) By rounding each of these numbers to the nearest whole number, estimate the value of (ii) Using a calculator, or otherwise, find the exact value of (iii) Using a calculator, or otherwise, write and as decimals. 8 5 Hence, or otherwise, put the following numbers in order, starting with the smallest And finishing with the largest 3 9, 8 5, 0.37 Part (b)(i) 10 marks Att is approximately equal to: = = * and stops 7 marks. 6 * No penalty if the intermediate step between approximations and correct final answer is not 18 shown i.e. not shown * Special Case: = 3.6 or 3 in this part Attempt 3 marks * and stops 7 marks.(-3) 6 B1 Error(s) in rounding off to the nearest whole number (once only if consistent) B Decimal error in calculation of final value B3 An arithmetic operation other than indicated 9 18 B4 Error(s) in the manipulation of the denominator e.g. = B5 Incorrect cancellation S1 Numerical errors to a max of 3 Attempts (3 marks) A1 Only one approximation made to the given numbers and stops A Ans. 3 with no preceding rounding off Page 8

86 W1 Incorrect answer without work (b)(ii) 5 marks Att = = 3.6 or or or = B1 Decimal error or early rounding off B Treats as = = B , 954 Reads as = = B Reads as = B5 5, Treats as = = B Treats as = = B Treats as = = B Leaves as 5,5815 S1 Numerical errors to a max of 3 Attempts ( marks) A1 Any correct relevant calculation and stops e.g = or = or = A Any of the following; (see above) , , , or merit marks (minimum 4 decimal places) (with or without work) W1 Incorrect answer without work but see A Page 9

87 (b)(iii) 10marks Att = = * Accept: 0.36, 0.37, for 10 marks. * Note: 3 0,375 9 = or = merits 4 marks 8 5 B1 Fails to write a fraction as a decimal (each time) B Writes fraction as incorrect decimal (each time) B3 Decimal error (once only if consistent) B4 Inverts fraction and continues. (each time) B5 Incorrect order or fails to order Attempts (3 marks) A = and stops 100 A Attempt at ordering using all 3 given numbers A Any of the given numbers in the correct order i.e.(, 0.37 ), ( 0.37, ), (, ) W1 Nothing correct W or or similar 8 5 Page 10

88 Part(c) 15(5, 5, 5) Att(,, ) (i) Using a calculator, or otherwise, divide 110 by n Express your answer in the form a 10, where1 a < 10 and n N (ii) Simplify a a 3 a a 5, Give your answer in the form n a, where n N (iii) Using your answer to part (ii), or otherwise, find the value of (c)(i) 5 marks Att 110 = = 3. * 3. ( without work) 4 marks B1 Decimal error B Inverts fraction = B3 Multiplies fraction i.e =39. (3.9 10) S1 Numerical errors to a max of 3 S Incorrect format, where a < 1 or a 10 S and stops S4 Error in index when forming Scientific Notation Attempts ( marks) A1 110 Any relevant step. e.g. Partial long division e.g. = A = = A Page 11

89 (c)(ii) 5 marks Att. 5 a a a. a. a. a. a. a. a = = a.a.a = a 3 3 a a a. a. a. a 5 7 a a a or = = a 7-4 = a 3 5 a a or = a = a 3 a a a a a a * a a a (as answer) 4 marks * 7 a 4 a and stops marks * a 7 and stops marks * a 3 a and stops marks B1 Correct answer, without work B Each error in calculation involving indices B3 Each incorrect number of a s in the extended form B4 Each incorrect elimination of a s in the extended form S1 7 a 1 = 3 or 4 3 a a as final answer Attempts ( marks) A1 Some correct manipulation of indices Worthless(0) W1 Incorrect answer with no work shown (c)(iii) 5 marks Att = =16 or = =6 3 = or = = or 6 3 =16 or = = *Accept candidate s answer from c(ii) unless it oversimplifies the question. B1 Correct answer, without work B Each error in calculation involving indices B3 Each incorrect number of 6 s in the extended form B4 Each incorrect elimination of 6 s in the extended form B5 Fails to finish Attempts ( marks) A1 Some correct manipulation of indices A Writes answer from c(ii) in c(iii) Worthless(0) W1 Incorrect answer with no work shown Page 1

90 QUESTION 3 Part (a) 15 marks Att 5 Part (b) 0(10, 5, 5) marks Att(3,, ) Part (c) 15(10, 5) marks Att(3, ) Part (a) 15 marks Att 5 Carol buys a magazine which costs 83. In her purse she only has the following Three 50 cent coins Four 0 cent coins Seven 10 cent coins How much money will she have left after paying for the magazine? (a) 15 marks Att = = =70 Total amount ( ) = 300 Change = 0.17 * Accept 17c, ( 0.17) * Final subtraction step subject to maximum deduction of 3 marks. * No penalty for the omission of symbol. B1 Correct answer without work B Fails to find the change. B3 Fails to find total cost i.e. no addition B4 Operation other than subtraction when finding the change B5 Operation other than addition when finding total cost B6 Decimal error (Note: * above) B7 Each missing multiplication or addition S1 Numerical errors to a max of 3 Attempts (5 marks) A1 Any attempt at addition / subtraction /multiplication of the given numbers W1 Incorrect answer without work Page 13

91 Part (b) 0(10, 5, 5) marks Att(3,, ) (i) A bicycle costs 305. There is a 15% discount on the cost during a sale What is the sale price of the bicycle? (ii) David wishes to get some bars for a party A packet of 1 bars cost 4.08 in Shop A. A packet of 7 bars costs 17 in Shop B. Find the unit cost (cost of one bar) in each shop (iii) If David buys 84 bars, how much will he save by buying the bars in the shop offering the better value? (b)(i) 10 marks Att = %-15% = 85% = = = * % = marks * % = and stops 7 marks * % and stops 4 marks or % and stops 4 marks * without work and stops merits 4 marks B1 Correct answer without work B Decimal error B Inverts as or and continues (giving answers or ) B4 Mishandles85 % or 15% e.g or or similar Note: {305 must be used} B5 305 taken as 85% or 15% B6 No subtraction (as per candidates work) B7 Addition of discount ( as per candidates work) B = S1 Numerical errors to a max of 3 Misreadings (-1) M1 Reads as 350 instead of 305 Attempts (3 marks) A or or and stops 100 A 100% = 305 and stops A and stops 305 A4 305 or similar and stops 85 W1 Incorrect answer without work W =90 and stops or continues Page 14

92 (b)(ii) 5 marks Att 408 Shop A. unit cost = Shop B Unit cost 7 = 0.31 * Accept 34c and/or 31c B1 Correct answers without work. B Operation other than division when finding the unit cost (once only) B3 Finds only one unit cost B4 Decimal error S1 Numerical errors to a max of 3 Attempts ( marks) A1 Any attempt at division W1 Incorrect answer without work W Addition or subtraction of the given numbers (b)(iii) 5 marks Att = = = = 604 =.5 Savings = = 5 =.5 * Accept.5 or 5 or 5 c B1 Correct answer without work B Operation other than subtraction or multiplication where appropriate B3 Finds only one unit saving B4 Decimal error B5 Fails to subtract S1 Numerical errors to a max of 3 Attempts ( marks) A1 Any attempt at subtraction or multiplication A Any one correct step W1 Incorrect answer without work Page 15

93 Part (c) 0(10, 10) marks Att(3, 3) (i) 1000 is invested at % per annum What is the amount of the investment at the end of the first year? (ii) Using central heating oil for 6 hours a day, a tank full of oil will last for 90 days If the oil were used for only 5 hours a day, how much longer would it last? (c)(i) 10 marks Att is invested at %per annum. What is the amount of the investment at the end of the first year? (c)(i) 10 marks Att = 140 or 1000 = P R 1000 = 140 I = = = 100 = = 140 * Finds interest only, 40 and stops 7 marks * 40 without work and stops 4 marks * % = marks * 1000 % = 40 and stops 7 marks * % and stops 4 marks * 1000 % and stops 4 marks B1 Correct answer, without work B Mishandles % of 1,000. {Must use 1,000} B3 Decimal error B4 Fails to finish S1 Numerical errors to a max of 3 Attempts (3 marks) A1 Some use of 100 in attempt to find percentage e.g. % = and stops 100 A Correct formula (with or without substitution) and stops W1 Incorrect answer without work W (=100) and stops Page 16

94 (c)(ii) 5 marks Att Using central heating oil for 6 hours a day, a tank full of oil will last for 90 days If the oil were used for only 5 hours a day, how much longer would it last? (c)(ii) 5 marks Att 6 90 = : 5= x : = = x = 6(90) = 540 x = = = = = 18 * 90 Answer = 18 att. only marks 5 i.e.(no mention of 6) * Answer 108 (work shown) 4 marks B Adds instead of subtracts B3 Incorrect ratio B4 Incorrect division or similar B5 Fails to finish (method 3) S1 Numerical errors to a max of 3 Attempts ( marks) A1 Some indication of subtraction or ratio A Some correct use of 6 or 5 but see W A3 Answer 108 or 18 (no work shown) A4 6-5 [=1] W1 Incorrect answer without work W 90 [=15] and stops 6 Page 17

95 QUESTION 4 Part (a) 10 marks Att(, ) Part (b) 0(10, 10) marks Att(3, 3) Part (c) 0(5, 5, 10) marks Att(,, 3) Part (a) 10(5, 5)marks Att(, ) If a = 3 and b = 5, find the value of (i) a +b (ii) ab 6 (a)(i) 5 marks Att a + b =3+(5) =3+10 =13 * marks * 5+(3) = 5+6=11 4 marks B1 Correct answer, without work B Leaves (5), in the answer B3 Incorrect substitution and continues B4 Breaks order i.e. 3+(5) = (5)(5) = 5 B5 Treats (5) as 7 or 5 S1 Numerical errors to a max of 3 S Treats as a b Misreadings (-1) M1 a and b interchanged see * above Attempts ( marks) A1 Any number substituted for a or b and stops e.g. (8) A Writes 5 or 3 in this part A3 Any correct step W1 Incorrect answer with no work Page 18

96 (a)(ii) 5 marks Att ab 6 3(5) 6 = 15 6 = 9 * marks B1 Correct answer without work B Leaves 3(5) in the answer B3 Incorrect substitution and continues B4 Breaks order e.g. 3(5 6) B5 Treats 3(5) as 8 or 35 S1 Numerical errors to a max of 3 S Treats as ab + 6 Misreadings (-1) M1 a and b interchanged but no penalty here if already penalised in a(i) Attempts ( marks) A1 Any substitution for either a or b and stops A writes 5 or 3 in this part A3 Any correct step Worthless (0 marks) W1 Incorrect answer, with no work Page 19

97 Part (b) 0(10, 10) marks Att(3, 3) (i) Write in it s simplest form (3x + y) (x + 3y 4) (ii) Solve 3x 7, x N (b)(i) 10 marks Att 3 (i) (3x + y) (x + 3y 4) = 3x + y x 6y + 8 = x 4y + 8 * 3x + y x 6y + 8 (stops or continues) 7 marks (at least ) B1 Correct answer without work B Error in distributive law and continues (each time) B3 Fails to finish Slip (-1) S1 Numerical errors to a max of 3 Attempts (3 marks) A1 Any correct step. A Combines x s to numbers and continues with any correct step Worthless (0 marks) W1 Combines x s to numbers and stops W Incorrect answer, with no work Page 0

98 (b)(ii) 10 marks Att3 3x 7 3x 7 + 3x 9 x 3 {1,, 3} (not necessary) * Do not penalize for inclusion of 0 in answer B1 Correct answer without work B Error in transposition B3 No plotting on number line B4 Mishandles the direction of inequality e.g. 3x 9 B5 Treats inequality as equality and continues S1 Numerical errors to a max of 3 S taken as < S3 Missing elements or incorrect elements on the number line (each time), but see S4 S4 Correct range but shaded in Misreadings (-1) M1 3x + 7, and continues M x Z or x R correctly mapped Attempts (3 marks) A1 Attempt at transposition and stops A 0 or 1 or or 3 substituted for x A3 Number line drawn with one of the correct elements only clearly indicated A4 Combines x's to numbers e.g. x 7 and continues W1 Incorrect answer with no work e.g.{1,,3,4,5,6,7,8,..}. Page 1

99 Part (c) 0(5, 5, 10) marks Att(,, 3) (i) Eoin is t years of age Katie is 4 years older than Eoin Laura is twice as old as Eoin Write Katie s age and Laura s age in terms of t (ii) From part(i), the sum of Eoin s age, Katie s age and Laura s age is 5 Write down an equation in t to represent this information Solve your equation to find Eoin s age in years (iii) Solve for x and for y: 7x + y = 11 4x + y = 7 (c)(i) 5 marks Att Katie s age = t + 4 Laura s age = t B1 Each incorrect expression Misreading (-1) M1 Substitutes x (or similar) for t M Treats Laura s age as twice Katie s age i.e. (t + 4) Attempts ( marks) A1 Any attempt at forming an expression but numbers written on their own are worthless (c)(ii) 5marks Att t + t t = 5 4t + 4=5 4t = 5 4 4t = 48 t = 1 i.e.( Eoin s age = 1 ) (not necessary) * Accept candidates expression from previous work. B1 Correct answer without work (t = 1 stated or substituted) B Errors in transposition B3 Stops at 4t = 48 B4 error in forming equation B5 Fails to solve equation Slip (-1) S1 Numerical errors to a max of 3 S Leaves as 48 or similar 4 Attempts ( marks) A1 Answer from part c(i) written down and stops A Any correct step Worthless (0 marks) W1 Incorrect answer, with no work Page

100 (c)(iii) 10 marks Att3 Solve for x and y: 7x + y = 11 4x + y = 7 (c)(iii) 10 marks Att3 I I II III 7x+ +y = = x 7x+ + y y = = y = y 7 = 7-4x 4x + y = 7 4x 4x + + y = y = 7 7 7x+(7-4x) = 11 7x +y = 11 8x+8y = 44 7x x = 11-8x 7x+ y = x-7y +8y = x + (7 4x) = 11 -x 8x y = x 7y = 49 7x + -x 14 = x = 11 x x = 3 3 y = y 5 = -5 x = 3 4(3)+y x = = 37 = 3 4x-5 = 7 x = 3-x = y = 1 7 4x =7+5 1 y = 7-1 4x = 1 x = 3 y y = = -5 5 x = x 3 = 3 y y = = 5-5 * Apply only one blunder deduction (B or B3) to any error(s) in establishing the first equation; in terms of x only or the first equation in terms of y only. * Finding the second variable is subject to a maximum deduction of ( 3). B1 Correct answers without work (stated or substituted) B Error or errors in establishing the first equation in terms of x only (-x = -3) or the first equation in terms of y only (y = 5) through elimination by cancellation (but see S1) B3 Error or errors in establishing the first equation in terms of x only (-x = -3) or the first equation in terms of y only (y = 5) through elimination by substitution (but see S1) B4 Errors in transposition when finding the first variable B5 Errors in transposition when finding the second variable B6 Incorrect substitution when finding second variable B7 Finds one variable only Slip (-1) S1 Numerical errors to a max of 3 Attempt (3 marks) A1 Attempt at transposition and stops A Multiplies either equation by some number and stops A3 Incorrect value of x or y substituted correctly to find his correct variable Worthless (0 marks) W1 Incorrect values for x or y substituted into the equations Page 3

101 QUESTION 5 Part (a) 10 marks Att 3 Part (b) 0(5, 5, 10) marks Att(,, 3) Part (c) 0(5, 5, 10) marks Att(,, 3) Part (a) 10 marks Att 3 (a) Solve the equation 3(x ) =x + 5. (a) 10 marks Att 3 3x 6 = x + 5 3x x = x =11 B1 Correct answer without work B Error(s) in distribution (each time) B3 Combining unlike terms (each time) and continues B4 Fails to group like terms B5 Error(s) in transposition (each time) B6 Fails to finish S1 Numerical errors to a max of 3 Misreadings (-1) M1 3(x + ) and continues. Attempts (3 marks) A1 Any correct multiplication. A Any correct step W1 combining unlike terms before attempting multiplication and stops e.g. 3(-x) = x + 5 Page 4

102 Part (b) 0(5, 5, 10) marks Att(,, 3) (i) (ii) (iii) Factorise 5 Factorise ab ax + mb mx Factorise + 4x 1. Hence solve the equation + 4x 1 = 0. (b) (i) 5 marks Att x x 5 5 ( x 5)( x + 5) * Accept also (with or without brackets) for 5 marks any of the following ( x 5) and ( x + 5) [The word and is written down.] ( x 5) or ( x + 5) [The word or is written down.] ( x 5), ( x + 5) [A comma is used] * Quadratic equation formula method is subject to slips and blunders. x 5 x + 5 merits 5 marks * ( )( ) B1 Incorrect two term linear factors of x and ± 5.e.g ( x 5 )( x +1) B Incorrect factors of 5 B3 Incorrect factors of x B4 ( 5 x )( 5 + x). B5 ( x 5 )( x + 5). B6 Answer left as roots. (x = ±5 ) 5 formed from correct (but inapplicable) factors of 5 S1 ( x ) ± ( x + 5) Attempts ( marks) A1 Correct factors of x only A Correct factors of ± 5 only A3 ± x or ± 5 appears. A4 x 5 = x x 5 5 A5 Mention of the difference of two squares.e.g. x 5 A6 Correct quadratic equation formula quoted and stops A7 5 Worthless (0 marks) W1 Combines xs to numbers and continues or stops Page 5

103 (b) (ii) 5 marks Att ab ax + mb mx or ab ax + mb mx b(a+ m) x(a+m) a(b x)+ m(b x) (a + m)(b x) (a + m)(b x) * Accept also (with or without brackets) for 5 marks any of the following (a+ m) and (b x) [The word and is written down.] (a+ m) or (b x) [The word or is written down.] (a+ m), (b x) [A comma is used] B1 Correct answer without work B Stops after first line of correct factorisation. e.g. a(b x) + m(b x)or equivalent. B3 Error(s) in factorising any pair of terms B4 Correct first line of factorisation but ends as (a+ m).-bx or equivalent S1 (a + m) ± (b x) Attempts ( marks) A1 Pairing off, or indication of common factors and stops A Correctly factorises any pair and stops (b) (iii) 10 marks Att 3 x x + 4x 1 = 0 x + 6x x 1 = 0 x( x + 6) ( x + 6) ( x + 6)( x ) = 0 = 0 x = 6 and x = ( x + 6 )( x ) = 0 x = 6 and x = + 6 x ( 4) ± ( 4) 4( 1)( 1) ( 1) 4 ± ± 8 = 1 4 = 6 and = x = 6 and x = x + 6 x = ( )( ) 0 * correct roots without work or by substitution 4 MARKS Page 6

104 Factor Method B1 Correct answers without work B Incorrect two term linear factors of + 4x 1 formed from correct (but inapplicable) factors of x and/or ±1. e.g. (x + 1)(x 1) B3 No roots given. B4 Incorrect factors of x and/or ±1 B5 Correct cross method but factors not shown and stops [Note: B3 applies also]. B6 x(x+6) (x+6) or similar and stops [Note: B3 applies also]. B7 Error(s) in transposition S1 Numerical errors to a max of 3 S One root only Attempts (3 marks) A1 Some effort at factorisation A States one correct root without work Worthless (0 marks) W1 x + 4x = 1 or similar and stops W Incorrect Trial and error W3 Oversimplification, resulting in a linear equation Formula Method B1 Error in a,b,c substitution (apply once only) B Sign error in substituted formula (apply once only) B3 Error in square root or square root ignored B4 4 ± 8 Stops at B5 Incorrect quadratic formula and continues B6 No factors from roots S1 Numerical errors to a max of 3 S p Roots left in the form q S3 One root only Attempts (3 marks) A1 Correct formula and stops A One correct substitution and stops Page 7

105 Part (c) 0(5, 5, 10) marks Att(,, 3) (i) (ii) 5x 1 4x 9 Express + as a single fraction. 3 Give your answer in its simplest form. Verify your answer to part (i) by substituting x = 3 into and into your answer to part (i). (iii) Multiply ( x ) by ( 3x +11) 5x 1 + 4x 9 3 Write your answer in its simplest form (c)(i) 5 marks Att 3(5x 1) + (4x 9) (i) 6 15x 3 + 8x 18 3x 1 = 6 6 * 5x 1 4x 9 9x 10 + = 3 5 Zero marks B1 Correct answer without work B Error(s) in distribution e.g. 3( 5x 1) = 15x 1 B3 Mathematical error e.g. 3 18= +1 B4 Incorrect common denominator and continues B5 5x x 9 Incorrect numerator from candidate's denominator e.g. 6 B6 No simplification of numerator B7 Omitting denominator S1 Drops denominator S Numerical errors to a max of 3 S3 Answer not in simplest form. e.g. Attempts ( marks) A1 6 only or a multiple of 6 only appears A Any correct step 46x 4 1 ( ) ( ) 5x 1 4x 9 W1 3 and stops Page 8

106 (c) (ii) 5 marks Att 5(3) 1 4(3) (3) = = = + and = = 6 = 8 = 8 * Accept candidates answer from previous section [May result in inequality]. * Accept usage of a value other than 3 for verification. B1 Correct answer without work B Substitutes into one expression only B3 Manipulation to force equality S1 Numerical errors to a max of 3 S Conclusion missing if unequal Attempts ( marks) A1 Writes answer from previous part in this section A Substitutes a value into one expression and stops (c) (iii) 10 marks Att 3 ( x ) ( 3x +11) x( 3x +11) ( 3x +11) +11x +6x +17x * If +11x +6x is correct (minimum 7 MARKS) B1 Errors in distribution each time B Errors in multiplication of powers B3 Errors in grouping of terms (apply once) B4 Mathematical errors eg -. 3x = -6x B5 Correct answer without work S1 Numerical errors to a max of 3 Attempts (3 marks) A1 One correct multiplication Page 9

107 QUESTION 6 Part(a) 10(5, 5) marks Att(, ) Part (b) 30(15, 15) marks Att(5, 5) Part (c) 10(5, 5) marks Att(, ) Part(a) (5, 5)marks att(,) P = {(1, 5), (, 8), (,9), (3, 10)} Write out the domain and range of P (a)(i) 5marks att Domain = {1,, 3} * Accept {1,,, 3} for full marks. S1 Each incorrect element omitted / included other than the misreading below Misreadings (-1) M1 Correct range {5, 8, 9, 10} given Attempts ( marks) A1 Any one correct element of the Domain W1 No element of the domain appears, but note M1. (a)(ii) 5marks att Range = {5, 8, 9, 10} S1 Each incorrect element omitted / included other then the misreading below Misreadings (-1) M1 Correct domain {1,, 3} or {1,,, 3} given Attempts ( marks) A1 Any one correct element of the Range A Range 5 10 W1 No element of the range appears but note M1 Page 30

108 Part(b) 30(15, 15)marks att (5, 5) Draw the graph of the function f : x 3 + x x in the domain 1 x 3, where x R. Table 15 marks Att 5 A f ( 1) = 3 + ( 1) -(-1) = 0 f (0) = 3 + (0) - ( 0) = 3 f (1) = 3 + (1) - ( 1) = 4 f () = 3 + () - ( ) = 3 f (3) = 3 + (3) - ( 3) = 0 B x x x f (x) * Error(s) in each row/column calculation attracts a maximum deduction of 3marks B1 Correct answer, without work i.e. 5 correct couples only and no graph B - taken as + all the way but see M1 B3 + x taken as all the way. [In the row headed + x by candidate] B4 3 calculated as 3 x all the way. [In the row headed 3 by candidate] B5 Adds in top row when evaluating f (x) in B B6 Omits 3 row B7 Omits + x row B8 Omits a value in the domain (each time). B9 Each incorrect image without work i.e. calculation through the function method (A) S1 Numerical errors to a max of 3 in any row / column Misreadings (-1) M1 Misreads x as + x and places + x in the table or function M Misreads + x as x and places x in the table or function M3 Misreads 3 as -3 and places -3 in the table or function Attempts (5 marks) A1 Omits x row from table or treats x as ± x or ± x A Any effort at calculating point(s) A3 Only one point calculated and stops Page 31

109 Graph 15 marks Att 5 4 (i) 3 (ii) * accept candidates values from previous work ( 5 co-ordinates needed ) but see S * Only one correct point graphed correctly Att 5 + Att 5 * Correct graph but no table full marks i.e. ( ) marks. Blunders ( -3) B1 Reversed co-ordinates plotted against non-reversed axes (once only) {See S3 below}. B Scale error (once only) B3 Points not joined or joined in incorrect order (once only). S1 Each point of candidate graphed incorrectly. {Tolerance ± 0. 5 } S Each point { 5 points needed } from table not graphed [ See * above ] S3 reversed co-ordinates if (i) axes not labelled or (ii) axes are reversed to compensate (see B1 above) Attempts (5 marks) A1 Graduated axes (need not be labelled) A Some effort to plot a point { See * above} Page 3

110 Part (c) 10(5, 5) marks Att(, ) (c) (i) Draw the axis of symmetry of the graph you have drawn in part (b) above. Work to be shown on the graph. (ii) Use the graph you have drawn in part (b) to estimate the value of 3 + x x when x =. 5 (c) (i) 5 marks Att 4 (i) 3 1 (ii) * Accept any vertical line (parallel to candidate s y-axis) within tolerance of ± Blunders ( -3) B1 Any vertical line (parallel to the candidate s y-axis) outside of the tolerance. B Marks x = 1 on the x-axis and stops. B3 States x = 1 but no line is indicated on the graph Attempts ( marks) A1 Any attempt at axial symmetry of f (x). A y-axis indicated as the axis of symmetry (See B1). (c) (ii) 5 marks Att Work to be shown on the graph and answer to be written here. 1.8 * Correct answer (clearly consistent with student s graph) inside the tolerance without graphical indication marks. B1 Correct answer without work B Answer on the diagram but outside of tolerance ( ± 0. 5 ) B3 Fails to write down the answer, when indicated correctly on graph S1 Answer not written in box when written on graph Attempts ( marks) A1 Attempt at algebraic evaluation or calculator. A Marks.5 in any way on either axis and stops. W1 Answer outside of tolerance without graphical indication. Page 33

111 JUNIOR CERTIFICATE EXAMINATION 009 MARKING SCHEME MATHEMATICS ORDINARY LEVEL PAPER 1 Page 1

112 GENERAL GUIDELINES FOR EXAMINERS 1. Penalties of three types are applied to candidates work as follows: Blunders - mathematical errors/omissions (-3) Slips- numerical errors (-1) Misreadings (provided task is not oversimplified) (-1). Frequently occurring errors to which these penalties must be applied are listed in the scheme. They are labelled: B1, B, B3,, S1, S,, M1, M, etc. These lists are not exhaustive.. When awarding attempt marks, e.g. Att(3), 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 is awarded zero marks. Some examples of such work are listed in the scheme and they are labelled as W1, W, etc. 4. The phrase hit or miss means that partial marks are not awarded the candidate receives all of the relevant marks or none. 5. The phrase and stops means that no more work is shown by the candidate. 6. Special notes relating to the marking of a particular part of a question are indicated by an asterisk. These notes immediately follow the box containing 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, verifications and answers derived from diagrams (unless requested) qualify for attempt marks at most. 11. A serious blunder, omission or misreading results in the attempt mark at most. 1. Do not penalise the use of a comma for a decimal point, e.g may be written as 5,50. Page

113 QUESTION 1 Part (a) 10 marks Att 3 Part (b) 0(5,5,5,5) marks Att,,, Part (c) 0(10,5,5) marks Att 3,, Part (a) 10 marks Att 3 P = {w, x, y, z} Q = {v, w, x} Fill the elements of P and Q into the following diagram. P Q (a) 10 marks Att 3 P Q y z w x v * Not necessary S1 Each element incorrectly filled into the diagram S Each element omitted from the diagram but see W1 S3 Each unlisted element used Misreadings (-1) M1 Interchanging P and Q totally Attempts (3 marks) A1 Totally incorrect filling of the Venn diagram using given elements Worthless W1 No filling in of the Venn diagram or use of unlisted elements only Page 3

114 (b) 0(5,5,5,5) marks Att,,, (b) U is the universal set. A = {1, 5, 6, 9, 10} B = {1, 3, 5, 8} C = {4, 5, 8, 10} U A C 7 B (i) List the elements of B C. (ii) List the elements of A', the complement of the set A. (iii) List the elements of ( B C) \ A. (iv) Write down #B. (b)(i) 5 marks Att B C = {1, 3, 4, 5, 8, 10} B1 Any incorrect set of the elements of B and C other than the misreading as below Misreadings (-1) M1 B C giving {5, 8} Attempts ( marks) A1, 6, 9 or 7 appear in the answer Page 4

115 (b) (ii) 5 marks Att A' = {, 3, 4, 7, 8} B1 Any incorrect set of elements of A' other than the misreadings below. Misreadings (-1) M1 A\B giving {6,9,10}.A\C giving {6,9,1} or A\( B C ) giving{6,9}. Attempts ( marks) A1, 4, 7, 8 or 3 appear in the answer. A A or any proper subset of A. (b) (iii) 5 marks Att ( B C) \ A = 8 B1 Any incorrect set of elements of A and B and C other than the misreading as below. Misreadings (-1) M1 ( B C )/A giving {3, 4, 8,}, A\(B C) giving {1,6,9,10} Attempts ( marks) A1 or 7 appear in the answer. (b) (iv) 5 marks Att #B.= 4 B1 Any incorrect cardinal number of B 10 other than the misreading as below. Misreadings (-1) M1 Set B giving {1, 3, 5, 8}. M #B = 6 i.e, # B ' Attempts ( marks) A1 Some understanding of notation e.g. Cardinal numbers or number of elements A #B = 17 or 10 Worthless W1 Any number greater than 10, but see A Page 5

116 Part(c) 0(10,5,5) marks Att3,, 1(c) In a survey, a group of students were asked if they were studying French or German at school. 80 of these students said they were studying French (F). 4 of these students said they were studying German (G). 15 of these students said they were studying both French and German. 11 of these students said they were studying neither of the two languages. (i) Represent this information in the Venn diagram below. (ii) How many students were in the group? (iii) How many students did not study German? (c)(i) 10 marks Att 3 (c)(i) F G *.Failing to subtract 15 from 80 and/or 4 is one blunder only(-3) B1 Each incorrect or omitted entry but see S1 and M1 below and * above S1 Numerical errors, where work is clearly shown to a max of 3 Misreadings (-1) M1 Interchanges French and German Attempts (3 marks) A1 Any one correct relevant entry Page 6

117 (c)(ii) 5 marks Att c(ii) = 100 * Any correct answer written here in the space provided takes precedence over an incorrect Venn diagram (Subject to S1) * Accept candidate s work from previous part c(i) B1 Any incorrect use of the given numbers or the numbers from an incorrect Venn diagram (Subject to S1) B Number of students = = 130 S1 Numerical errors where work is clearly shown, to a max of 3 S Fails to finish Attempts ( marks) A1 Any one correct relevant sum where work is clearly shown Worthless W1 Incorrect answer with no work shown (c)(iii) 5 marks Att c(iii) = 76 or * Any correct answer written here in the space provided takes precedence over an incorrect Venn diagram (Subject to M1) * Accept candidate s work from previous part c(i) B1 Any incorrect use of the given numbers or the numbers from an incorrect Venn diagram (Subject to S1) S1 Numerical errors, where work is clearly shown to a max of 3 S Fails to finish. Misreadings (-1) M1 German read as French (Ans. = 0). Attempts ( marks) A1 Mention of 65 or 11 or candidate s work from c(i) Worthless W1 Incorrect answer with no work shown Page 7

118 QUESTION Part (a) 10 marks Att 3 Part (b) 0(5,10,5) marks Att,3, Part (c) 0(10,5,5) marks Att 3,, Part (a) 10 marks Att 3 (a) 9 metres of cloth cost Find the cost of 0 metres of the same cloth. (a) 10 marks Att 3 Method (1) Method () Method (3) Method (4) 9m = :0 9:0 = 13.95:x m = = x m=1.55 0= =31 9x = = x = = 31 9 * Correct answer without work 7 marks * 9 Special Case = marks 0 * Stops at 1.55 or [=1.55] 4 marks (no use of 0(-3) and B4 or B5 9 * Stops at [= 79] 4 marks (no use of 9 and possible slips) * Incorrect answer without work 0 marks except 79,155 or equivalent 31 0 B1 Divisor 9 and continues but see nd * B Incorrect multiplier i.e. 0 and continues but see nd * B3 0 : 9 =13.95 : x and continues B4 Error in decimal point (once only) B5 Fails to finish S1 Numerical errors where work is clearly shown to a max of 3 Attempts (3 marks) A1 Indicates 9 0 or 9 : 0 or : x,only, and stops A 1 79 or 1.55 or, only, appears 0 A3 1 only appears 9 A or and stops or continues A is multiplied or divided by any wrong number correctly Worthless W =.95 or similar Page 8

119 Part (b) 0(5,10,5) marks Att,3, (i) a Simplify a 9 6 a a 3, giving your answer in the form a n, where n N. (ii) By rounding each of these numbers to the nearest whole number, estimate the value of (iii) Using a calculator, or otherwise, find the exact value of (b)(i) 5 marks Att (i) a a a 4 a a = a or = 3 4 a a = a a a a a a 9 3 a a or 6 a a = a a * 1 a 8 a and stops marks * 1 a and stops marks * Correct answer without work marks * a 3 a and stops marks * a a a a as answer marks B1 Correct answer, without work B Each error in calculation involving indices B3 Each incorrect number of a s in the extended form B4 Each incorrect elimination of a s in the extended form a a a a a a a a a a a a a a a a a = a 4 a 1 a S1 8 a 1 = 4 or 4 a as final answer Attempts ( marks) A1 Some correct manipulation of indices Worthless W1 Incorrect answer with no work shown Page 9

120 (b)(ii) 10 marks Att = = 6 18 * and stops 4 marks. 4 * No penalty if the intermediate step between approximations and correct final answer is not 18 shown i.e. not shown * Special Case: = 3.15 or presented in this part Attempt 3 marks. 18 * and stops 7 marks. 6 B1 Error(s) in rounding off to the nearest whole number (once only) B Decimal error in calculation of final value B3 An arithmetic operation other than indicated B4 Error(s) in the manipulation of the denominator B5 Incorrect cancellation S1 Numerical errors to a max of 3 Attempts (3 marks) A1 Only one or two approximations made to the given numbers and stops. A Ans. 3 with no preceding rounding off W1 Incorrect answer without work (b)(iii) 5 marks Att or * Any of the following; or merit marks (with or without work) B1 Decimal error B Fails to finish S1 Numerical errors to a max of 3 S Any rounding off. Attempts ( marks) A1 Any correct relevant calculation and stops e.g or similar 3.7 W1 Incorrect answer without work but see * Page 10

121 Part (c) 0(10,5,5) marks Att 3,, (c) (i) 1 13 Using a calculator, or otherwise, write and as decimals Hence or otherwise, put the following numbers in order, starting with the smallest and finishing with the largest: 1 13,, (ii) Using a calculator, or otherwise, find the exact value of ( 3 61). 1 (iii) Using a calculator, or otherwise, evaluate ( 75) Give your answer correct to two decimal places. 1 * Note: or merits 4 marks (c)(i) 10marks Att = 0.15 = * Accept: 0.15, 0.155, 0.165, merits 10 marks. * Note: or merits 4 marks B1 Fails to write a fraction as a decimal (each time) B Writes fraction as incorrect decimal (each time) B3 Decimal error (once only if consistent) B4 Inverts fraction and continues. (each time) B5 Incorrect order or fails to order. Attempts (3 marks) A and stops A Attempt at ordering Worthless(0) W1 Nothing correct Page 11

122 (c)(ii) 5 marks Att 1.9 or B1 Squares B Decimal error Attempts ( marks) A1 mentions square root or power Worthless(0) W1 Dividing by or multiplying by (c)(iii) 5 marks Att = = * answer marks * answer = marks 45 * [ ] as final answer 0 marks but = marks 3 * Ans (no work shown) marks B1 Correct answer, without work B Decimal error B3 Inverts fraction B4 Incorrect operator S1 Numerical errors to a max of 3 S Fails to give answer to dec. places S3 Each premature rounding off, that effects final answer,( to a maximum of 3marks) Attempts ( marks) A1 Any relevant step. e.g. Partial long division or similar Page 1

123 QUESTION 3 Part (a) 10 marks Att 3 Part (b) 0(10,10) marks Att 3,3 Part (c) 0(10,10) marks Att 3,3 Part (a) 10 marks Att 3 (a) Aideen owns 6000 shares in a certain company. She sells two-thirds of her shares. How many shares does she now own in the company? (a) 10 marks Att = 000 or Number of shares sold: 6000 x ⅔ = 4000 Shares now owned: = 000 B1 Correct answer without work B 6000 ⅔ B3 Calculates the number of shares sold and stops B4 Operation other than subtraction in final step S1 Numerical errors (to max -3) S Early rounding off Attempts (3 marks) A1 Any attempt at getting ⅔ of 6000 A Writes down or Page 13

124 Part (b) 0(10,10) marks Att 3,3 (i) Brian s gross annual pay is His annual tax credit is 800. He pays income tax at the rate of 0%. What is his annual take-home pay? (ii) A dealer buys a car for He sells the car for Calculate his profit as a percentage of the cost price. (b)(i) 10 marks Att 3 (i) Brian s gross annual pay is His annual tax credit is 800. He pays income tax at the rate of 0%. What is his annual take-home pay? (b)(i) 10 marks Att 3 Gross Pay % 500 Tax Credit 800 Tax Due 400 Take-home Pay = = * Finds Tax Due 400 and stops 7 marks (at least out 3 boxes filled in) B1 Correct answer, without work. B Mishandles 0% of 6,000. {Must use 6,000} B3 Decimal error B4 Misuse of Tax Credit B5 Incorrect use of Tax Amount e.g B6 Fails to finish. {B4 may apply} S1 Numerical errors to a max of 3 Attempts (3 marks) A1 Some use of 100 in attempt to find percentage e.g. 0% = 0/100 and stops. W1 Incorrect answer without work Page 14

125 (b) (ii) 10 marks Att 3 (b) (ii) A dealer buys a car for He sells the car for Calculate his profit as a percentage of the cost price. (b)(ii) 10 marks Att = % or 3800 Method : = 36% * Answer marks * marks 100 B1 Correct answer without work B Adds to B3 Calculates profit as percentage of selling price. B4 Divisor not equal to B Mishandles the calculation of profit as a percentage e.g B6 Incorrect cancellation(s) B7 Fails to multiply by 100 B8 Fails to finish S1 Numerical errors to a max of 3 Attempts (3 marks) A1 Some indication of subtraction A Some use of 100 Page 15

126 Part (c) 0(10,10) marks Att 3,3 3(c) (i) is invested at 5 % per annum. What is the amount of the investment at the end of one year? (ii) 5000 is withdrawn from this amount at the beginning of the second year. The interest rate for the second year is 6 5% per annum. What is the amount of the investment at the end of that year? (c)(i) 10 marks Att = 1040 or = 1040 or 5.% = % = Interest =1040 Amount = Amount = 1040 or P R I I = Interest =1040 Amount = Amount = 1040 or Amount = Amount = 1040 * 1040 (without work) and stops 4 marks. * Writes down % = marks * Writes down % = 1040 and stops 7 marks. * Writes down % and stops, or % and stops 4 marks. B1 Correct answer without work B 0000 Mishandles 5. %. e.g Note: {0000 must be used}. B3 Decimal error (once only) B4 Stops at interest i.e. fails to calculate amount. B5 Subtracts to calculate amount. B treated as 1 5. S1 Numerical errors to a max of 3 Misreadings (-1) M1 Reads as 000 Page 16

127 Attempts (3 marks) A1 Correct formula with or without substitution and stops 5. A Some use of 100 in attempt to find percentage e.g. 5.% or 1 05 and stops. 100 W1 Incorrect answer without work W = and stops or continues. (c)(ii) 10 marks Att = = [ or = ] * Accept candidates answer from (i) * (without work) and stops 4 marks. * Writes down % = marks * Writes down % = and stops 7 marks. * Writes down % and stops, or % and stops 4 marks. * Uses 5000 (-3)(-3). Uses 0000 (-3) B1 Correct answer without work B Fails to subtract 5000 B3 Mishandles 6.5% B4 Decimal error (once only). B5 Stops at interest i.e. fails to calculate amount. B6 Subtracts to calculate amount. B7 Incorrect Principal S1 Numerical errors to a max of 3 Misreadings (-1) M1 Reads as 500 or similar. Attempts (3 marks) A1 Correct formula with or without substitution and stops A Some use of 100 in attempt to find percentage and stops. A = and stops W1 Incorrect answer without work W and stops or continues Page 17

128 QUESTION 4 Part (a) 10 marks Att, Part (b) 0(10,10) marks Att 3,3 Part (c) 0(5,5,10) marks Att,,3 Part (a) 10 (5,5)marks Att, (a) If a = 5, find the value of (i) 4a + 1 (ii) a 3a + 6 (a)(i) 5 marks Att (i) 4(5) + 1 = 1 * => 4 marks B1 Correct answer, without work B Leaves 4(5), in the answer B3 Incorrect substitution and continues B4 Breaks order i.e. 4(5 + 1) = 4.6 = 4 B5 Treats 4(5) as 9 or 45 S1 Numerical errors to a max of 3 S Treats as 4a - 1 Attempts ( marks) A1 Any number substituted for a and stops e.g. 4(8). A Writes 5 in this part A3 Any correct step. W1 Incorrect answer with no work. Page 18

129 (a)(ii) 5 marks Att (ii) (5) 3(5) + 6 = 16 or = = 16 * or => 4 marks B1 Correct answer without work B Leaves 5 or -3(5) in the answer B3 Incorrect substitution and continues. B4 Breaks order e.g. -3(5+6). B5 Treats -3(5) as or -35. B6 Fails to finish but see * above S1 Numerical errors to a max of 3 S Treats as a 3a 6 Attempts ( marks) A1 Any substitution for either a or -3a and stops e.g. (8) etc. A writes 5 in this part. A Any correct step. Worthless (0 marks) W1 Incorrect answer, with no work. Page 19

130 Part (b) 0(10,10) marks Att 3,3 4(b) (i) Solve the equation 5x 10 3( x ). (ii) Multiply ( x 3) by ( x 1). Write your answer in its simplest form. (b)(i) 10 marks Att 3 (i) 5x 10 = 3x + 6 => 5x 3x = => x = 16 => x = 8 B1 Correct answer without work (x = 8 stated or substituted). B Error in distributive law and continues, e.g. 5x 10 = 3x +. B3 Errors in transposition (each time) B4 Stops at x = 16 or similar. S1 Numerical errors to a max of 3 S Leaves as16 or similar. Attempts (3 marks) A1 Any substitution for values of x other than x = 8. A Any correct step. A3 Combines x s to numbers and continues with any correct step e.g. 5x 10 = -5x. Worthless (0 marks) W1 Combines x s to numbers and stops. W Incorrect answer, with no work Page 0

131 (b)(ii) 10 marks Att3 (ii) x(x 3) + 1(x 3) or x(x + 1) 3(x + 1) => x 6x + x 3 => x + x 6x 3 => x 5x 3 => x 5x 3 * x + x 6x 3 => 7 marks B1 Correct answer without work B Error(s) in distribution.(each time) B3 Fails to group or groups incorrectly Slip (-1) S1 Numerical errors to a max of 3. Attempts (3 marks) A1 Any correct multiplication e.g. x etc. A Any correct grouping of terms. A3 Any correct step. A4 Substitutes a value of x and continues correctly. A5 Treats as (x 3)±(x + 1) to give 3x or x 4 A6 Combines x s to numbers and continues with correct step e.g. x 3= 3x or x + 1 = 3.x Worthless (0 marks) W1 Combines x s to numbers and stops. W No distribution but A or A5 may apply to subsequent work e.g. gathering of terms. Page 1

132 Part (c) 0(5,5,10) marks Att,,3 (i) The cost of a cinema ticket is t for an adult and 5 for a child. The cost of tickets for adults and 3 children is 33. Write down an equation in t to represent this information. (ii) Solve the equation you formed in part (i) above, for t. (iii) Solve for x and for y: 5x 4y = 16 x + 3y = 11 (c)(i) 5 marks Att t + 3(5) = 33 or t + 15 = 33 B1 Each incorrect term in equation Misreading (-1) M1 Substitutes x (or similar) for t Attempt ( marks) A1 Any attempt at forming an equation but numbers written on their own (except 15 or 33) are worthless (c)(ii) 5marks Att t + 15 = 33 => t = 18 => t =9 * Accept candidates equation from previous work. B1 Correct answer without work (t = 9 stated or substituted). B Errors in transposition B3 Stops at t = 18 or similar Slip (-1) S1 Numerical errors to a max of 3 S Leaves as18 or similar. Attempts ( marks) A1 Answer from part c(i) written down and stops A Any correct step e.g. 3.5 = 15 Worthless (0 marks) W1 Incorrect answer, with no work Page

133 (c)(iii) 10 marks Att 3 I II 5x 4y = 16 5x 4y = 16 4y = 5x 16 x + 3y = 11 x + 3y = 11 y = 5x x 1y = 48 10x 8y = 3 x + 3( 5x 16 ) = x + 1y = 44 10x 15y = 55 8x + 15x 48 = 44 3x = 9 3y = 3 3x = 9 x = 4 x = 9 = 4 y = 3 = => y = 1 => y = 1 => x = 4 * Apply only one blunder deduction (B or B3) to any error(s) in establishing the first equation; in terms of x only or the first equation in terms of y only. * Finding the second variable is subject to a maximum deduction of (3). B1 Correct answers without work(stated or substituted) B Error or errors in establishing the first equation in terms of x only (3x = 9) or the first equation in terms of y only ( 3y = 3) through elimination by cancellation (but see S1) B3 Errors in transposition when finding the first variable. B4 Errors in transposition when finding the second variable B5 Incorrect substitution when finding second variable B6 Finds one variable only S1 Numerical errors to a max of 3 Attempt (3 marks) A1 Attempt at transposition and stops A Multiplies either equation by some number and stops A3 Incorrect value of x or y substituted correctly to find his correct nd variable Worthless (0 marks) W1 Incorrect values for x or y substituted into the equations Page 3

134 QUESTION 5 Part (a) 10 marks Att 3 Part (b) 15(5,5,5) marks Att,, Part (c) 5(5,10,10) marks Att,3,3 Part (a) 10 marks Att 3 (a) Write in its simplest form 3(x + ) + 4(3x + 1). (a) 10marks Att 3 3x x + 4 = 15x + 10 * Stops after correct removal of brackets 7marks 5 3x * Ignore excess work B1 Correct answer without work B Error(s) in distribution (each time) B3 Combining unlike terms Attempts (3 marks) A1 Any correct multiplication B4 Fails to group like terms S1 Numerical errors to a max of 3 Misreadings (-1) M1 3 x 4 3x 1 and continues W1 combining unlike terms, before attempting multiplication and stops Page 4

135 Part (b) 15(5,5,5) marks Att,, 5(b) Factorise (i) 5cd 7d (ii) ax 3ay 4x 1y (iii) x 49 (b)(i) 5 marks Att d 5c 7 B1 Removes factor incorrectly. Attempts ( marks) A1 Indication of common factor e.g. underline ds and stops. (b) (ii) 5marks Att ax 3ay 4x 1y ax 4x 3ay 1y a x 3y 4 x 3y or x a 4 3y a 4 a 4 x 3y x 3y a 4 * Accept also (with or without brackets) for 5 marks any of the following a 4 and x 3 y [The word and is written down.] a 4 or x 3 y [The word or is written down.] a 4, x 3 y [A comma is used] B1 Correct answer without work B Stops after first line of correct factorisation e.g. a x 3y 4 x 3y or equivalent. B3 Error(s) in factorising any pair of terms (each time) B4 Incorrect common factor and continues. e.g. x a 4 y 3a 1 (B will apply) B5 Correct first line of factorisation but ends as x 3y 4a. x 3y a S1 4 Attempts ( marks) A1 Pairing off, or indication of common factors and stops. A Correctly factorises any pair and stops. Page 5

136 (b) (iii) 5 marks Att x 49 x 7 x 7 x 7 * Accept also (with or without brackets) for 5 marks any of the following x 7 and x 7 [The word and is written down.] x 7 or x 7 [The word or is written down.] x 7, x 7 [A comma is used] * Quadratic equation formula method is subject to slips and blunders. x 49 x 49 merits 5 marks * B1 Incorrect two term linear factors of x 49 formed from correct (but inapplicable) factors of x and 49.e.g. x 49 x 1 B Incorrect factors of 49 B3 Incorrect factors of x B4 7 x 7 x. B5 x 49 x 49. B6 Answer left as roots. ( x 7 ) S1 7 ± x x 7 Attempts ( marks) A1 Correct factors of x only A Correct factors of 49 only A3 x or 7 appears. A4 x 49 x x 7 7 A5 Mention of the difference of two squares.e.g. x 49 A6 Correct quadratic equation formula quoted and stops. A7 49 Worthless (0 marks) W1 Combines xs to numbers and continues or stops. Page 6

137 Part (c) 5(5,10,10) marks Att,3,3 (i) 5x 1 x 6 Express as a single fraction. 3 5 Give your answer in its simplest form. (ii) Verify your answer to part (i) by substituting x = 4 into and into your answer to part (i). 5x 1 3 x 6 5 (iii) Solve the equation x 4x 1 0. (c)(i) 5 marks Att (i) 5(5x 1) 3( x 6) 15 = 5x 5 3x = x * 5x 1 x 6 4x Zero marks B1 Correct answer, without work B Error(s) in distribution e.g. 5 5x 1 5x 1. B3 Mathematical error e.g. 5-18=13, -3(6) =18 B4 Incorrect common denominator and continues B5 3 5x 1 5 x 6 Incorrect numerator, from candidate's denominator e.g. 15 B6 No simplification of numerator B7 Omitting denominator. S1 Drops denominator S Numerical errors to a max of 3 S3 Answer not in simplest form. e.g. 44x Attempts ( marks) A1 15 only or a multiple of 15 only appears. A Any correct step. 5x 1 x 6 W1 3 5 and stops. Page 7

138 Part(c) (ii) 10 marks Att 3 5(4) x (4) and * Accept candidates answer from previous section [May result in inequality]. * Accept usage of a value other than 4 for verification. B1 B B3 B4 Correct answer, without work Substitutes into one expression only (B4 will also apply) Manipulation to force equality Conclusion missing S1 Numerical errors to a max of 3 Attempts (3 marks) A1 Writes answer from previous part in this section A Substitutes a value into one expression and stops Page 8

139 (c) (iii) 10 marks Att 3 x x 4x 1 0 7x 3x 1 0 x x 7 3 x 7 x 3 x x 3 and x 7 x + 3 x 7 x 3 x 7 0 x 7 and x and 3 x 7 and x 3 Factor Method B1 Correct answers without work B Incorrect two term linear factors of x 4x 1 formed from correct (but inapplicable) factors of x and/or ±1. e.g. (x+1)(x-1) B3 No roots given.(each time) B4 Incorrect factors of x and/or ±1. B5 Correct cross method but factors not shown and stops [Note: B3 applies also]. B6 x(x-7) +3(x-7) or similar and stops [Note: B3 applies also]. B7 Error in transposition (each time) S1 Numerical errors, to a max of 3 Attempts (3 marks) A1 Some effort at factorisation A One correct answer without work Worthless (0 marks) W1 x 4x 1, or similar, and stops. W Trial and error W3 Oversimplification, resulting in a linear equation Formula Method B1 Correct answers without work. B Error in a,b,c substitution (apply once only) B3 Sign error in substituted formula (apply once only) B4 Error in square root or square root ignored. Page 9

140 B5 B Stops at Incorrect quadratic formula and continues. S1 Numerical errors to a max of 3 S p Roots left in the form q Attempts (3 marks) A1 Correct formulas and stops A One correct substitution and stops Page 30

141 QUESTION 6 Part (a) 10(5,5) marks Att, Part (b) 5(10,15) marks Att 3,5 Part (c) 15(10,5) marks Att 3, Part (a) 10 (5,5)marks Att, (a) f (x) = 4x 5. Find: (i) f(3) (ii) f(-) (a)(i) 5 marks Att f (3) 4(3) * Answer marks B1 Correct answer no work. B Leaves 4(3) in the answer B3 Mathematical error e.g. treats 4(3) as 43. B4 Breaks order i.e. [ ( ) 8]. S1 Numerical errors to a max of 3 S Leaves x in the answer e.g. 7x Misreadings (-1) M1 Correct substitution of any number other than 3 and continues. Attempts ( marks) A1 Substitutes for x and stops e.g. 4(3) A Treats as an equation and continues or stops 4x 5 3 A3 Combines "x"s to numbers and continues. e.g. 4x 5 x (3) W1 Combines "x"s to numbers and stops. W Ignores x giving W3 3 f (x) 1x 15 W4 Replaces coefficient i.e. 4x 3x W5 Incorrect answer, without work Page 31

142 (a) (ii) 5 marks Att f ( ) 4( ) * Answer marks (stops or continues) * -8x -5x = 13x 4marks but -8x -5x = 13 5marks (rectified error) B1 Correct answer no work B Leaves 4(-) in the answer B3 Mathematical error e.g. treats 4(-) as 4. B4 Breaks order i.e. [ 4 5 4( 7) 8]. S1 Numerical errors to a max of 3 S Leaves x in the answer e.g. -13x A3 Combines "x"s to numbers and continues. e.g. 4x 5 x ( ) A4 Substitutes positive value for x and continues correctly Misreadings (-1) M1 Correct substitution of any negative number other than - and continues Attempts ( marks) A1 Substitutes for x and stops e.g. 4(-) A Treats as an equation and continues or stops 4x 5 W1 Combines "x"s to numbers and stops W Ignores x giving W3 f (x) 8x 10 W4 Replaces coefficient i.e. 4x x W5 Incorrect answer, without work Page 3

143 Part (b) 5(10,15) marks Att 3,5 (b) Draw the graph of the function f : x x x 1 in the domain 1 x 3, where x R. Table 10marks Att 3 f(-1) = (-1) -(-1) -1 = x f(0) = (0) -(0) -1 = -1 x f(1) = (1) -(1) -1 = - f() = () -() -1 = -1 - x f(3) = (3) -(3) -1 = f x * Error(s) in each row /column calculation attracts a maximum deduction of 3 marks B1 Correct answer, without work i.e. 5 correct couples only and no graph B x taken as all the way. [In row headed x by candidate] B3-1 calculated as x all the way. [In row headed -1 by candidate] B4 Adds in top row when evaluating f (x). B5 Omits -1 row B6 Omits - x row B7 Omits a value in the domain (each time). B8 Each incorrect image without work i.e. calculation through the function method S1 Numerical errors to a max of 3 in any row / column Misreadings (-1) M1 Misreads x as x and places ''- x '' in the table or function. M Misreads -x as x and places x in the table or function. M3 Misreads -1 as 1 and places 1 in the table or function Attempts (3 marks) A1 Omits x row from table or treats x as A Any effort at calculating point(s). A3 Only one point calculated and stops. xor x. Page 33

144 Graph 15 Marks Att 5 y-axis 3 1 x-axis * Accept candidate's values from previous work.( 5 co-ordinates needed ) but see S * Only one correct point graphed correctly Att 3 + Att 5 * Correct graph but no table full marks i.e. (10+15) marks. * Accept reversed co-ordinates if (i) if axes not labelled or (ii) if axes are reversed to compensate (see B1 below) B1 Reversed co-ordinates plotted against non-reversed axes (once only) {See 4th * above}. B Scale error (once only). B3 Points not joined or joined in incorrect order (once only). S1 Each point of candidate graphed incorrectly. {Tolerance 0. 5 } S Each point ( 5 points needed ) from table not graphed [See nd * above]. Attempts (5 marks) A1 Graduated axes (need not be labelled) A Some effort to plot a point {See nd * above} Page 34

145 Part (c) 15(10,5) marks Att 3, Use the graph drawn in 6(b) to estimate: (i) the values of x for which x x 1 0 (ii) the value of f(x) when x 1 5. (c) (i) 10 marks Att 3 x.4 and x 0. 4 work to be shown on graph for correct answer * Accept candidate's values from previous work. * indications on graph and values written down (blunder each time) Blunders(-3) B1 Answers beyond tolerance. {Tolerance 0. 5 } Misreading (-1) M1 Answers not presented in designated box (but elsewhere). Attempts (3marks) A1 One point of intersection indicated only or one value of x written down A Algebraic evaluation ( x = 1 ) W1 Answers outside of tolerance without graphical indication W f(0) gives -1 as answer. Page 35

146 (c) (ii) 5 marks Att f(x) = work to be shown on graph for correct answer * Accept candidate's values from previous work. B1 Answer beyond tolerance. {Tolerance 0. 5 }. B Correct answers no work B3 Sign error Misreading (-1) M1 Answers not presented in designated box (but elsewhere) Attempts ( marks) A1 Point indicated only. A Algebraic evaluation or correct calculator calculation. A3 Testing x value for y = 1.5 Worthless(0) W1 Answers outside of tolerance without graphical indication. Page 36

147 JUNIOR CERTIFICATE 008 MARKING SCHEME MATHEMATICS ORDINARY LEVEL PAPER 1

148 GENERAL GUIDELINES FOR EXAMINERS 1. Penalties of three types are applied to candidates work as follows: Blunders - mathematical errors/omissions (-3) Slips- numerical errors (-1) Misreadings (provided task is not oversimplified) (-1). Frequently occurring errors to which these penalties must be applied are listed in the scheme. They are labelled: B1, B, B3,, S1, S,, M1, M, etc. These lists are not exhaustive.. When awarding attempt marks, e.g. Att(3), 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 is awarded zero marks. Some examples of such work are listed in the scheme and they are labelled as W1, W, etc. 4. The phrase hit or miss means that partial marks are not awarded the candidate receives all of the relevant marks or none. 5. The phrase and stops means that no more work is shown by the candidate. 6. Special notes relating to the marking of a particular part of a question are indicated by an asterisk. These notes immediately follow the box containing 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, verifications and answers derived from diagrams (unless requested) qualify for attempt marks at most. 11. A serious blunder, omission or misreading results in the attempt mark at most. 1. Do not penalise the use of a comma for a decimal point, e.g may be written as 5,50. Page 1

149 QUESTION 1 Part (a) 10(5, 5) marks Att 4(, ) Part (b) 0 (5, 5, 5, 5) marks Att 8 (,,, ) Part (c) 0 (5, 5, 5, 5) marks Att 8 (,,, ) Part (a) (i) 5 marks Att S = {a, b, c} 1(a) (i) Write down a subset of S that has one element Part (a) (i) 5 marks Att {a} or {b} or {c}. * No penalty for the omission of brackets. * No penalty for use of Venn Diagram to show subsets. B1 Any incorrect set of elements of S other than the misreading below. Misreadings (-1) M1 Subset of S with two elements. e.g. S = {a, b}. Attempts ( marks) A1 Draws a single bracket & stops. A { } Null set. Part (a) (ii) 5 marks Att S = {a, b, c} 1(a) (ii) Write down a subset of S that has two elements Part (a) (ii) 5 marks Att {a,b} or {a,c} or {b,c} * No penalty for omission of brackets. * No penalty for use of Venn Diagram to show subsets. B1 Any incorrect set of elements of S other than the misreading below. Misreadings (-1) M1 Subset of S with one element. e.g. S = {a}. Attempts () A1 Draws a single bracket & stops. A { } Null set. Page

150 Part (b) 0 (5, 5, 5, 5) Att 8 (,,,) 1(b) U is the universal set. P = {, 3, 6, 7, 8} Q = {1, 6, 8, 10} R = {, 4, 6} P R Q 9 Part (b) (i) 5 marks Att 1 (b) (i) List the elements of: P Q Part (b) (i) 5 marks Att P Q = 6,8 { } B1 Any incorrect set of elements of P and Q other than the misreading below. Misreadings (-1) M1 P Q = 1,,3, 6, 7,8,10. { } Attempts ( marks) A1 4 or 5 or 9 appear in the answer. Part (b) (ii) 5 marks Att 1 (b) (ii) List the elements of: Q \ R Part (b) (ii) 5 marks Att Q \ R = 1,8,10 { } B1 Any incorrect set of elements of Q and R other than the misreading below. Misreadings (-1) M1 R \ Q = {, 4 }. Attempts ( marks) A1 3 or 5 or 7 or 9 appear in the answer. Page 3

151 Part (b) (iii) 5 marks Att 1 (b) (iii) List the elements of: ( Q R ) / Part (b) (iii) 5 marks Att Q R / = 3,5, 7,9 ( ) { } B1 Any incorrect set of elements of Q and R other than the misreadings below. B ( Q R) = { 1,, 4, 6,8, 10} Misreadings (-1) / M1 ( Q R) = { 1,,3, 4,5, 7,8,9, 10} / / M Q R = { 1,,3, 4,5, 7,8,9,10} Attempts ( marks) A1 Any incorrect listing of element(s) other than the misreadings above. Part (b) (iv) 5 marks Att P R \ 1 (b) (iv) List the elements of: ( ) Q Part (b) (iv) 5 marks Att P R \ Q = ( ) { } B1 Any incorrect set of elements of P and Q and R other than the misreadings below. P R =,6 B { } Misreadings (-1) M1 Q \ ( P R) = { 1,8, 10 }. P R \ Q = {,3,4,7 M ( ) } Attempts ( marks) A1 5 or 9 appear in the answer. Page 4

152 Part(c) 0(5, 5, 5, 5) Att 8 (,,, ) M is the set of natural numbers from 1 to 36, inclusive Part (c) (i) 5 marks Att 1(c) (i) List the elements of M that are multiples of 6. Part (c) (i) 5 marks Att 6, 1, 18, 4, 30, 36. S1 Each missing or incorrect element subject to a max of 3. Attempts ( marks) A1 Any one correct element identified. A Any correct factor(s) of 6. W1 Elements listed are not multiples or factors of 6. Part (c) (ii) 5 marks Att 1(c) (ii) List the elements of M that are multiples of 9. Part (c) (ii) 5 marks Att 9, 18, 7, 36. S1 Each missing or incorrect element subject to a max of 3. Attempts ( marks) A1 Any one correct element identified. A Any correct factor(s) of 9. W1 Elements listed are not multiples or factors of 9. Page 5

153 Part (c) (iii) 5 marks Att 1(c) (iii) Write down the lowest common multiple of 6 and 9. * Accept candidate's L.C.M. from incorrect answers in part (i) and part (ii) for full marks. * Accept indication of candidate's L.C.M. Part (c) (iii) 5 marks Att L.C.M. = 18 B1 An incorrect common multiple that is not the lowest.e.g. {36}. S1 Answer written as 3 3 and stops. Misreadings (-1) M1 Writes down H.C.F. = 3. Attempts ( marks) A1 Any multiple of either 6, or 9 written in this part. W1 Elements listed are not multiples of 6 or 9. Part (c) (iv) 5 marks Att 1(c) (iv) Express 30 as the product of three prime numbers, Part (c) (iv) 5 marks Att 30 = 3 5 *, 3, 5, listed merits full marks. B1 Each correct prime constituent omitted and/or each incorrect prime constituent included. Misreadings (-1) M1 Write as sum of 3 primes (30 = ) or (30 = ). Attempts ( marks) A1 Writes a prime number M. W1 No prime number appears. Page 6

154 QUESTION Part (a) 10 marks Att 3 Part (b) 0 (5, 10, 5) Att 7 (, 3, ) Part (c) 0 (10, 5, 5) Att7 (3,, ) Part (a) 10 marks Att 3 (a) 60 is shared between Mark and Una in the ratio 6:7. How much does each receive? Part (a) 10 marks Att 3 6 parts : 7parts 6 x : 7 x = Parts = x = 60 1 = 0 x = Part = = 0 6 x = 10 6 = 10 (Mark) Mark = 0 6 = 10 7 x = = 140 (Una) Una = 0 7 = 140 or =140 7 or = = = 10 B1 Correct answers without work. B Divisor 13only and continues. B3 Incorrect multiplier or fails to multiply. (each time). B4 Error in transposition. B5 Fails to find second amount. B6 Adds instead of subtracts.e.g = 380 or = 400. B7 Finds 6% of 60 ( 15 6) and 7% of 60 ( 18 ) S1 Numerical errors to a max of 3. Misreadings (-1) M1 Interchanges Mark and Una. Attempts (3 marks) A1 A Divisor 13 e.g and/or and stops. 6 7 Indicates 6 parts or 7 parts or 13 parts or 13 6 or 13 7 or 6 + 7=13 and stops. A3 A4 Indicates multiplication of 60 by 6 and/or 7 and stops. Both answers added together equal 60. (No work shown). W1 Incorrect answer without work. {Subject to A4}. Page 7

155 Part (b) (i) 5 marks Att (b) (i) On a day when 1 = 0 68, find the value in euro of 816. Part (b) (i) 5 marks Att 1 = = 1? = = ? = = = 816 = * No penalty for the omission of or symbols. 1 * Note: Natural Display calculator gives = 0.68 B1 Correct answer without work. B Incorrect operation. i.e = B3 Incorrect ratio. 816 or , = B4 B5 Decimal error. Fails to finish, leaves as and stops. S1 Numerical errors to a max of 3. S 1 Rounds off too early. e.g. = = 1 47 ( ) or 1 5 ( 14 ) 0 68 Attempts ( marks) 1 A1 1= and stops W1 Adds or subtracts 816 to 0 68 and stops. W Incorrect answer without work. Page 8

156 Part (b) (ii) 10 marks Att 3 (b) (ii) By rounding each of these numbers to the nearest whole number, estimate the value of Part (b) (ii) 10 marks Att is approximately equal to = = * and stops 4 marks. 1 * No penalty if the intermediate step between approximations and final answer is not 48 shown. i.e not shown * Special Case: = presented in this part Attempt 3 marks B1 Correct answer without work. B Error(s) in rounding off to the nearest whole number. (once only) B3 Decimal error in calculation of final value. B4 An arithmetic operation other than indicated. B Error(s) in the manipulation of the denominator. e.g. = B Incorrect cancellation e.g. = = = S1 Numerical errors to a max of 3. Attempts (3 marks) A1 Only one or two approximations made to the given numbers and stops. W1 Incorrect answer without work. Page 9

157 Part (b) (iii) 5 marks Att (b) (iii) Using a calculator, or otherwise, find the exact value of Part (b) (iii) 5 marks Att = = * Note: =, = B1 Decimal error. B Leaves answer as B Treats as = = B Reads as = = B5 Reads as = = B Treats as = = B Treats as = = B Treats as = = S1 Numerical errors to a max of 3. S Any rounding off. Attempts ( marks) A1 Any correct relevant calculation and stops e.g = , = = , W1 Incorrect answer without work = = 0 7. Page 10

158 Part(c) (i) 10 marks Att (c) (i) Using a calculator, or otherwise, write and as decimals Hence, or otherwise, put the following numbers in order, starting with the smallest and finishing with the largest: ,, Part(c) (i) 10 marks Att = 0 = , 11, , * Accept: 0, 0, 0 5, merits 10 marks. * Accept: No penalty for writing 0 5 as 4 1. * Note: 1 11 = 0 or = merits 4 marks B1 Fails to write a fraction as a decimal. B Writes fraction as incorrect decimal. B3 Decimal error. B4 Inverts fraction and continues. B5 Incorrect order or fails to order. Attempts ( 3 marks) 1 A1 0 5 = and stops. 4 Worthless(0) W1 0 5, or or similar. Page 11

159 Part(c) (ii) 5 marks Att (c) (ii) Using a calculator, or otherwise, divide 1170 by 0 45 and express your answer in the form a 10 n, where 1 a < 10 and n N Part(c) (ii) 5 marks Att = 600 = B1 Correct answer without work. B Decimal error. B3 1 Inverts = B Multiplies = 56 5 = S1 Numerical errors to a max of 3. S Incor rect format, where a < 1 or a 10 and n N. S3 600 and stops. Attempts ( marks) A1 A Any relevant step. e.g. Partial long division.e.g. 1 = = 0 45 Page 1

160 Part(c) (iii) 5 marks Att (c) ( iii ) U sing a calculator, or otherwise, evaluate 5 3 = ( 3 9) Give your answer correct to two decimal places Part(c) (iii) 5 marks Att 5 3 = = = = = 3 17 * Correct answer (without work) incorrectly rounded off marks. e.g. = , = * Note: Na tural Display calculator gives ( 3 9) =, =, B1 Correct answer wit hout work. B Mishandles ( 3 9). B3 Mishandle s B4 Error in or B5 Error in or B6 Decimal error. B7 Subtracts before Division = =. ( ) B8 Subtracts before Multiplication ( ) 1 81= = B9 Use of mathematical operator other than that which is indicated. 5 3 B10 Works as = = = B11 Works as = = = S1 Numerical errors to a max of 3. S Each premature rounding off that effects the final answer to a max of 3. S3 Fails to round off or rounds off incorrectly when giving final answer. S Leaves as. 000 Attempts ( marks.) A 1 Any correct relevant step e.g. ( 3 9) = 15 1, 0 64 = 0 8. W1 Incorrect answer without work. (Note 1 st *). Page 13

161 QUESTION 3 Part (a) 10 marks Att 3 Part (b) 0 (10, 10) marks Att 6(3, 3) Part (c) 0 (5, 5, 5, 5,) marks Att 8(,,, ) Part (a) 10 marks Att 3 3(a) Kate went to the cinema. She bought a ticket at 8 50 and then bought popcorn costing How much change did Kate get from a 0 note? Part (a) 10 marks Att 3 * ** *** = = = = = = 7 10 * Accept 710c, ( 7 10 ) * Final subtraction step subject to maximum deduction of 3 marks. * No penalty for the omission of symbol. B1 Correct answer without work. B Calculates the cost of the ticket and the popcorn but fails to find the change. B3 Fails to find total cost i.e. no addition. B4 Operation other than subtraction when finding the change. B5 Operation other than addition when finding total cost. B6 Decimal error e.g. 1 9 (Note: 1 st * above). S1 Numerical errors to a max of 3. Attempts (3 marks) A1 Any attempt at addition /subtracation. W1 Incorrect answer without work. W Multiplication or division of the given numbers. Page 14

162 Part (b) (i) 10 marks Att3 3(b) (i) VAT at 13 5 % is added to a bill of 860 Calculate the total bill. Part (b) (i) 10 marks Att3 100 % = % = % = VAT = % = = = Total Bill = Total Bill = Total Bill = * without work and stops merits 4 marks. Total Bill = B1 Correct answer without work. B Decimal error. B Inverts as or and continues (giving answers or ) B4 Mishandles 13 5% e.g or Note: {860 must be used}. B5 860 taken as %. B6 No addition of VAT (as per candidates work) to the bill. B7 Subtraction of VAT ( as per candidates work) from the bill. S1 Numerical errors to a max of 3. Misreadings (-1) M1 Reads as 15 3% or 680. Attempts (3 marks) A and stops. or 100 A 100% = 860 and stops. A and stops. 860 or 860 and stops and stops A % and stops. W1 Incorrect answer without work. W = and stops or continues. Page 15

163 Part (b) (ii) 10 marks Att 3 3(b) (ii) 4750 is invested at 3 7 % per annum What is the amount of the investment at the end of one year? Part (b) (ii) 10 marks Att % = % = P R I = I = Interest = Interest = Amount = Amount = Amount = * (without work) and stops 4 marks. Amount = Amount= B1 Correct answer without work. B Mishandles 3 7 %. e.g. 3 7 {No penalty if already penalised in b (i)} B3 Decimal error (once only). B4 Stops at interest i.e. fails to calculate amount. B5 Subtracts to calculate amount. B Mathematical error(s) working with B treated as S1 Numerical errors to a max of 3. Misreadings (-1) M1 Reads as M 3 7% written as 7 3%. Attempts (3 marks) A1 Correct formula with or without substitution and stops. A 3 7 Some use of 100 in attempt to find percentage e.g. 3 7% = or and stops. 100 A % and stops. W1 Incorrect answer without work. W = and stops or continues. Page 16

164 Part (c) 0 (5, 5, 5, 5) marks Att 8 (,,, ) 3(c) Darragh s annual wage is He pays income tax at the rate of 0% on the first of his wage and income tax at the rate of 41% on the remainder of his wage. Darragh has an annual tax credit of Part (c) (i) 5 marks Att 3(c) (i) Find the tax on the first of his wage, calculated at the rate of 0%. Part (c) (i) 5 marks Att 100% = Tax = % = Tax = % = 340 Tax = 6800 Tax = % = 6800 Tax = 6800 Tax = 6800 * No penalty for missing symbol. B1 Correct answer without work. B Mishandles 0%, e.g = or = 1700 B3 Uses instead of B4 Decimal error. S1 Numerical errors to a max of 3 Attempts ( marks) 0 A1 Some use of 100 in attempt to find percentage e.g. 0% = or 0 and stops A 0 % = 5 A3 Writes % and stops. W1 Incorrect answer without work 1 5 Page 17

165 Part (c) (ii) 5 marks Att 3(c) (ii) Find the tax on the remainder of his wage, calculated at the rate of 41%. Part (c) (ii) 5 marks Att (ii) Remainder of wage = = 14, % = Tax = Tax = 41 = % = 140 Tax = % = 5740 Tax = 5740 * No penalty for missing symbol. B1 Correct answer without work. B Mishandles 41%, e.g [ No penalty if already penalised in (c) (i)]. B3 Does not use B4 Decimal error. S1 Numerical errors to a max of 3. Attempts ( marks) A1 Some use of 100 in attempt to find percentage e.g. A or and stops % = = 0 41 and stops. 100 W1 Incorrect answer without work. Page 18

166 Part (c) (iii) 5 marks Att 3(c) (iii) Hence calculate Darragh s gross tax. Part (c) (iii) 5 marks Att (iii) Darragh s gross tax = = * Allow candidates incorrect answers from parts (i) and (ii). * No penalty for missing symbol. B1 Correct answer without work. B Subtracts to find gross tax e.g = 1060 B3 Misuse of tax credit. S1 Numerical errors to a max of 3. Attempts () A1 Answer from c (i) and /or c (ii) written in this part. W1 Incorrect answer without work Part (c) (iv) 5 marks Att 3(c) (iv) Calculate Darragh s take home pay. Part (c) (iv) 5 marks Att Tax due = = 8940 Take home pay = Take home pay = * Allow candidate s incorrect gross tax figure from (c) (iii). * No penalty for missing symbol. B1 Correct answer without work. B Misuse or no use of tax credit e.g B3 Decimal error. S1 Numerical errors to a max of 3. Attempts () A1 Answer from c (iii) written in this part. W1 Incorrect answer without work. Page 19

167 QUESTION 4 Part (a) 10 (5, 5) marks Att 4(, ) Part (b) 0 (10, 10) marks Att 6(3, 3) Part (c) 0 (5, 5, 5, 5) marks Att 8(,,, ) Part (a) 10(5, 5) marks Att 4(,) 4(a) If a = 5 and b = 7, find the value of: (i) 9a + b (ii) ab + 13 Part (a) (i) 5 marks Att 4(a) (i) If a = 5 and b = 7, find the value of: (i) 9a + b Part (a) (i) 5 marks Att (i) 9a + b = 9(5) + 7 = = 5 * marks. B1 Correct answer without work. B Leaves 9(5) in the answer. B3 Incorrect substitution and continues B4 Breaks order i.e. [ 9 ( 5+ 7) = 9( 1) = 108]. B5 Treats 9(5) as 14 or 95. B6 Combines 9a + b to give 9ab and continues. S1 Numerical errors to a max of 3. S Treats as 9 a b. S3 Treats as 9 b + a. Attempts ( marks) A1 Substitutes for either a or b and stops e.g. 9(5). A Writes 7 or 5 in this part. A3 Any correct step. W1 Incorrect answer with no work. W Writes as 9ab and stops. Page 0

168 Part (a) (ii) 5 marks Att 4(a) (ii) If a = 5 and b = 7, find the value of: (ii) ab + 13 Part (a) (ii) 5 marks Att ab + 13 = 5(7 )+13 = = 48 * marks. B1 Correct answer without work. B Leaves 5(7) in the answer. B3 Incorrect substitution and continues B4 Breaks order i.e. [ 5 ( ) = 5( 0) = 100 ]. B5 Treats 5(7) as 1 or 57. B6 Combines ab +13 to give 13ab and continues. S1 Numerical errors to a max of 3. S Treats as ab 13. Attempts ( marks) A1 Substitutes for either a or b and stops e.g. a (7). A Writes 7 or 5 in this part. A3 Any correct step. W1 Incorrect answer with no work. W Writes as 13ab and stops. Page 1

169 Part (b) (i) 10 marks Att 3 4(b) (i) Solve the equation 3 ( x 1 ) = 4x + 9 Part (b) (i) 10 marks Att 3 ( x 1) 3 = 6x 3 = 6x 4x = x = x = x = 4x + 9 4x B1 Correct answer without work. ( x = 6 stated or substituted) B Error in distributive law and continues. e.g. 6 x 1 = 4x + 9 B3 Error in transposition. (each time). B4 Combines x's to numbers and continues. e.g. 6 x 3= 3x or 4 x + 9 = 13or x 1= x B5 Stops at x = 1 or similar. S1 Numerical errors to a max of 3. S 1 Leaves as x = or similar. Attempts (3 marks) A1 Any substitution for values of x other than x = 6. A Any correct step. e.g. ( x 1) 4x + 9 = 3 W1 Combines x's to numbers and stops W Incorrect answer with no work. Page

170 Part (b) (ii) 10 marks Att 3 4(b) (ii) Multiply ( 5x ) by (3x + 4). Write your answer in its simplest form. Part (b) (ii) 10 marks Att 3 ( 5x )( 3x + 4) ( 3x + 4)( 5x ) ( 5x )( 3x + 4) 5x( 3x + 4) ( 3x + 4) 15x 15x + 0x 6x x 8 15x 15x 6x + 0x x 8 15x 15x + 0x 6x x 8 * 15x + 0x 6x 8 merits 7 marks. B1 Correct answer without work. B Error in distribution. B3 Combines x's to numbers and continues. e.g. 5 x = 3 or 3 x + 4 = 7x. B4 Fails to group or groups incorrectly. S1 Numerical errors to a max of 3. Attempts (3 marks) A1 Any correct multiplication. e.g. 15x. A Any correct grouping of terms. A3 A correct step. A4 Substitutes a value of x and continues correctly. A5 Treats as ( 5 x ) ± ( 3x + 4) to give 8 x + or x 6. W1 Combining unlike terms and stops. W No distribution but A or A5 may apply to subsequent work e.g. gathering of terms. Page 3

171 Part (c) 0 (5, 5, 5, 5) marks Att 8 (,,, ) 4(c) Shane is x years old. Eileen is three years younger than Shane. Part (c) (i) 5 marks Att 4(c) (i) Find Eileen's age in terms of x. Part (c) (i) 5 marks Att x 3 * Algebraic work required to earn marks. B1 Incorrect expression of Eileen's age other than mireadings below. Misreadings (-1) M1 Answer given as x +3. M Answer given as 3x. Attempts ( marks) A1 Any effort at forming an expression.e.g. Eileen s age = y. Worthless (0 marks) W1 Eileen's age given as a constant or x. W Inequality appears. Part (c) (ii) 5 marks Att 4(c) (ii) If the sum of Shane s age and Eileen s age is 47, write down an equation in x to represent this information. Part (c) (ii) 5 marks Att x + x 3 = 47 * Accept candidates answer from previous work. B1 Error in forming equation. Attempts ( marks) A1 Answer from part c (i) written down and stops. A Any effort at forming an expression. A3 Writes Shane's age and stops. i.e. x Page 4

172 Part (c) (iii) 5 marks Att 4(c) (iii) Solve the equation that you formed in part (ii) above, for x. Part (c) (iii) 5 marks Att x + x 3 = 47 x = x = x = x = 5 * Accept candidates answer from previous work. B1 Correct answer without work. ( x = 5 stated or substituted) B Error in transposition. B3 Error in grouping terms or fails to group. B4 Combines x's to numbers and continues. e.g. x + x 3 = x B5 Stops at x = 50 or similar. S1 Numerical errors to a max of 3. S 50 Leaves as x = or similar. Attempts ( marks) A1 Any substitution for values of x other than x = 5. A Answer from part c(ii) written down and stops. A3 Any effort at solving equation. A4 Any correct step. W1 Combines x's to numbers and stops. ( Note A3 above) W Incorrect answer with no work. Page 5

173 Part (c) (iv) 5 marks Att 4(c) (iv) When Eileen is x + 5 years old, find the sum of Shane s age and Eileen s age Part (c) (iv) 5 marks Att Allow Eileen's age x + 5= ( 5) + 5= = 55 Shane's age 55+ 3= 58 Sum of Shane's age and Eileen's age = = 113 * Accept candidates answer from previous work. Eileen's age = x + 5 Shane's age = x + 8 Sum of Shane's age and Eileen's age = x + 5+ x + 8= 4 x + 13 B1 Correct answer without work. B Error in grouping terms or fails to group. B3 Combines x's to numbers and continues. e.g. x + 5+ x + 8= 7 x B4 Only one age calculated. Misreadings (-1) M1 Takes Shane's age as x and continues x + 5+ x = 3x + 5. S1 Numerical errors to a max of 3. S Leaves as or similar. Attempts ( marks) A1 Any correct step. A Writes answer from c (iii) and stops. W1 Incorrect answer with no work. W Works with x + 5 = 47 Page 6

174 QUESTION 5 Part (a) 5 marks Att Part (b) 0(5, 5, 5, 5) marks Att 8(,,, ) Part (c) 5(5, 5, 15) marks Att 9(,, 5) Part (a) 5 marks Att 5 (a) Find the values of x for which 3x + 8, x N Part (a) 5 marks Att 3x + 3x 8 3x 6 x {0,1, } 8 or 0 1 B1 Correct answer without work. B Error in transposition. B3 Combines x's to numbers. e.g.. 5x 8 and continues. B4 Mishandles the direction of inequality e.g. 3x 6 B5 Treats inequality as equality and continues. S1 Numerical errors to a max of 3. S taken as < S3 Each missing or incorrect value of x subject to a max of 3. Misreadings (-1) M1 x + 3 8, and continues. Attempts ( marks) A1 Attempt at transposition and stops. A 0, 1 or substituted for x. A3 Number line with one or more of the correct elements clearly indicated. (Ignore arrows) W1 Incorrect answer with no work.e.g.{1,, 3, 4, 5, 6, 7, 8,..}. Page 7

175 Part (b) (i) 5 marks Att 5(b) (i) Factorise: 4 a + ab Part (b) (i) 5 marks Att a 4 + b B1 Removes factor incorrectly. ( ) Attempts ( marks) A1 Indication of common factor. e.g. underline a 's and stops. Part (b) (ii) 5 marks Att 5(b) (ii) Factorise: x y + cx cy Part (b) (ii) 5 marks Att x y + cx cy x y + cx cy ( x y) + c( x y) or x( + c) y( + c) + c x y x y + c ( )( ) ( )( ) * Accept also (with or without brackets) for 5 marks any of the following ( + c) and ( x y) [The word and is written down.] ( + c) or ( x y) [The word or is written down.] ( + c), ( x y) [A comma is used] B1 Correct answer without work. B Stops after first line of correct factorisation. e.g. ( x y) + c( x y) B3 Error(s) in factorising any pair of terms. or equivalent. B4 Incorrect common factor and continues. e.g. y ( c) + x( + c) S1 ( + c) ± ( x y) S Correct first line of factorisation but ends as c( x y) or xy ( + c) Attempts ( marks) A1 Pairing off, or indication of common factors and stops. A Correctly factorises any pair and stops.. Page 8

176 Part (b) (iii) 5 marks Att 5(b) (iii) Factorise: x x 4 Part (b) (iii) 5 marks Att x x x 4 6x + 4x 4 6 x( x 6) + 4( x ( x + 4)( x 6) ) x x ( x + 4)( x 6) 1 ( ) ± ( ) 4( 1)( 4) () 1 ± = and ( x 6 )( x + 4) * Accept also (with or without brackets) for 5 marks any of the following ( x 6) and ( x + 4) [The word and is written down.] ( x 6) or ( x + 4) [The word or is written down.] ( x 6), ( x + 4) [A comma is used] B1 Incorrect two term linear factors of. x x 4 formed from correct (but inapplicable) factors of x and 4.e.g ( x 1 )( x + ). B Incorrect factors of x. B3 Incorrect factors of 4. B4 Correct cross method but factors not shown and stops. B5 x ( x 6) + 4( x 6) or similar and stops. B6 Incorrect common factor and continues. B7 Incorrect quadratic formula and continues. B8 Error in quadratic formula. B9 Answer left as roots. B10 Sign error in substituted formula. B11 Error in square root or square root ignored. S1 Numerical errors to a max of 3. S ( x + 4) ± ( x 6 ). Attempts ( marks) A1 Correct quadratic equation formula quoted and stops A Correct factors of either x or ± 4. A3 Any correct step. Worthless (0 marks) W1 x x = 4 or similar and stops. W Combines x's to numbers and continues or stops ± 10 = 8 = 4 Page 9

177 Part (b) (iv) 5 marks Att 5(b) (iv) Factorise: 144 y Part (b) (iv) 5 marks Att ( 1 y )( 1 + y) * Accept also (with or without brackets) for 5 marks any of the following ( 1 y) and ( 1 + y) [The word and is written down.] ( 1 y) or ( 1 + y) [The word or is written down.] ( 1 y), ( 1 + y) [A comma is used] * Quadratic equation formula method is subject to slips and blunders. 144 y y merits 5 marks. * ( )( ) B1 Incorrect two term linear factors of 144 y formed from correct (but inapplicable) factors of 144 and y.e.g ( 144 y )( 1 + y). B Incorrect factors of 144. B3 Incorrect factors of y. B4 ( y 1 )( y + 1). B5 ( y)( 144 y). B6 Answer left as roots. ( y = ±1 ) S1 ( 1 y ) ± ( 1 + y) Attempts ( marks) A1 Correct factors of 144 only. A Correct factors of ± y only. A3 ±1 or ± y appears. A4 144 y = 1 1 y and stops. ( )( ) ( )y A5 Mention of the difference of two squares.e.g. {( 144) y } A6 Correct quadratic equation formula quoted and stops. A7 144 Worthless (0 marks) W1 Combines y's to numbers and continues or stops. Page 30

178 Part(c) (i) 5 marks Att 5(c)(i) x 1 x Express as a single fraction 5 7 and give your answer in its simplest form. Part(c) (i) 5 marks Att x 1 x x 7 5x x ( x 1) 5( x ) B1 Correct answer without work. B Error(s) in distribution. e.g. 7( x 1) = 7x 1 B3 Mathematical error e.g = -3. (-1) =. B4 Incorrect common denominator and continues. B5 5 x 1 7 x Incorrect numerator from candidate's denominator e.g. 35 B6 No simplification of numerator. B7 Omitting denominator. ( ) ( ). S1 Drops denominator. S Numerical error to a max of 3. S3 4 x + 6 Answer not in simplest form. e.g.. 70 Attempts ( marks) A1 35 only or a multiple of 35 only appears. A Any correct step. x 1 x W1 and stops. 5 7 x 1 x x 3 W = 5 7 Page 31

179 Part(c) (ii) 5 marks Att x 1 x 5(c) (ii) Hence, or otherwise, solve the equation = Part(c) (ii) 5 marks Att x + 3 = 1 35 x + 3 = 35 x = 35 3 x = 3 3 x = x = 16 * Accept candidates answer from previous work. B1 Correct answer without work. B Error in transposition. (each time) S1 Numerical error to a max of 3. S 3 Leaves as. Attempts ( marks) A1 Answer from (c) (i) written in this part or worked again in this part. A Any correct step and stops. Page 3

180 Part(c) (iii) 15 marks Att 5 5(c) (iii) Solve for x and for y:. 3x + y = 73 4x + y = 59 Part(c) (ii) 15 marks Att 5 3x + y = 73 4x + 3x + y = 73 8x y = 118 5x y = 59 = 45 5x = x = = 9 5 y = 3 I 3x + y 4 x + 1x + 8y 1x 3y = 177 5y = 115 = 115 y = 5 x = 9 73 y = 59 = 9 = 3 y II = 73 3x 73 3x y = 73 3x 4x + = 59 8x x = 118 5x = x = x = = 9 5 y = 3 * Apply only one blunder deduction (B or B3) to any error(s) in establishing the first equation in terms of x only or the first equation in terms of y only. * Finding the second variable is subject to a maximum deduction of (3). B1 Correct answers without work. B Error(s) in establishing the first equation in terms of x only [ 5 x = 45 ]or the first equation in terms of y only [ 5 y = 115 ] through elimination by cancellation. B3 Error(s) in establishing the first equation in terms of x only [ 5 x = 45 ]or the first equation in terms of y only [ 5 y = 115 ] through elimination by substitution. B4 Errors in transposition in solving the first one variable equation. B5 Errors in transposition when finding the second variable. B6 Incorrect substitution when finding second variable. B7 Finds one variable only. S1 Numerical errors to a max of 3 Attempts (5 marks) A1 Attempt at transposition and stops. A Multiplies either equation by some number and stops. Worthless (0 marks) W1 Incorrect values for x or y substituted into the equations. Page 33

181 QUESTION 6 Part (a) 10 (5, 5) marks Att 4(,,) Part (b) 5 (15, 10) marks Att 8(5, 3) Part (c) 15 (5, 5, 5) marks Att 6(,, ) Part (a) 10(5, 5) marks Att 4(, ) 6 (a) f (x) = 3x 1. Find: (i) f (5) (ii) f ( 4) Part (a) (i) 5 marks Att 6(a) (i) f (x) = 3x 1. Find: f (5) Part (a) (i) 5 marks Att f 5 = = 15 1 = (i) () () 14 B1 Leaves 3 5 in the answer () B Combines x's to numbers and continues e.g. 3 x 1= x = ( 5) = 10 B3 Mathematical error. e.g = 14 B4 Breaks order i.e = 3 4 = ( ) ( ) 1 S1 Numerical errors to a max of 3. S Leaves x in the answer e.g. 14x Misreadings (-1) M1 Correct substitution of any number other than 5 and continues. Attempts (marks) A1 Substitutes for x and stops. i.e. 3 ( 5) A Treats as equation and continues or stops. i.e. 3 x 1= 5 W1 Combines x's to numbers and stops. W Ignores x giving 3 1 =. W3 5[ f ( x) ] = 15x 5 W4 Replaces coefficient i.e. 3x 5x. W5 Incorrect answer without work. Page 34

182 Part (a) (ii) 5 marks Att 6(a) (ii) f (x) = 3x 1. Find: f ( 4) Part (a) (ii) 5 marks Att f 4 = = 1 1 = (ii) ( ) ( ) 13 B1 Leaves 3( 4) in the answer B Combines x's to numbers and continues e.g. 3x 1= x = ( 4) = 8 B3 Mathematical error. e.g. 1 1 = = 3 5 = B4 Breaks order i.e. ( ) ( ) 15 S1 Numerical errors to a max of 3. S Leaves x in the answer e.g. 13x. Misreadings (-1) M1 Correct substitution of any negative number other than 4 and continues. Attempts (marks) A1 Substitutes for x and stops. i.e. 3( 4) A Treats as equation and continues or stops i.e. 3x 1= 4 W1 Combines x's to numbers and stops. W Ignores x giving 3 1 =. W3 4[ f ( x) ] = 1x 4. W4 Replaces coefficient.i.e. 3x 4x. W5 Incorrect answer without work. Page 35

183 Part (b) 5 (15, 10) marks Att 8 (5, 3) 6(b) Draw the graph of the function f : x x 3x 1 in the domain 1 x 4, where x R Part (b) Table 15 marks Att 5 ( x) = x 3x -1 x = ( 1) -1 = 3 x 3 x = 0 3( 0) -1 = = 1 3() 1-1 = -3 f ( x) = 3( ) -1 = -3 = 3 3() 3-1 = -1 = = 3 f f ( 1) ( ) f ( 0) ( ) f () 1 () f ( ) ( ) f () 3 () f ( 4) ( ) 4 ( ) * Error(s) in each row /column attract a maximum deduction of 3. B1 3x taken as 3 all the way. [In row headed 3x by candidate] B 1 calculated as x all the way.[in row headed 1 by candidate] B3 Adds in top row when evaluating f (x). B4 Omits 1 row. B5 Omits 3x row. B6 Omits a value in the domain (each time). B7 Each incorrect image without work. S1 Numerical errors to a max of 3 in any row / column. Misreadings (-1) M1 Misreads x as x and places x in the table or function M Misreads 3x as 3x and places 3x in the table or function. M3 Misreads 1 as 1 and places 1 in the table or function Attempts (5 marks) A1 Omits x row from table or treats x as A Any effort at calculating point(s). A3 Only one point calculated and stops. ± x. Page 36

184 Part (b) Graph 10 marks Att * Accept candidate's values from previous work. * Only one correct point graphed correctly Att 5 + Att 3 * Correct graph but no table full marks i.e. (15+10) marks. * Accept reversed co-ordinates if (i) if axes not labelled or (ii) if axes are reversed to compensate (see B1 below) B1 Reversed co-ordinates plotted against non-reversed axes (once only) {See 4th * above}. B Scale error (once only). B3 Points not joined or joined in incorrect order (once only). S1 Each point of candidate graphed incorrectly. {Tolerance ± 0. 5 } S Each point from table not graphed [See nd * above]. Attempts (3 marks) A1 Axes drawn (need not be labelled). A Some effort to plot a point{see nd * above}. Page 37

185 Part(c) (i) 5 marks Att 6(c) (i) Given that y = x +, complete the table below Part(c) (i) 5 marks Att x y * Accept candidate's values without work. Slips(-1) S1 Each y value omitted or incorrect. Misreadings (-1) M1 Misreads y = x + as y = x. Attempts(marks) A1 Any one correct y value. A Any effort at calculating points. W1 Copies x values into y row. W All y values incorrect with no work shown. (See M1 above). Page 38

186 Part(c) (ii) 5 marks Att 6(c) (ii) On the grid below, the graph of the line y = 4 x is drawn. Using your answers from (i), draw the graph of y = x + on the same grid. Part(c) (ii) Graph 5 marks Att y = x + 1 y = 4 - x * Accept candidate's values from previous work. B1 Reversed co-ordinates ( y, x ) plotted. B Points not joined or joined in incorrect order. S1 Each point of candidate graphed incorrectly. {See B1} S Each point from table not graphed or not contained on the candidate s graph. Attempts ( marks) A1 Any one correct point plotted. W1 No correct point plotted. (See B1 above). Page 39

187 Part(c) (iii) Intersection 5 marks Att 6(c) (iii) Use the graphs drawn in 6 (c) (ii) to write down the coordinates of the point of intersection of the two lines y = 4 x and y = x +. Part(c) (iii) Intersection 5 marks Att Point of intersection = (1, 3) * Accept correct answer based on candidate's graph fully plotted. i.e. 4 points correctly plotted from c(i) otherwise attempt marks at most. Blunders(-3) B1 Answer not presented in designated box. B Answer beyond tolerance. {Tolerance ± 0. 5 }. B3 Reversed co-ordinates ( y, x ) plotted. Attempts(marks) A1 Point of intersection highlighted on graph. Worthless(0) W1 Answers outside of tolerance without graphical indication. W Incorrect answer from candidate's graph. Page 40

188 MARKING SCHEME JUNIOR CERTIFICATE EXAMINATION 007 MATHEMATICS - ORDINARY LEVEL - PAPER 1 GENERAL GUIDELINES FOR EXAMINERS 1. Penalties of three types are applied to candidates work as follows: Blunders - mathematical errors/omissions (-3) Slips- numerical errors (-1) Misreadings (provided task is not oversimplified) (-1). Frequently occurring errors to which these penalties must be applied are listed in the scheme. They are labelled: B1, B, B3,, S1, S,, M1, M, etc. These lists are not exhaustive.. When awarding attempt marks, e.g. Att(3), 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 is awarded zero marks. Some examples of such work are listed in the scheme and they are labelled as W1, W, etc. 4. The phrase hit or miss means that partial marks are not awarded the candidate receives all of the relevant marks or none. 5. The phrase and stops means that no more work is shown by the candidate. 6. Special notes relating to the marking of a particular part of a question are indicated by an asterisk. These notes immediately follow the box containing 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, verifications and answers derived from diagrams (unless requested) qualify for attempt marks at most. 11. A serious blunder, omission or misreading results in the attempt mark at most. 1. Do not penalise the use of a comma for a decimal point, e.g may be written as 5,50. Page 1

189 QUESTION 1 Part (a) 10(5, 5) marks Att 4(, ) Part (b) 0(5, 5, 5, 5) marks Att 8(,,, ) Part (c) 0(5, 5, 5, 5) marks Att 8(,,, ) Part (a) (i) 5 marks Att 1(a) (i) Using the Venn diagram below, shade in the region that represents A B. A B Part (a) (i) 5 marks Att A B B1 Any incorrect indication other than the misreading below. Misreadings (-1) M1 A B indicated. Page

190 Part (a) (ii) 5 marks Att 1(a) (ii) Using the Venn diagram below, shade in the region that represents A B. A B Part (a) (ii) 5 marks Att A B B1 Any incorrect indication other than the misreading below. Misreading (-1) M1 A B indicated. Page 3

191 Part (b) 0(5, 5, 5, 5) marks Att(,,,) 1(b) U is the universal set P = {4, 7, 8} Q = {1,, 5, 7, 8} R = {1,, 3, 6, 7} P Q R 9 Part (b) (i) 5 marks Att 1(b) (i) List the elements of: P Q. Part (b) (i) 5 marks Att P Q = 1,, 4, 5, 7, 8. { } B1 Any incorrect set of elements of P and Q other than the misreading as below. Misreadings (-1) M1 P Q giving {7, 8}. Attempts ( marks) A1 3 or 6 or 9 appear in the answer. Part (b) (ii) 5 marks Att 1(b) (ii) List the elements of: P \ R. Part (b) (ii) 5 marks Att P \ R = 4, 8 { } B1 Any incorrect set of elements of P and R other than the misreading as below. e.g. { P \ Q R } = {4. ( ) } Misreadings (-1) M1 R \ P giving {1,, 3, 6}. Attempts ( marks) A1 5 or 9 appear in the answer. Page 4

192 Part (b) (iii) 5 marks Att 1(b) (iii) List the elements of: ( P R) Q Part (b) (iii) 5 marks Att P R Q = 1,, 7, 8 ( ) { } B1 Any incorrect set of elements of P and Q and R other than the misreading as below. B P R = {1,,3,4,6,7,8}. Misreadings (-1) M1 ( P R) Q giving {1,, 5, 7, 8}. M ( P R) Q giving {1,, 3, 4, 5, 6, 7, 8}. M3 P R giving {7}. ( ) Q Attempts ( marks) A1 9 appears in the answer. Part (b) (iv) 5 marks Att 1(b) (iv) List the elements of: ( P Q) / Part (b) (iv) 5 marks Att ( P Q) / = {3, 6, 9} B1 Any incorrect set of elements of P and Q other than the misreading as below. B ( P Q) = {1,, 4, 5, 7, 8} in this part. Misreadings (-1) / M1 ( P Q) giving {1,, 3, 4, 5, 6, 9,}. / / M P Q giving {1,, 3, 4, 5, 6, 9}. Attempts ( marks) A1 Any incorrect listing of elements other than the misreadings above. Page 5

193 Part (c) 0(5,5,5,5)marks Att(,,,) 1(c) In a class, all the students study Science (S) or Technical Graphics (T). A number of the students study both of these subjects. students study Science. 1 students study Technical Graphics 8 study both subjects. Part (c) (i) 5 marks Att 1(c) (i) Represent this information in the Venn diagram below. S T Part (c) (i) 5 marks Att S T B1 Incorrect Venn diagram subject to S1 below. S1 Numerical errors where work is clearly shown to a max of 3. Misreadings (-1) M1 Interchanges Technical Graphics and Science. Attempts ( marks) A1 Any one correct relevant entry. A Incorrect work with numbers 8, 1, and. (work shown) Page 6

194 Part(c) (ii) 5 marks Att 1(c) (ii) How many students study Science only? Part(c) (ii) 5 marks Att 14 * A correct answer written here in the space provided takes precedence over an incorrect Venn diagram. * Accepts candidates work from previous part c (i). * If no work appears here, award marks if the correct answer appears in the Venn diagram. B1 Any incorrect use of the given numbers or the numbers from the candidates incorrect Venn diagram. {Subject to S1}. S1 Numerical errors where work is clearly shown to a max of 3. Misreadings (-1) M1 Science read as Technical Graphics. Attempts ( marks) A1 Incorrect work with numbers 8, and/or. ( work shown) Part(c) (iii) 5 marks Att 1(c) (iii) How many students are there in the class? Part(c) (iii) 5 marks Att 6 * A correct answer written here in the space provided takes precedence over an incorrect Venn diagram. * Accepts candidates work from previous part c (i), c (ii). Note: Answer c (ii) + 1 added correctly merits full marks. B1 Any incorrect use of the given numbers or numbers from the candidates incorrect Venn diagram. {Subject to nd * above}. S1 Numerical errors where work is clearly shown to a max of 3. S Written as Attempts ( marks) A1 Incorrect work with numbers 14, 8,4,1 or. Page 7

195 Part(c) (iv) 5 marks Att 1(c) (iv) How many students study only one of the two subjects? Part(c) (iv) 5 marks Att 18 * A correct answer written here in the space provided takes precedence over an incorrect Venn diagram. * Accepts candidates work from previous part c (i), c (ii) and c (iii). Note: Answer c (iii) - 8 merits full marks. B1 Any incorrect use of the given numbers or numbers from the candidates incorrect Venn diagram. {Subject to nd * above}. S1 Numerical errors where work is clearly shown to a max of 3. S Written as Attempts ( marks) A1 Incorrect work with numbers 14, 8, 4, 1, or. Page 8

196 QUESTION Part (a) 10 marks Att 3 Part (b) 0(5, 10, 5) marks Att 7(, 3, ) Part (c) 0(5, 5, 10) marks Att 7(,, 3) Part (a) 10 marks Att 3 (a) 6650 was shared between Ciarán and Sheila in the ratio :5. How much did each receive? Part (a) 10 marks Att 3 parts : 5 parts + 5 = 7 x : 5 x x = 6650 = 950 = x = 950 Ciaran = 950 = 1900 x = 1900 (C) = 1900 (C) 7 Sheila = = = 4750 (S) 5 x = 4750 (S) B1 Correct answer without work. B Divisor 7 only and continues. B3 Incorrect multiplier or fails to multiply. (each time). B4 Error in transposition. B5 Fails to find second amount. B6 Adds instead of subtracts. e.g = S1 Numerical errors to a max of 3. Misreadings (-1) M1 Interchanges Ciaran and Sheila. Attempts (3 marks) A1 A Divisor 7 e.g and/or and stops. 5 Indicates 7 parts or parts or 5 parts or 7 or 7 5 or +5=7 and stops. A3 A4 Indicates multiplication of 6500 by and/or 5 and stops. Both answers added together equal (No work shown). W1 Incorrect answer without work. {Subject to A4}. Page 9

197 Part (b) (i) 5 marks Att (b) (i) 8 10 a a n Simplify, giving your answer in the form, a where n N. 5 7 a a Part (b) (i) 5 marks Att 8 a a 5 a a 10 7 = a a 18 1 = 6 a a a 8 5 a a 10 7 = a 3 a 3 = a 6 a a.. a = a a a 6 a B1 Correct answer without work. B Error in calculation involving indices. B3 Error in number of a s in the extended form. B4 Error in elimination in the extended form. B5 Fails to finish. 18 a S1 = 6 1 a S Answer left a a a a a a. Attempts ( marks) A1 Some correct manipulation of indices. e,g, 8+10, 18, a 3, a 5, or a and stops. 1 W1 Incorrect answer without work. Page 10

198 Part (b) (ii) 10 marks Att 3 (b) (ii) By rounding each of these numbers to the nearest whole number, 4 09 estimate the value of Part (b) (ii) 10 marks Att is approximately equal to = = 8 _ * and stops 4 marks. 6 3 * No penalty if the intermediate step between approximations and final answer is not shown. i.e 3 4 not shown * Special Case: = 7 6 presented in this part Attempt 3 marks B1 Correct answer without work. B Error(s) in rounding off to the nearest whole number. B3 Decimal error in calculation of final value. B4 An arithmetic operation other than indicated. 4 4 B5 Error(s) in the manipulation of the denominator. e.g. or similar. 6 3 S1 Numerical errors to a max of 3. Attempts (3 marks) A1 Only one or two approximations made to the given numbers and stops. A No rounding off applied to given numbers. W1 Incorrect answer without work. Page 11

199 Part (b) (iii) 5 marks Att (b)(iii) Using a calculator, or otherwise, find the exact value of Part (b) (iii) 5 marks Att = = B1 Decimal error. B 4 09 Treats as.93 = = {B1 may occur}. 6 1 B Treats as 4 09 = = {B1 may occur}. 93 B4 Treats as = = {B1 may occur} B5 Treats as = = {B1 may occur} B Treats as 4 09 = = {B1 may occur} B Treats as = = {B1 may occur} S1 Numerical errors to a max of 3. S Any rounding off. Attempts ( marks) A1 Any correct relevant calculation and stops. W1 Incorrect answer without work. Page 1

200 Part(c) (i) 5 marks Att (c) (i) Using a calculator, or otherwise, find the exact value of ( 5). 1 Part(c) (i) 5 marks Att 1 3 ( 5) = =1 5 1 ( ) B1 Mishandles ( 5) e.g. 5 = B Decimal error. Attempts ( marks) A1 5 and stops. 1 A 5 = Worthless(0) W1 5 or 75. Part(c) (ii) 5 marks Att (c) (ii) Using a calculator, or otherwise, multiply 54 5 by 60 n and express your answer in the form a 10, where1 a < 10 and n N Part(c) (ii) 5 marks Att = 370 = B1 Correct answer without work. B Decimal error. S1 Numerical errors to a max of 3. S 3 3 Rounds off to, or S3 3 Incorrectly rounds off. e.g also attracts S. S4 Incorrect format, where a < 1 or a 10 and n N. Attempts ( marks) A1 Any relevant step. e.g. Partial multiplication. Page 13

201 Part(c) (iii) 10 marks Att 3 (c) (iii) Using a calculator, or otherwise, evaluate ( 6 9) Give your answer correct to two decimal places Part(c) (iii) 10 marks Att 3 = = = = 44 8 * Correct answer (without work) incorrectly rounded off 6 marks B1 Correct answer without work. B Mishandles ( 6 9). B3 Mishandles B4 Error in or candidate's equivalent from previous work. B5 Error in or candidate's equivalent from previous work. B6 Decimal error. B7 Subtracts before Division = = {Note S,S3 } B8 Use of mathematical operator other than that which is indicated. B9 Works as = = 1 59.{Note S,S3 } S1 Numerical errors to a max of 3. S Each premature rounding off that effects the final answer to a max of 3. S3 Fails to round off or rounds off incorrectly when giving final answer. Attempts (3 marks.) A1 Any correct relevant step e.g. ( 6 9) = 47 61, = Page 14

202 QUESTION 3 Part (a) 10 marks Att 3 Part (b) 0(10, 10) marks Att 6(3, 3) Part (c) 0(10, 10) marks Att 6(3, 3) Part(a) 10 marks Att 3 3(a) In one week Bríd sent 6 text messages on her mobile phone 11 of these messages cost 8c each The rest of the text messages cost 1c each. Find the total cost of Bríd s texting. Part (a) 10 marks Att = = = Times = = Times = 180. Total Cost = 68c ( 68 ) Total Cost = 68c ( 68 ) * No penalty for omission of symbol. * Accept 68c, ( 68 ) * Adds = 0 and stops merits 3 marks (Oversimplification). B1 Correct answer without work. B Fails to subtract 11 from 6. B3 Each missing product when finding each cost e.g. 11 not multiplied by 8. B4 Each missing item when finding total cost e.g. Expensive texts omitted. B5 Fails to find total cost i.e. no addition. B6 Operation other than addition when finding total cost. B7 Decimal error e.g. 6 8 (Note: 1 st * above). S1 Numerical errors to a max of 3. Misreadings (-1) M1 15 8c and 11 1c. Attempts (3 marks) A1 Any attempt at addition /multiplication. W1 Incorrect answer without work. Page 15

203 Part (b) (i) 10 marks Att 3 3(b) (i) John s gross pay is His tax credit is 3400 He pays income tax at the rate of 0% Find John s take-home pay. Part (b) (i) 10 marks Att 3 Gross Pay % 4600 Tax Credit 3400 Tax-Due 100 Take-home Pay 1,800 0 Tax = 3,000 = or 3, = 4600 Tax Due = = 100 Take-home Pay = = 1,800 B1 Correct answer without work. B Mishandles 0% of 3,000. {Must use 3 000} B3 Decimal error. B4 Misuse of Tax Credit B5 Incorrect use of Tax Amount. B6 Fails to finish. {B4 may apply} S1 Numerical errors to a max of 3 Attempts (3 marks) 0 A1 Some use of 100 in attempt to find percentage e.g. 0 % = and stops. 100 W1 Incorrect answer without work Page 16