2018 Mathematics. National 5 - Paper 1. Finalised Marking Instructions

Transcription

1 National Qualifications Mathematics National 5 - Paper 1 Finalised Marking Instructions Scottish Qualifications Authority 018 The information in this publication may be reproduced to support SQA qualifications only on a noncommercial basis. If it is reproduced, SQA should be clearly acknowledged as the source. If it is to be used for any other purpose, written permission must be obtained from permissions@sqa.org.uk. Where the publication includes materials from sources other than SQA (secondary copyright), this material should only be reproduced for the purposes of examination or assessment. If it needs to be reproduced for any other purpose it is the centre s responsibility to obtain the necessary copyright clearance. SQA s NQ Assessment team may be able to direct you to the secondary sources. These ing instructions have been prepared by examination teams for use by SQA appointed ers when ing external course assessments. This publication must not be reproduced for commercial or trade purposes.

2 General ing principles for National 5 Mathematics Always apply these general principles. Use them in conjunction with the detailed ing instructions, which identify the key features required in candidates responses. For each question, the ing instructions are generally in two sections: generic scheme this indicates why each is awarded illustrative scheme this covers methods which are commonly seen throughout the ing In general, you should use the illustrative scheme. Only use the generic scheme where a candidate has used a method not covered in the illustrative scheme. (a) (b) (c) (d) (e) (f) (g) Always use positive ing. This means candidates accumulate s for the demonstration of relevant skills, knowledge and understanding; s are not deducted for errors or omissions. If you are uncertain how to assess a specific candidate response because it is not covered by the general ing principles or the detailed ing instructions, you must seek guidance from your team leader. One is available for each. There are no half s. If a candidate s response contains an error, all working subsequent to this error must still be ed. Only award s if the level of difficulty in their working is similar to the level of difficulty in the illustrative scheme. Only award full s where the solution contains appropriate working. A correct answer with no working receives no, unless specifically mentioned in the ing instructions. Candidates may use any mathematically correct method to answer questions, except in cases where a particular method is specified or excluded. If an error is trivial, casual or insignificant, for example 6 x 6 = 1, candidates lose the opportunity to gain a, except for instances such as the second example in point (h) below. page 0

3 (h) If a candidate makes a transcription error (question paper to script or within script), they lose the opportunity to gain the next process, for example This is a transcription error and so the is not awarded. This is no longer a solution of a quadratic equation, so the is not awarded. x x x x 4x 0 x 1 The following example is an exception to the above This error is not treated as a transcription error, as the candidate deals with the intended quadratic equation. The candidate has been given the benefit of the doubt and all s awarded. x x x x 4x 0 ( x )( x1) 0 x 1 or (i) Horizontal/vertical ing If a question results in two pairs of solutions, apply the following technique, but only if indicated in the detailed ing instructions for the question. Example: x x 4 y 5 y 7 Horizontal: 5 x and x 4 Vertical: 5 x and y 5 6 y 5 and y 7 6 x 4 and y 7 You must choose whichever method benefits the candidate, not a combination of both. (j) In final answers, candidates should simplify numerical values as far as possible unless specifically mentioned in the detailed ing instruction. For example 15 1 must be simplified to 5 4 or must be simplified to must be simplified to must be simplified to 8* must be simplified to 4 15 *The square root of perfect squares up to and including 100 must be known. page 0

4 (k) (l) Commonly Observed Responses (COR) are shown in the ing instructions to help common and/or non-routine solutions. CORs may also be used as a guide when ing similar non-routine candidate responses. Do not penalise candidates for any of the following, unless specifically mentioned in the detailed ing instructions: working subsequent to a correct answer correct working in the wrong part of a question legitimate variations in numerical answers/algebraic expressions, for example angles in degrees rounded to nearest degree omission of units bad form (bad form only becomes bad form if subsequent working is correct), for example ( x x x )( x 1 ) written as ( x x x ) x 1 4 x 5x 8x 7x gains full credit repeated error within a question, but not between questions or papers (m) In any Show that question, where candidates have to arrive at a required result, the last is not awarded as a follow-through from a previous error, unless specified in the detailed ing instructions. (n) (o) (p) You must check all working carefully, even where a fundamental misunderstanding is apparent early in a candidate s response. You may still be able to award s later in the question so you must refer continually to the ing instructions. The appearance of the correct answer does not necessarily indicate that you can award all the available s to a candidate. You should legible scored-out working that has not been replaced. However, if the scoredout working has been replaced, you must only the replacement working. If candidates make multiple attempts using the same strategy and do not identify their final answer, all attempts and award the lowest. If candidates try different valid strategies, apply the above rule to attempts within each strategy and then award the highest. For example: Strategy 1 attempt 1 is worth s. Strategy 1 attempt is worth 4 s. From the attempts using strategy 1, the resultant would be. Strategy attempt 1 is worth 1. Strategy attempt is worth 5 s. From the attempts using strategy, the resultant would be 1. In this case, award s. page 04

5 Detailed ing instructions for each question Question Generic scheme Illustrative scheme 1. 1 identify common denominator or answer 15 or Correct answer without working award 0/. Do not penalise incorrect conversion of 47 to a mixed number 15 Question Generic scheme Illustrative scheme. 1 start expansion complete expansion collect like terms (see Note ) 1. Correct answer without working award / x x x 1 or 1 x 10 x x x 1 x 10 5x x 11. Terms in the expression to be simplified must include a constant and two different powers of x eg (a) x x x 1 x 10 x x x 11 award / (b) ( x 1) 1( x 1) ( x 5) x x 1 x 10 x 4x 14 award / (c) x 11 x 10 5x 10 award 1/. For subsequent incorrect working, the final is not available page 05

6 . ¹ evidence of scaling (match x or y coefficients) ¹ eg 8 x 10 y 6 0x10y5 ² follow a valid strategy through to produce values for x and y ² values for x and y ³ calculate correct values for x and y ³ x 05, y 1 1. Correct answer without working award 0/. Answer obtained by guess and check award 0/ page 06

7 4. 1 evidence of subtraction 1 eg all components correct Correct answer without working award /. The maximum available is 1/ where (a) brackets are omitted from the final answer (b) final answer is given in coordinate form. Accept one correct component for evidence of subtraction 5 eg or 9 or award 1/ (a) award / (b) v 4 1 award 1/ u ( u v ) no working necessary award 1/ u ( u v) no working necessary award 1/ 4. 5, 9, or 5, 9, award 1/ page 07

8 1. Question Generic scheme Illustrative scheme 5. 1 correct factorisation solve for x 1. Correct answer without working award 0/ 1 ( x)( x8 ) x x, 8. For an answer obtained by guess and check award 0/. BEWARE is only available if,8 are clearly stated as solutions to ( x )( x 8) 0 and not as factors of 4 4. Where quadratic formula is used award s as follows ,8 v 1. ( x )( x 8),8 award 0/. ( x 4)( x6) 4,6 award 1/ page 08

9 6. ¹ state the value of a ¹ a 5 state the value of b b 4 1. (a) y 5cos 4x award / (b) 5,4 award 1/. (a) a 4, b 5 or y 4cos 5x award 1/ (b) 4,5 award 0/. (a) y 5cos x award 1/ (b) 5, award 0/ page 09

10 7. (a) Method 1: y b mx a Method 1 ¹ calculate gradient ¹ 6 4 or equivalent ² substitute gradient and a point y b m x a into ³ state equation in simplest form in terms of P and d ² eg y 0 x1 ³ P d or equivalent 6 4 Method : y mx c Method ¹ calculate gradient ² substitute gradient and a point into y mx c ¹ 6 4 ² eg or equivalent c 4 ³ state equation in simplest form in terms of P and d ³ P d or equivalent page 10

11 1. Correct answer without working award 0/. Gradient need not be simplified for the award of. Where 6 4 is simplified incorrectly is still available 6 eg m y 0 ( x 1) P d 1 award / 4 4. ³ is not available where the calculated gradient is an integer 5. ³ is not available where a decimal approximation is used for the gradient 4 eg m 0 67 y ( x 1) P 0 67d award 1/ 6 6. ³ is not available for invalid subsequent working eg P d P d award / Working must be shown. 1. y x award /. (a) P d P d 4 award / 6 (b) P d 4P 6d 8 4 award /. 6 m 0 67 y ( x 1) P 0 67d award / 4. 6 m 0 75 y ( x 1) P 0 75d award / 5. 4 m y 0 ( x 1) P d 1 6 award / 6. m 0 75 y ( x 1) P 0 75d 11 4 award / page 11

12 (b) ¹ calculate cost ¹ Correct answer without working award 1/1. Do not penalise omission of. Do not accept 95 or Follow through from part (a) is only available if 5 is multiplied or divided by a non-unitary fraction (or decimal equivalent) followed by an addition or subtraction page 1

13 8. ¹ find discriminant ¹ 4 state nature of roots 1. Correct answer without working award 0/ no real roots, award / no real roots. Do not accept no roots, no real (or) distinct roots, no real (and) distinct roots 4. Expected answer(s) for the award of, when (a) b 4ac 0 : two real (and) distinct roots (b) b 4ac 0 : one repeated real root or two equal real roots award 1/ award 1/ Question Generic scheme Illustrative scheme 9. 1 calculate the size of an interior angle of the decagon or angle JKL 1 interior angle 7 7 or JKL 6 calculate the size of angle KJL Correct answer without working award /. Degree signs are not required. Full s may be awarded for information ed on the diagram 4. For a final answer of 6 which is not named or shown at JKL award 0/ 5. Where JKL has been calculated incorrectly is only available where there is clear evidence that JKL has been calculated by using the interior angle or exterior angle of the decagon page 1

14 10. 1 correct substitution into cosine rule calculate XY 144 calculate XY 1 (cm) 1. Correct answer without working award 0/ 1 1. For cos cos where cos is scored out in each line of working award /. is available for (a) expressing eg x as a surd in its simplest form award 1/ [ is not available where x cannot be simplified] (b) calculating x where x is a perfect square greater than 100 eg award 0/ cos 1 award / cos award 0/ 8 1 page 14

15 11. ¹ express as equivalent fraction with rational denominator ¹ express in simplest form ² 6 1. Correct answer without working award 0/. Accept For subsequent incorrect working, the final is not available eg 6 award 1/ page 15

16 1. ¹ state value ¹ 05 1 Question Generic scheme Illustrative scheme 1. 1 state coordinates of B 1 4, 8, 5 state coordinates of C 6, 8, 0 1. The maximum available is 1/ where (a) brackets are omitted (unless already penalised in Q4) (b) answers are given in component form (unless already penalised in Q4). (a) For 4, 8, 5 and (b) For 6, 8, 0 award / C 4, 8, 5 and B6, 8, 0 award 1/. For eg 5, 4, 8 and 0, 6, 8 [repeated error] award 1/ 4. is available for answers of the form B x, y, z,cx, y, z 5 eg, 8, 5 and 4, 8, ,8,5 and 6,8,0 award 1/ (award / if omission of brackets has already been penalised in Q4) and 8 or 8 and 8 award 1/ (award / if use of coordinates instead of components has already been penalised in Q4) and award 0/ page 16

17 14. 1 subtract h 1 y h g x divide by g y h x g square 1. Correct answer without working award 1/ y h x g. For subsequent incorrect working, the final is not available Working must be shown. y h 1. x g g y h. x g award / award / page 17

18 15. ¹ start process ¹ 4 9 or p 8 complete process p 1. For a correct answer without working award /. For subsequent incorrect working, the final is not available. BEWARE : For p p p award 0/ 4 4. Award ¹ for an incorrect expansion leading to... or... p eg p p p p p p p award 1/ 9 9 page 18

19 16. 1 identify roots 1 4 AND 6 identify turning point OR y-intercept (1, 5) OR 4 identify the turning point AND the y-intercept and sketch a consistently annotated parabola (1, 5) AND 4 and consistently annotated parabola (see Note 1). 1. is only available where the roots, turning point AND y-intercept are clearly ed and consistently annotated on the sketch. Accept correctly calculated roots and/or y-intercept annotated as (0,4, (0, 6) and (4, 0) as evidence for the award of (treat as bad form). is not available if graph is not a parabola eg roots 6and4 ( 1, 1) or 4 award 1/ page 19

20 17. Method 1 1 start substitution into formula Method h or 1 18 Ah 1 complete substitution into formula calculate height 1 6 h (cm) Method 1 change subject to h Method V 1 A or V 1 A correct substitution into formula or calculate height 11 5 (cm) page 0

21 1. For a correct answer without working award 0/. For an answer obtained by guess and check award 0/. Accept or For subsequent incorrect working, the final is not available eg Calculation must involve division by a number greater than 10 for the award of Working must be shown (a) 6 h 18 1 h 18 4h 18 h 4 5 award / (b) 1 1 h 18 4h 18 h 4 5 award 1/ h 18 6 h 18 h [ 6 A ] award 1/. 1 4 h 18 8 h 18 h 17 5 [ A 4 6] award 1/ (a) V Ah 6 h 18 h 18 h 69 award 1/ V 18 (b) h h h 69 A 6 award 1/ (c) 1 6h 18 h 18 h 69 award 0/ page 1

22 18. ¹ correct substitution for tan x ¹ sin x sin xcos x cos x ² express in simplest form ² sin x 1. For sin x without working award 0/. Degree signs are not required. Accept ( sin x) but not sin x 4. ¹ is not available if there are no variables sin eg sin cos sin award 1/ cos 5. ¹ is not available if candidate simply states sin x tan x then proceeds no further cos x 6. ² is not available if there is invalid subsequent working sin x eg (a) sin xcos x sin x 1 cos x award 1/ cos x sin x (b) sin xcos x sin x 1 cos x award / cos x 7. Alternative acceptable strategy: o a o ¹ h h a o ² sin x award / h 1... sin x sin xcos x sin x cos x award 1/ sin x 1 tan x cos x award 1/ cos x sin xcos x cos x award 1/ sin x page

23 19. (a) (i) 1 correct bracket with square 1 x complete process x Correct answer without working award /. If the solution to (a)(i) only appears in (a)(ii) then both s are available No working necessary: 1. Award / for (a) x 90 or x (b) x x 90. Award 1/ for (a) x. (a) (b) x x (c) x x 90 (d) x (e) x award 1/ (b) x 6 90 award 0/ 90 (ii) 1 state equation of axis of symmetry 1 x 1 1. For or axis of symmetry award 0/1. Answer must be consistent with answer to 19(a)(i) [unless no answer given to 19(a)(i)] page

24 19. (b) Ans: d, e 10 4 Method 1 1 equate complete square form to zero start to solve solve equation Method 1 1 ( x ) 90 0 x 90 x 90 4 complete process 4 d, e 10 or 10 Method 1 correct substitution into quadratic formula evaluate discriminant Method (stated or implied by ) express discriminant in simplest form 4 complete process d, e10 or Correct answer without working award 0/4. Where ab, and c are all positive is not available. Where b 4ac is calculated incorrectly, and 4 are only available if (See CORs 5) b 4ac is only available where a correct simplification of discriminant leads to a final answer of the form d d e (See COR 6) page 4

25 Working must be shown award / (BEWARE: candidate may get 88 then change it to 88 ) award 1/ award / (BEWARE: candidate may get 88 then change it to 88 ) award 1/ (BEWARE: candidate may get 88 then change it to 88 ) award 0/ award 1/ award / award /4 [END OF MARKING INSTRUCTIONS] page 5

26 National Qualifications Mathematics National 5 - Paper Finalised Marking Instructions Scottish Qualifications Authority 018 The information in this publication may be reproduced to support SQA qualifications only on a noncommercial basis. If it is reproduced, SQA should be clearly acknowledged as the source. If it is to be used for any other purpose, written permission must be obtained from permissions@sqa.org.uk. Where the publication includes materials from sources other than SQA (secondary copyright), this material should only be reproduced for the purposes of examination or assessment. If it needs to be reproduced for any other purpose it is the centre s responsibility to obtain the necessary copyright clearance. SQA s NQ Assessment team may be able to direct you to the secondary sources. These ing instructions have been prepared by examination teams for use by SQA appointed ers when ing external course assessments. This publication must not be reproduced for commercial or trade purposes.

27 General ing principles for National 5 Mathematics Always apply these general principles. Use them in conjunction with the detailed ing instructions, which identify the key features required in candidates responses. For each question, the ing instructions are generally in two sections: generic scheme this indicates why each is awarded illustrative scheme this covers methods which are commonly seen throughout the ing In general, you should use the illustrative scheme. Only use the generic scheme where a candidate has used a method not covered in the illustrative scheme. (a) (b) (c) (d) (e) (f) (g) Always use positive ing. This means candidates accumulate s for the demonstration of relevant skills, knowledge and understanding; s are not deducted for errors or omissions. If you are uncertain how to assess a specific candidate response because it is not covered by the general ing principles or the detailed ing instructions, you must seek guidance from your team leader. One is available for each. There are no half s. If a candidate s response contains an error, all working subsequent to this error must still be ed. Only award s if the level of difficulty in their working is similar to the level of difficulty in the illustrative scheme. Only award full s where the solution contains appropriate working. A correct answer with no working receives no, unless specifically mentioned in the ing instructions. Candidates may use any mathematically correct method to answer questions, except in cases where a particular method is specified or excluded. If an error is trivial, casual or insignificant, for example 6 x 6 = 1, candidates lose the opportunity to gain a, except for instances such as the second example in point (h) below. page 0

28 (h) If a candidate makes a transcription error (question paper to script or within script), they lose the opportunity to gain the next process, for example This is a transcription error and so the is not awarded. This is no longer a solution of a quadratic equation, so the is not awarded. x x x x 4x 0 x 1 The following example is an exception to the above This error is not treated as a transcription error, as the candidate deals with the intended quadratic equation. The candidate has been given the benefit of the doubt and all s awarded. x x x x 4x 0 ( x )( x1) 0 x 1 or (i) Horizontal/vertical ing If a question results in two pairs of solutions, apply the following technique, but only if indicated in the detailed ing instructions for the question. Example: x x 4 y 5 y 7 Horizontal: 5 x and x 4 Vertical: 5 x and y 5 6 y 5 and y 7 6 x 4 and y 7 You must choose whichever method benefits the candidate, not a combination of both. (j) In final answers, candidates should simplify numerical values as far as possible unless specifically mentioned in the detailed ing instruction. For example 15 1 must be simplified to 5 4 or must be simplified to must be simplified to must be simplified to 8* must be simplified to 4 15 *The square root of perfect squares up to and including 100 must be known. page 0

29 (k) (l) Commonly Observed Responses (COR) are shown in the ing instructions to help common and/or non-routine solutions. CORs may also be used as a guide when ing similar non-routine candidate responses. Do not penalise candidates for any of the following, unless specifically mentioned in the detailed ing instructions: working subsequent to a correct answer correct working in the wrong part of a question legitimate variations in numerical answers/algebraic expressions, for example angles in degrees rounded to nearest degree omission of units bad form (bad form only becomes bad form if subsequent working is correct), for example ( x x x )( x 1 ) written as ( x x x ) x 1 4 x 5x 8x 7x gains full credit repeated error within a question, but not between questions or papers (m) In any Show that question, where candidates have to arrive at a required result, the last is not awarded as a follow-through from a previous error, unless specified in the detailed ing instructions. (n) (o) (p) You must check all working carefully, even where a fundamental misunderstanding is apparent early in a candidate s response. You may still be able to award s later in the question so you must refer continually to the ing instructions. The appearance of the correct answer does not necessarily indicate that you can award all the available s to a candidate. You should legible scored-out working that has not been replaced. However, if the scoredout working has been replaced, you must only the replacement working. If candidates make multiple attempts using the same strategy and do not identify their final answer, all attempts and award the lowest. If candidates try different valid strategies, apply the above rule to attempts within each strategy and then award the highest. For example: Strategy 1 attempt 1 is worth s. Strategy 1 attempt is worth 4 s. From the attempts using strategy 1, the resultant would be. Strategy attempt 1 is worth 1. Strategy attempt is worth 5 s. From the attempts using strategy, the resultant would be 1. In this case, award s. page 04

30 Detailed ing instructions for each question Question Generic scheme Illustrative scheme 1. 1 know how to decrease by % know how to calculate new total carry out calculations correctly within a valid strategy 1. Correct answer without working award / (tonnes). Where an incorrect percentage is used, the working must be followed through to give the possibility of awarding /, with working eg , with working award /. Where division is used (a) along with 0 98, 1 is not available eg (...) award / (b) along with an incorrect percentage, 1 and are not available eg (...) award 1/ Working must be shown: award / award 1/ award 1/ award 1/ award 0/ page 05

31 . Method 1 1 appropriate fraction Method correct substitution into arc length formula calculate arc length (...) cm Method 1 appropriate fraction correct substitution into arc length formula calculate arc length Method (...) cm page 06

32 1. Correct answer without working award 0/. Do not penalise variations in eg ( 0...) award / 60. Premature rounding: rounded working must be to at least significant figures eg (a) ( 8...) 60 award / (b) or 41 8 (...) 60 award / 4. Accept or 41(...) award / For subsequent incorrect working, the final is not available 0 eg or 5( 16...) award / 60 Working must be shown: or 0 6(64...) award / or 15 9(18...) award / ( 16...) award / or 6 or 5(8...) award 1/ ( 11...) award 1/ ( 49...) award 0/ page 07

33 . 1 start process solution 8 1. Correct answer without working award /. Accept for the award of 1. For eg 4 ( 1) (...) or 1 5 award 0/ No working necessary: or 784 award 1/ 7 or (...) award 1/. 496 award 0/ award 0/ page 08

34 4. 1 start to process right hand side collect like terms solve for x 1. Correct answer without valid working award 0/ Treat guess and check as invalid working 1 6x 6 1 x 18 or 18 x x6 or 6 x. For subsequent incorrect working final is not available eg 6 x x 6 award / 1. x 6x 6 1 x 18 x 6 award 1/. 1 x 6x 11 x 1 x award /. (a) x 6x 6 1 x 18 x 6 x 6 award / (b) x 6x 6 1 x 18 x 6 award / page 09

35 1. Question Generic scheme Illustrative scheme 5. (a) Method 1 1 calculate mean calculate x x Method , 0, 1, 5, 16, 4 4 substitute into formula 4 calculate standard deviation ( 049 ) Method Method 1 calculate mean 1 16 calculate x and x 756, 958 substitute into formula calculate standard deviation 4 4( 049 ) 1. For 16 and 4( 04 ) without working award 1/4. Accept (standard deviation =) 4 04 with working. (a) For 16 and 8 5 4( ) award 4/4 (b) For 16 and 8 5 1( ) award /4 v page 10

36 5. (b) 1 compare means 1 eg on average the number of customers was higher on Saturday compare standard deviations eg the number of customers was less varied on Saturday 1. Answers must be consistent with answers to part (a). Statements must involve (a) reference to number of customers and a chronological comparison or reference to Saturday and/or Sunday Accept eg there were more customers on Saturday on average the number of customers decreased Do not accept eg there were more visits on Saturday, the customers were more consistent on Saturday on average the number of customers was more. For the award of 1 (a) eg Accept Saturday s average number of customers was more The amount of people was higher on Saturday (b) eg Do not accept The mean number of customers on Saturday was more There were more customers at each stall on Saturday The average number of people visiting the stalls was better 4. For the award of (a) eg Accept The spread of customer numbers on Saturday was less The number of customers on Saturday was more consistent Saturday s customer numbers were less varied (b) eg Do not accept The standard deviation on Saturday was less The range of customer numbers on Saturday was less The customers on Saturday were less varied On average the number of customers on Saturday was less varied The standard deviation is more consistent v page 11

37 6. 1 valid strategy 1 54a 7 or 5417 state value of a ( a )17 1. Correct answer without working award /. Accept f ( 17) 7 without working award /. Accept use of x in place of a award 0/. 54a 7 or 5417 a 17 f ( a) 17 award /. 54a 7 or 5417 f ( a) 17 award 1/ 4. f ( 7) 5 4x x 17 award / page 1

38 7. 1 substitute into formula 1 4 π calculate volume round to significant figures cm 1. Correct answer without working award 0/. Accept variations in π eg π award /. 4 π award /. 4 π award / 4. 4 π award 1/ page 1

39 1 Question Generic scheme Illustrative scheme 8. 1 rearrange equation 1 1 sin x 7 calculate value of x 8(1...) calculate nd value of x 1718 or 1717 ( 8...) 1. Correct answers without working award 1/. Accept 8 and 17 with valid working. Degree signs are not required 4. Premature rounding: rounded working must be to at least decimal places 1 eg (a) sin x 0 14 x 8( 04...),17 or 171 ( 95...) award / 7 1 (b) sin x 0 1 x 6 or 5( 7...),174 ( 6...) award / 7 5. Inappropriate use of RAD or GRAD should only be penalised once in Q8, Q9, Q1 or Q17 (a) , (RAD) (b) , (GRAD) v 5 1. sin x x 456, award / 1. (a)sin x x 188,518 7 award / 1 (b) sin x x 8, award 0/ page 14

40 9. 1 correct substitution into sine rule 0 1 DC sin 7 sin105 re-arrange formula 0sin105 sin 7 calculate length 1 cm 1. Correct answer without working award 0/. Degree signs are not required 0 DC. BEWARE ( 1...) award / sin 7 sin Disregard errors due to premature rounding provided there is evidence 5. Inappropriate use of RAD or GRAD should only be penalised once in Q8, Q9, Q1 or Q17 (a) 0( 16...) (RAD) (b) 6( 1...) (GRAD) 1. 0? 0sin 8 0( 46...) sin 7 sin 8 sin 7 award /. 0 DC 57 or 56( 7...) award 0/ page 15

41 10. 1 express ED in terms of u and DC in terms of w express BC in terms of u and w in simplest form 1. Correct answer without working award / 1 1 ED uand DC w u 1 w. Accept 1 u + w award /. Evidence for the award of 1 may appear on the diagram 4. For the award of 1 accept 1 (a) u w u w 1 (b) u w AB EA 5. BA AE ED DC alone is not enough for the award of 1 6. For u 1 w (a) without working award 0/ (b) but may be worth 1/ if there is valid working page 16

42 11. 1 know that 85% % begin valid strategy 910 1% complete calculation within valid strategy (km ) or (km ) (a) with valid working award / (b) without working award 0/. BEWARE 115% of or (a) and evidence of 1 award 1/ (b) otherwise award 0/ % of ( 05) (a) and evidence of 1 award 1/ (b) otherwise award 0/ 4. Do not accept eg for the award of award / % award / % award / 0 15 page 17

43 1. Method 1 1 marshal facts and recognise right angled triangle 1 A 4 O consistent Pythagoras statement x 1 10 calculate x 4 calculate width cm Method 1 marshal facts and recognise right angled triangle 1 consistent Pythagoras statement x 6 0 calculate x calculate width cm page 18

44 1. Correct answer without working award 0/4. In the absence of a diagram accept of 1 and x 1 10 or x 6 0 as evidence for the award. BEWARE Where a diagram is shown, working must be consistent with the diagram is not available for an incorrect diagram leading to x 1 10 or x is available for a valid trig. method 5. Where a candidate assumes the sizes of one or both of the smaller angles in the right-angled triangle, only 1 and 4 are available 6. 4 is only available following a Pythagoras (or trig.) calculation within a right-angled triangle involving 1 and 10 or 6 and 0 7. Disregard errors due to premature rounding provided there is evidence 1. x 1 10 x 16 4 ; width 9 4 (a) working inconsistent with correct diagram (b) working consistent with candidate s diagram (cosine rule may be used to calculate x ) (c) no diagram award /4 award /4 award /4.. x 6 0 x 8 ; width 9 4 (a) working inconsistent with correct diagram (b) working consistent with candidate s diagram (cosine rule may be used to calculate x ) (c) no diagram x 0 1 x 15 ; width 8 (a) working consistent with candidate s diagram (cosine rule may be used to calculate x ) (b) no diagram award /4 award /4 award /4 award /4 award 1/4 page 19

45 1. 1 correct substitution into cosine rule correct calculation of cos YTF calculate angle YTF 4( ) 4 calculate bearing 4 8( ) page 0

46 1. Correct answer without working award 0/4. For subsequent invalid working 4 is not available eg Degree signs are not required 4. Where an incorrect angle has been calculated 4 can only be awarded where there is clear evidence of an intention to calculate angle T eg cos T OR T 1 OR angle ed at T on the diagram can only be awarded for adding 40 to a value previously calculated using trig. 6. Disregard errors due to premature rounding provided there is evidence 7. Inappropriate use of RAD or GRAD should only be penalised once in Q8, Q9, Q1 or Q17 (a) (RAD) (b) (GRAD) Working must be shown (a) cos T (b) award /4 (see Note 4) award / (a) cos T (b) award /4 (see Note 4) award /4 page 1

47 14. Method 1 1 isolate term in y or divide throughout by 5 state coordinates (must use brackets) Method 1 1 5y... 0 or y or x 5 y , 4 Method 1 substitute x 0 into equation state coordinates (must use brackets) 1. Correct answer without working award / Method 1 0 5y 0 0, 4. Disregard errors in the x term for the award of 1. For finding where the line crosses the x-axis, (10, 0), with working award 1/ Commonly Observed Responses 1. 0, 4 (no working necessary) award 1/. y 4 (no working necessary) award 1/ page

48 15. ¹ start to divide fractions n n ¹ n 4 ² factorise ² nn ³ multiply and simplify ³ n n or n n 6 S 1. Correct answer without working 0/. For subsequent incorrect working, the final is not available n n eg (a) award / n n (b) n n ( ) award / n ( n ) 1. n n 4 n n 4 n 4 n 4 ( n ) ( n ) award / page

49 16. 1 start valid strategy or (stated or implied by ) 1 4 continue strategy calculate length of space diagonal 4 valid conclusion with comparison 90 4 Yes, since page 4

50 1. Correct answer without working 0/4. Accept correct use of cosine rule. Accept eg ( 40) for the award of 1 and 4. 4 can only be awarded for a valid conclusion and comparison made with a value obtained from a Pythagoras (or trigonometric) calculation 5. Award of 4 : eg (a) ; Yes, since the umbrella is only 85 award 4/4 (b) ; Yes award /4 (c) ; No, since the locker is only award /4 1. (a) ; No, since award /4 (b) ; No, since award /4. (a) ; Yes, since award 4/4 (b) ; Yes, since award 4/4. (a) (b) ; Yes,since award 4/ ; Yes,since award /4 4. (a) (b) (c) (d) ; No,since award / ; No,since award 1/ ; No,since award / ; No,since award 1/ ; No,since award 1/4 page 5

51 17. 1 substitute correctly into area of triangle formula sin... 5 appropriate fraction for sector substitute correctly into area of sector formula know to subtract area of sector from area of triangle 4 evidence of area of triangle area of sector 5 calculate area of shaded region and state correct units cm page 6

52 1. Correct answer without working award 0/5. Accept variations in. Disregard errors due to premature rounding provided there is evidence 4. Inappropriate use of GRAD leading to... Q9, Q1 or Q cm should only be penalised once in Q8, 5. The following answers should be awarded 4/5 if the use of RAD has already been penalised in Q8, Q9 or Q1 /5 if the use of RAD has not already been penalised in Q8, Q9 or Q1 405(...) 589( 04) cm (a) (b) (...) ( )... (c) ( ) (...)... (d) ( ) ( (...)) cm cm cm 6. 5 is only available for calculating the sum or difference of the area of a triangle and the area of a sector, where the area of the triangle is calculated using trigonometry Working must be shown: sin ( 1...) cm 60 award 4/ sin ( 4...) cm 60 award 4/ sin ( 6...) cm 60 award / ( 0...) cm 60 award / sin (...) award 1/ ( 4...) cm award 0/5 page 7

53 18. (a) Method 1 1 state linear scale factor know to multiply volume by cube of linear scale factor Method 1 1 eg or equivalent correct calculation (must involve a power of the scale factor), valid comparison and conclusion , so the cartons are not similar Method Method 1 state linear scale factor 1 eg 16 4 or equivalent know to divide volume by cube of linear scale factor correct calculation (must involve a power of the scale factor), valid comparison and conclusion , so the cartons are not similar Method Method 1 state volume scale factor 1 eg or equivalent know to multiply depth by cube root of volume scale factor correct calculation (must involve a root of the volume scale factor), valid comparison and conclusion Method 4 0 4, so the cartons are not similar Method 4 1 state linear scale factor 1 eg 4 16 or equivalent know to compare cube of linear scale factor with volume scale factor correct calculation (must involve a power of the scale factor), valid comparison and conclusion and , so the cartons are not similar page 8

54 18. (a) Method 5 1 state volume scale factor Method 5 1 eg or equivalent know to compare cube root of volume scale factor with linear scale factor correct calculation(must involve a root of the volume scale factor), valid comparison and conclusion and , so the cartons are not similar 1. Correct answer without working award 0/. For the award of 1 accept a rounded or truncated decimal equivalent (to at least decimal places) for evidence of the scale factor eg 195, 0 66 ; but see Note Where premature rounding leads to an inaccurate answer, is not available eg ; , so the cartons are not similar award / 4. For the award of there must be a numerical comparison or a statement such as the two scale factors are different ; a simple statement of not similar is not sufficient ; 576, so the cartons are not similar 4 award /. 4 (a) ; , so the cartons are not similar 16 award 1/ 16 (b) ; , so the cartons are not similar 4 award 1/. (a) (b) ; , so the cartons are not similar award / ; , so the cartons are not similar award / (a) , ; , so the cartons are not similar 16 4 award / (b) 5, ; , so the cartons are not similar 16 4 award / (c) 6, ; , so the cartons are not similar 16 4 award 1/ page 9

55 18. (b) Method 1 1 find volume scale factor Method correct calculation to find height (must involve a root of the volume scale factor) cm 576 Method 1 find volume scale factor Method correct calculation to find height (must involve a root of the volume scale factor) 1. Correct answer without working award 0/ cm Disregard errors due to premature rounding provided there is evidence award 1/ award 1/ award 1/ 115 [END OF MARKING INSTRUCTIONS] page 0