GCSE (9–1) Mathematics J560/04: Paper 4 (Higher tier) General Certificate of Secondary Education Mark Scheme

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GCSE (9–1) Mathematics J560/04: Paper 4 (Higher tier) General Certificate of Secondary Education Mark Scheme Oxford Cambridge and RSA Examinations H GCSE (9–1) Mathematics J560/04: Paper... 4 (Higher tier) General Certificate of Secondary Education Mark Scheme for DRAFT November 2021Oxford Cambridge and RSA Examinations OCR (Oxford Cambridge and RSA) is a leading UK awarding body, providing a wide range of qualifications to meet the needs of candidates of all ages and abilities. OCR qualifications include AS/A Levels, Diplomas, GCSEs, Cambridge Nationals, Cambridge Technicals, Functional Skills, Key Skills, Entry Level qualifications, NVQs and vocational qualifications in areas such as IT, business, languages, teaching/training, administration and secretarial skills. It is also responsible for developing new specifications to meet national requirements and the needs of students and teachers. OCR is a not-for-profit organisation; any surplus made is invested back into the establishment to help towards the development of qualifications and support, which keep pace with the changing needs of today’s society. This mark scheme is published as an aid to teachers and students, to indicate the requirements of the examination. It shows the basis on which marks were awarded by examiners. It does not indicate the details of the discussions which took place at an examiners’ meeting before marking commenced. All examiners are instructed that alternative correct answers and unexpected approaches in candidates’ scripts must be given marks that fairly reflect the relevant knowledge and skills demonstrated. Mark schemes should be read in conjunction with the published question papers and the report on the examination. © OCR 2021 DRAFTJ560/04 Mark Scheme November 2021 2 Annotations available in RM Assessor. These must be used whenever appropriate during your marking. Annotation Meaning Correct Incorrect Benefit of doubt Follow through Ignore subsequent working (after correct answer obtained), provided method has been completed Method mark awarded 0 Method mark awarded 1 Method mark awarded 2 Accuracy mark awarded 1 Independent mark awarded 1 Independent mark awarded 2 Misread Special case Omission sign Blank page Seen DRAFTJ560/04 Mark Scheme November 2021 3 For a response awarded zero (or full) marks a single appropriate annotation (cross, tick, M0 or ^) is sufficient, but not required. For responses that are not awarded either 0 or full marks, you must make it clear how you have arrived at the mark you have awarded and all responses must have enough annotation for a reviewer to decide if the mark awarded is correct without having to mark it independently. It is vital that you annotate standardisation scripts fully to show how the marks have been awarded. Subject-Specific Marking Instructions 1. M marks are for using a correct method and are not lost for purely numerical errors. A marks are for an accurate answer and depend on preceding M (method) marks. Therefore M0 A1 cannot be awarded. B marks are independent of M (method) marks and are for a correct final answer, a partially correct answer, or a correct intermediate stage. SC marks are for special cases that are worthy of some credit. 2. The following abbreviations are commonly found in GCSE Mathematics mark schemes. - figs 237, for example, means any answer with only these digits. You should ignore leading or trailing zeros and any decimal point e.g. 237000, 2.37, 2.370, 0.00237 would be acceptable but 23070 or 2374 would not. - isw means ignore subsequent working after correct answer obtained and applies as a default. - nfww means not from wrong working. - oe means or equivalent. - rot means rounded or truncated. - soi means seen or implied. - dep means that the marks are dependent on the marks indicated. You must check that the candidate has met all the criteria specified for the mark to be awarded. - with correct working means that full marks must not be awarded without some working. The required minimum amount of working will be defined in the guidance column and SC marks given for unsupported answers. 3. Anything in the mark scheme which is in square brackets […] is not required for the mark to be earned, but if present it must be correct. 4. Unless the command word requires that working is shown and the working required is stated in the mark scheme, then if the correct answer is clearly given and is not from wrong working full marks should be awarded. Do not award the marks if the answer was obtained from an incorrect method, i.e. incorrect working is seen and the correct answer clearly follows from it. DRAFTJ560/04 Mark Scheme November 2021 4 5. Where follow through (FT) is indicated in the mark scheme, marks can be awarded where the candidate’s work follows correctly from a previous answer whether or not it was correct. For questions with FT available you must ensure that you refer back to the relevant previous answer. You may find it easier to mark these questions candidate by candidate rather than question by question. Figures or expressions that are being followed through are sometimes encompassed by single quotation marks after the word their for clarity, e.g. FT 180 × (their ‘37’ + 16), or FT 300 – (their ‘52 + 72’). Answers to part questions which are being followed through are indicated by e.g. FT 3 × their (a). 6. In questions with no final answer line, make no deductions for wrong work after an acceptable answer (i.e. isw) unless the mark scheme says otherwise, indicated by the instruction ‘mark final answer’. 7. In questions with a final answer line and incorrect answer given: (i) If the correct answer is seen in the body of working and the answer given on the answer line is a clear transcription error allow full marks unless the mark scheme says ‘mark final answer’. Place the annotation ✓ next to the correct answer. (ii) If the correct answer is seen in the body of working but the answer line is blank, allow full marks. Place the annotation ✓ next to the correct answer. (iii) If the correct answer is seen in the body of working but a completely different answer is seen on the answer line, then accuracy marks for the answer are lost. Method marks could still be awarded if there is no other method leading to the incorrect answer. Use the M0, M1, M2 annotations as appropriate and place the annotation  next to the wrong answer. 8. In questions with a final answer line: (i) If one answer is provided on the answer line, mark the method that leads to that answer. A correct step, value or statement that is not part of the method that leads to the given answer should be awarded M0 and/or B0. (ii) If more than one answer is provided on the answer line and there is a single method provided, award method marks only. (iii) If more than one answer is provided on the answer line and there is more than one method provided, award marks for the poorer response unless the candidate has clearly indicated which method is to be marked. 9. In questions with no final answer line: (i) If a single response is provided, mark as usual. DRAFTJ560/04 Mark Scheme November 2021 5 (ii) If more than one response is provided, award marks for the poorer response unless the candidate has clearly indicated which response is to be marked. 10. When the data of a question is consistently misread in such a way as not to alter the nature or difficulty of the question, please follow the candidate’s work and allow follow through for A and B marks. Deduct 1 mark from any A or B marks earned and record this by using the MR annotation. M marks are not deducted for misreads. If a candidate corrects the misread in a later part, do not continue to follow through, but award A and B marks for the correct answer only. 11. Unless the question asks for an answer to a specific degree of accuracy, always mark at the greatest number of significant figures even if this is rounded or truncated on the answer line. For example, an answer in the mark scheme is 15.75, which is seen in the working. The candidate then rounds or truncates this to 15.8, 15 or 16 on the answer line. Allow full marks for the 15.75. 12. Ranges of answers given in the mark scheme are always inclusive. 13. For methods not provided for in the mark scheme give as far as possible equivalent marks for equivalent work. If in doubt, consult your Team Leader. 14. If in any case the mark scheme operates with considerable unfairness consult your Team Leader. DRAFTJ560/04 Mark Scheme November 2021 6 Question Answer Marks Part marks and guidance 1 (a) 68921 1 (b) 2.86 3 B2 for 2.85[7…] OR B1 for 66.95 or 8.2 or 8.16[4…] and B1 for their answer written to more than 3 figures correctly rounded to 3 sf 2 71.4 2 M1 for ½×(12.3 + 8.7) ×6.8 oe 3 x8 DRAFT 1J560/04 Mark Scheme November 2021 7 Question Answer Marks Part marks and guidance 4 16 4 oe nfww 4 M2 for 16 correct outcomes shown or for [4 × 4 =] 16 [outcomes] or M1 for table, list etc, with at least 10 correct outcomes to a maximum of 16 (ignoring repeats) AND M2FT for correctly indicating all the primes in their outcomes (at least 6) and gives the correct response for their outcomes or M1FT for writing their correct response from their outcomes or for indicating all the primes in their outcomes with maximum one error to a maximum of 3 marks if 0 scored then SC3 for a correct response from adding 16 outcomes i.e. 9 16 or SC2 for a correct response from adding (at least 6 outcomes), primes must be indicated or SC1 for correct response from adding (at least 6 outcomes), primes are not indicated Note : an alternative method is M3 for [P(1 with 2,3 OR 2,3 with 1)=] 1 4 × 2 4 + 2 4 × 1 4 or M2 for the above method with one error or M1 for a correct tree diagram drawn M marks are for products The outcomes may be a list or table showing 16 outcomes which may have numbers or ticks and crosses to show primes etc, if just numbers with nothing above 8 assume addition By e.g. shading, underlining or ringing M1 implied by a correct numerator and a correct denominator for their list Note that 2 4 × 2 4 is an incorrect method DRAFTJ560/04 Mark Scheme November 2021 8 Question Answer Marks Part marks and guidance 5 (a) 18 07 [pm] or 6 07 pm 4 B3 for 18 07 am or 6 07 [am] OR B2 for listing the next three correct times of both fountains, i.e. 15 43, 16 07, 16 31 and 16 01, 16 43, 17 25 OR B1 for listing the next three correct times of one fountain, i.e. 15 43, 16 07, 16 31 or 16 01, 16 43, 17 25. Alternative method B3 for 2[h] 48[m] OR B2 for [LCM=] 168 OR B1 for listing the next three multiples of 24 and 42, i.e. 48, 72, 96 and 84, 126, 168 OR M1 for [24 =] 2 × 2 × 2 × 3 or [42 =] 2 × 3 × 7 allow in a factor tree or table or [LCM=] 168k (k ≠ 1) and M1 for correctly converting their time(mins) to hours and mins Condone use of 12 hour clock e.g. [0]3 43 and 3 43 am for B1 and B2 their time must be over 60 DRAFTJ560/04 Mark Scheme November 2021 9 Question Answer Marks Part marks and guidance (b) [size] 15 [number] 11 4 B3 for 15 and 11 OR B2 for [HCF or group size =] 15 or M2 for [60] = 2× 2 × 3 × 5 and [105] = 3 × 5 × 7 or for listing complete factors of both numbers allow in a factor tree or table OR M1 for one of 2 × 2 × 3 × 5 or 3 × 5 × 7 allow in a factor tree or table or for common factors 3 or 5 AND B1 for [size] 3 [number] 55 or [size] 5 [number] 33 accept any correct method [60] 1,2,3,4,5,6,10,12,15,20,30,60 [105] 1,3,5,7,15,21,35,105 6 (a) 500 ml with three correct comparisons 3 Allow any correct comparison e.g.(converting all to 500 ml) B2 for three correct figures to compare or B1 for two correct figures OR M1 for one correct appropriate calculation e.g. 1.96 ÷ 4 or 31 × 5 ÷ 3 oe See appendix for other values e.g. 49[p] is sufficient for B1 as it compares to 47[p] (b) 7 × 120 soi by 840 840 ÷ 300 soi by 2.8 or 3 and 3 × 31 = 93[p] 840 ÷ 500 soi by 1.68 or 2 and 2 × 47 = 94[p] M1 B1 B1 accept any correct argument If B0 then SC1 for 3 [of 300 ml] and 2[of 500 ml] Condone omitting one day so 6 × 120 soi by 720 for M1 3 × 31 = 93[p] is sufficient 2 × 47 = 94[p] is sufficient DRAFTJ560/04 Mark Scheme November 2021 10 Question Answer Marks Part marks and guidance 7 (a) (i) 0.2 and 0.8 in all the correct places 2 B1 for first branch correct or second branches correct Accept equivalent fractions and percentages (need % sign) (a) (ii) 0.64 or 16 25 oe or 64% 2 FT their tree for 1 or 2 marks ( their values < 1) M1 for 0.8 × 0.8 oe Allow long method : e.g. 1 – (0.04 + 0.16 + 0.16) (a) (iii) Suggestion of dependence between the trains or unexpected events or data may not be applicable 1 Accept any correct reason, e.g. if first train is late second train may be held up e.g. unexpected delays can occur e.g. changed schedule that day (implies data not applicable) (b) 0.73[4] or 734 1000 oe or 73.4% 3 M2 for 1 – 0.35 × 0.76 or 0.35 × 0.24 + 0.65 × 0.24 + 0.65 × 0.76 oe or M1 for two correct products or 0.35 × 0.76 e.g. common equivalent 367 500 products implied by 0.266, 0.084, 0.156, 0.494 8 3.25 4 B3 for 0.0325 OR M1 for 7170 – 6000 or 7170 6000 M1 for �ℎ��� 1170 6000 or �ℎ��� 1170 6 or 1.195 - 1 M1 for their 0.195 ÷ 6 or their 195 ÷ 6000 Accept any correct method and condone extra % symbol implied by 1170 or 1.195 implied by 0.195 or 195 implied by 0.0325 watch out for 6√1170 = 3.246… DRAFTJ560/04 Mark Scheme November 2021 11 Question Answer Marks Part marks and guidance 9 1240 3 M2 for 1426 ÷ 1.15 oe or B1 for 1.15 or 115 Accept 115 100 but not 115% 10 87 253 278 with correct working 7 B1 for 3n – 8 B1 for 3n – 8 + 25 or better M1 for writing a correct equation equal to 618 using their expressions e.g. n + 3n – 8 + 3n – 8 + 25 = 618 or better M1 for simplifying their equation e.g. 7n + 9 = 618 A1FT for correctly solving their equation e.g. n = 87 M1 for substituting their 87 into both expressions e.g.3×87 – 8 and 3×87 – 8 + 25 oe Trials: B1 for one complete trial with n ≥ 3 B1 for second complete trial n ≥ 3 If 0 or 1 scored SC3 for 87, 253, 278 with B1 only or SC2 for 87, 253, 278 with no working “Correct working” requires evidence of at least B1 B1 Expressions could start from B or C. See appendix for a more complete set of trials DRAFTJ560/04 Mark Scheme November 2021 12 Question Answer Marks Part marks and guidance 11 AED or DEA and corresponding common oe correct reason e.g. AAA or both triangles have the same angles oe 1 1 1 accept CED, DEC accept “same as angle DAE” oe ignore any reasons 12 (a) ≥ ≤ ≤ 2 B1 for two correct or “> < <” i.e correct but no equals (b) x + y ≥ 6 or y ≥ 6 – x oe 3 B1 for the correct straight line drawn B1 for correct equation for their line e.g. x + y = 6 oe implied by e.g. x + y = 6 oe and accept a ruled or good freehand line bold line shows minimum length implied by e.g. answer of x + y ≤ 6 oe 13 First bar(170≤ h<180) at ‘height’ 2.4 Second bar(180≤h<200) at height 0.5 6 M2 for 3.2×3×10 4×10 oe or M1 for 3.2 × 10 B1 for their bar correctly drawn at �ℎ���24 10 AND M2 for �ℎ��� 80 – �ℎ��� 24 – 32 – �ℎ��� 0.7 × 20 20 oe or M1 for their 80 – their 24 – 32 – their 0.7 × 20 oe AND B1 for their bar correctly drawn at �ℎ��� 10 20 M2 implied by ‘first bar height’ 2.4 M1 implied by 24, 32 or 80 M2 implied by ‘second bar’ height 0.5 M1 implied by 10 DRAFTJ560/04 Mark Scheme November 2021 13 Question Answer Marks Part marks and guidance 14 −3, 8 4 B1 for (x + 3)2 or −6 ÷ 2 B2FT for +8, correct or ft their (x + 3)2 or M1 for (their −3)2 + 6 × (their −3) + 17 B1FT for (−a, b) FT their{(x + a)2 + b} to a maximum of 3 marks If no working B2 for either ordinate correct accept any correct method(see appendix) B3 implied by (x + 3)2 + 8 15 [a=] 3 [b=] -5 [c=] 1 4 B2 for a = 3 or M1 for second differences = 6 M1 for revised terms of -4 -9 -14 -19 or B1 for either b = -5 or c = 1 Condone e.g. 3n2 at least two terms See appendix for alternative methods 16 (a) 6800 1 (b) 4.5 1 condone extra % (c) (i) 11500 or 11530 or 11532 2 M1 for 6800 × 1.04512 oe allow 11531 and 11531.9[9…] (ii) Any correct reason e.g. the rate may not continue DRAFT 1 see appendixJ560/04 Mark Scheme November 2021 14 Question Answer Marks Part marks and guidance 17 105 or 104.7 to 104.82 6 M2 for 85 48sin 53 oe implied by 26.8… or M1 for 48 ]sin[ 85 sin 53 B = oe M1 for [C=] 180 − 53 − their 26.807… or 100.19[…] M2 for sin 53 85sin theirC oe or M1 for sin 53 85 sin ][ = theirC AB oe Alternative method cosine rule (AB = x) M3 for quadratic with coefficients evaluated or M2 for x2 + (−2 × 48 × cos53)x + (482 – 852) [=0] oe or M1 for 852 = x2 + 482 – 2 × x × 48cos53 AND M2 for correct use of quadratic formula or M1 for quadratic formula with at most one error If 0 scored SC2 for 105 or 104.7 to 104.82 with no working or SC1 for 26.8… with no working “Correct working” requires evidence of at least M2 or M1M1 DRAFTi.e. x2 – 57.77…x – 4921 [= 0] oeJ560/04 Mark Scheme November 2021 15 Question Answer Marks Part marks and guidance 18 (a) (i) tan x 1 (ii) 3x 1 (b) (i) Graph of y = –x3 1 Mark intent (ii) y = x3 translated vertically 8 down y-intercept -8 x-intercept 2 1 1 1 Intercepts must be marked on graph and accept given as coordinates 19 (a) complete correct argument e.g. angle ABC = 40° [BO = ] e.g. 6 ���20 17.542… or 17.543 B1 M2 A1dep M1 for e.g sin 20 = 6 [��] dep. on at least M1 accept any correct method not using 17.54 could be on diagram and also accept ABO = 20°, BO’T’ = 70° i.e BO as subject for M2 and condone sine rule with sin 90° for M2 (b) 202 or 201.5 to 201.8 with correct working 5 Accept any correct method e.g. M1 for [height=] 17.54 + 6 or 23.54… M2 for [half base =] 6 tan 35 or 23.54 tan 70 or M1 for tan 35 = 6 ℎ��� ���� or tan 70 = 23.54 ℎ��� ���� M1 for ½ × their base × their height oe If 0 scored SC2 for 202 or 201.6 to 201.8 with no working or SC1 for 8.56 to 8.57 or 17.1 to 17.2 with no working “Correct working” requires evidence of at least M2 or M1M1 Condone use of 8.6 leading to an answer of 202.4… M2 implied by e.g. 8.56 to 8.57 or 17.1 to 17.2 e.g. ½ × (2 × 8.568…) × 23.54… DRAFTJ560/04 Mark Scheme November 2021 16 Question Answer Marks Part marks and guidance 20 [x=] −4 [y=] −1 [x=] 4 [y=] 7 with correct algebraic working 5 accept any correct method M1 for correct substitution e.g. (x – 3)2 + (x + 3)2 [= 50] M1 for expanding both brackets correctly e.g. x2 − 3x − 3x + 9 + x2 + 3x + 3x + 9 [=50] M1 for simplifying their equation e.g. 2x2 = 32 or 2x2 – 32 [= 0] A1FT for x = −4, 4 If 0 scored SC2 for [x=] −4 [y=] −1 [x=] 4 [y=] 7 with no working or SC1 for both x values with no working or a correct pair of x and y values with no working “Correct algebraic working” requires evidence of at least M1M1 implied by 2x2 + 18 [= 50] condoning one error or better to ax2 = b or to ax2 + bx + c [= 0] FT their quadratic equation See appendix for alternative DRAFTmethodsOCR (Oxford Cambridge and RSA Examinations) The Triangle Building Shaftesbury Road Cambridge CB2 8EA [Show More]

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