# Wednesday 3 June 2015 Morning

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2 Section A (36 marks) 1 (i) Differentiate 1 3 x. (ii) Integrate x 63. [] A sequence is defined by u 1 = and u 10 = k 1 u Calculate 4 u k k = 1 /.. + k 3 An arithmetic progression has tenth term 11.1 and fiftieth term 7.1. Find the first term and the common difference. Find also the sum of the first fifty terms of the progression. [5] 4 A sector of a circle has angle 1.5 radians and area 7 cm. Find the perimeter of the sector. [4] 5 Use calculus to find the set of values of x for which x 3-6x is an increasing function. [5] 6 (i) On the same axes, sketch the curves y = 3 x and y = 3 x, identifying clearly which is which. (ii) Given that 3 x = 79, find in either order the values of 3 x and x. [] 7 Show that the equation sin x = 3cos x - can be expressed as a quadratic equation in cos x and hence solve the equation for values of x between 0 and r. [5] 8 Fig. 8 shows the graph of log y against log x. It is a straight line passing through the points ^, 8h and ^0, h. log 10 y (, 8) (0, ) O log 10 x Fig. 8 Find the equation relating log y and log x and hence find the equation relating y and x. [4] OCR /01 Jun15

3 9 8 cm 3 Section B (36 marks) 13 cm 0 cm A Fig. 9.1 (i) Jean is designing a model aeroplane. Fig. 9.1 shows her first sketch of the wing s cross-section. Calculate angle A and the area of the cross-section. [5] (ii) Jean then modifies her design for the wing. Fig. 9. shows the new cross-section, with 1 unit for each of x and y representing one centimetre. y x Fig. 9. Here are some of the coordinates that Jean used to draw the new cross-section. Upper surface Lower surface x y x y Use the trapezium rule with trapezia of width 4 cm to calculate an estimate of the area of this cross-section. [6] OCR /01 Jun15 Turn over

5 475 Mark Scheme June 015 Question Answer Marks Guidance 1 (i) kx oe k is any non-zero constant 4x 3 isw cao [] ignore + c allow any equivalent exact simplified form 1 (ii) 3 1 kx + oe k is any non-zero constant 3x isw + c allow any equivalent exact simplified form u =, u 3 =, u 4 = isw their.5 their 1.6 * NB.5, 1.6, or,, u + u3 + u4 soi dep* must be the sum of 4 terms only may be implied by eg sight of 3.9 and answer of or or cao isw B3 if unsupported NB.5, 1.1, 0.65 scores M0M0 7

6 475 Mark Scheme June a + (10 1)d = 11.1 and a + (50 1)d = 7.1 may be implied by 40d = ±4 or embedded in attempt to solve condone one slip in coefficient of d d = 0.1 if unsupported, B for one of these and B3 for both a = ( their a + 7.1) with a > 11.1 or 50 ( a+ (50 1) d) with a > 11.1 and d < or 477½ or 955 cao if M0, B for any form of correct answer www [5] 4 7 = ½ r 1.5 oe or = π r 360 angle in degrees rounded to sf or more r = 6 soi may be embedded in formula for arc length may be implied by later work eg 9 or 1 their r 1.5 or their π their r if r is incorrect, we must see 360 their r 1.5 [ + r] for if r is correct, may be implied by 9 or 1 1 [cm] cao allow full marks for recovery from working with rounded value of θ in degree form [4] B4 for 1 unsupported www 8

7 475 Mark Scheme June x 6 seen their y = 0 or y > 0 or y 0 must be quadratic with at least one of only two terms correct and identified may be implied by use with inequalities or by ±1.41[41356] to 3 sf or more x = implies x < or x isw if A0A0, allow SC1 for fully correct answer in decimal form to 3 sf or more NB just > x> or < x< or x > or x [5] or A for x > or x x >± A0A0 implies the first then 6 (i) both curves with positive gradients in 1 st and nd quadrants; ignore labels for this mark do not award if clearly not exponential shape; condone touching negative x-axis but not crossing it consider each curve independently; ignore scales and points apart from (0, 1) both through (0, 1) allow if indicated in table of values or commentary if not marked on graph y = 3 x above y = 3 x in first quadrant and below it in second must be clearly labelled, A0 if wrongly attributed or if coincide for negative x from (0, 1) if M0 allow SC1 for one graph fully correct 6 (ii) x = 3 B0 if wrongly attributed 3 x = 7 B0 if wrongly attributed allow 3 3 = 7 with x = 3 stated [] 9

8 475 Mark Scheme June cos x = 3cos x oe * cos x + 3cos x 3 [= 0] *dep or cos x 3cos x + 3 = 0 condone one sign error or constant term of 1 (in LH version) or + 1 (in RH version) cos x = their or cos x = their 0.79 to soi dependent on award of previous method mark, must be correct for their quadratic ignore other values (eg ); condone recovery from x = but M0 if no recovery [x =] to 0.66 isw cao A0 for eg 0.66π if 0.66 not seen separately NB x = [x =] 5.65 to 5.63 isw cao if extra values in range incur a penalty of 1; ignore extra values outside range NB x = [5] if A0A0 allow SC1 for to 37.7 and 3 to 3.31 or for (0.09 to 0.1)π and (1.79 to 1.791)π no SC mark available if extra values in range 8 m = 3 seen log y = mlog x + or log y = mlog x + log100 or log y 8 = m(log x ) condone lack of base; c = is insufficient log y = log x 3 + or log y = log x 3 + log100 or better or 10 log y = 10 3log x + or 10 3log x + log100 or better condone lack of base, but not bases other than 10 unless fully recovered y = 100x 3 www isw or 3 3log x+ log x + y = 10 or y = 10 [4] 3 3log x + log100 log x + log100 y = 10 or y = 10 10

9 475 Mark Scheme June (i) [cos A = ] 13 0 * or 8 = cos A 505 [cos A = ] oe soi 50 or to A = to 13.8 or 14 or to 0.41 or 0.4 (radians); allow B3 if given to 3sf or more unsupported or 15.3 (grad) [Area = ] ½ 0 13 sin their A dep* or for eg ½ 0 8 sin.8, as long as angle calculated correctly from their A (other angles are.7984 and or ) or ( 8)( 13)( 0) NB 13sin A = if ½ b h used to isw or oe isw [5] allow B for unsupported answer within range 11

10 475 Mark Scheme June (ii) h = 4 soi their 4 ( ( )) or their 4 ( (±0.85± 0.76± 0.55± 0.30)) * shape of formula correct with, 3 or 4 y-values in inner bracket with their h; allow recovery from bracket errors M0 if any non-zero x-values used or if y-values used twice eg their 4 { ( ) } ; signs must be consistent in nd alternative all y-values correctly placed with their h, condone omission of zeros and/or omission of outer brackets either 1.8 or ± 9.84 their their dep* ignore subsequent rounding, but A0 if answer spoiled by eg multiplication by 0 or + B3* if area of triangles and 3 trapezia calculated to give correct answer www The final dep* may then be earned. NB or ± 1.7 ± 3. ±.6 ± 1.7 ± 0.60 with consistent signs throughout 1

11 475 Mark Scheme June (ii) alternatively h = 4 soi attempt to find all y-values y upper y lower M0 if values are added to obtain 0.60, 0.80 etc.3,.3, 1.8, 1.34 all y-values correct their 4 ( ( )) shape of formula correct with, 3 or 4 of their y-values in inner bracket with their h; allow recovery from bracket errors eg ½ 4 { (.3+1.8)} M0 if any non-zero x-values used or if y-values used twice FT all their y-values correctly placed, condone omission of zeros and/or omission of outer brackets 31.1 ignore subsequent rounding, but A0 if answer spoiled by eg multiplication by 0 or + B3 if area of triangles and 3 trapezia calculated to give correct answer www NB [6] 13

12 475 Mark Scheme June (i) dy = dx or 11 isw * 9 = their (4 + 3 ) + c dep* or y 9 = their (4 + 3 ) (x ) y = 11x 13 or y = 11x + c and c = 13 stated isw or y 9 = 11(x ) isw 10 (ii) 4x + 3x * [y =] x + 3x + c must see and + c ; may be earned later eg after attempt to find c 9 = c dep* must include constant, which may be implied by answer y = x + 3x 5 cao allow first 4 marks for y = x + 3x + c and c = 5 stated (1, 0) and (.5, 0) oe cao or for x = 1, y = 0 and x =.5, y = 0 B0 for just stating x = 1 and x =.5 3 x = 4 49 y = 8 [7] 6.15 or 6⅛ 14

13 475 Mark Scheme June (iii) substitution to obtain [ y =] f(x) in polynomial form f(x) must be the quadratic in x with linear and constant term obtained in part (ii), may be in factorised form or their x = 1 their 0.5 and their x =.5 their x = 1.5 y = (x 1)(4x + 5) or y = 8x + 6x or y = x+ 4 8 FT must be simplified to one of these forms, FT their quadratic in x with linear and constant term obtained in part (ii) hence y = (x 1)(4x + 5) FT their x-intercepts from their quadratic in x with linear and constant term obtained in part (ii) 3 49, 8 8 oe or FT their (both non-zero) co-ordinates for minimum point or their quadratic in x with linear and constant term obtained in part (ii) 11 (i) oe 6561 [] condone or B if unsupported do not award if only seen in sum of terms of GP if 0, SC1 for 187 unsupported 11 (ii) valid attempt to sum a GP with r = 3 and n = (3 1) 3 1 oe eg or B if M0 or B3 if unsupported must see at least first two terms and last term NB implies from

14 475 Mark Scheme June (iii) 3(3 n 1) > oe 3 1 * M0 for working backwards M0 if = or < used eg 3 n + 1 > or www n > correctly taking logs of both sides eg (n + 1) log 3 > log or nlog3 > log log 3 dep* eg log 3 n + 1 > log www or log 3 n + log 3 > log www; may be implied by next stage of working at least one previous progressive interim step needed with no wrong working; M0dep* for log(3 n 1) > log eg n + 1 > and completion to log 3 without any wrong working log n > 1 log 3 do not allow recovery from bracket errors at any stage n = 13 seen B0 for n 13 or n > 13 [4] 11 (iv) valid attempt to sum a GP with r = and n = 15 * dep* if correct eg = with their < their their their isw allow B3 for unsupported NB 3767 implies from

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### Thursday 25 May 2017 Morning Time allowed: 1 hour 30 minutes

Oxford Cambridge and RSA GCSE (9 1) Mathematics J560/04 Paper 4 (Higher Tier) H Thursday 25 May 2017 Morning Time allowed: 1 hour 30 minutes *7348865303* You may use: A scientific or graphical calculator

### Thursday 26 May 2016 Morning

Oxford Cambridge and RSA H Thursday 26 May 2016 Morning GCSE MATHEMATICS B J567/03 Paper 3 (Higher Tier) * 5 9 8 5 4 1 9 7 8 4 * Candidates answer on the Question Paper. OCR supplied materials: None Other

### H H * * MATHEMATICS FOR ENGINEERING H860/02 Paper 2 LEVEL 3 CERTIFICATE. Wednesday 9 June 2010 Afternoon. Duration: 1 hour 30 minutes.

LEVEL 3 CERTIFICATE MATHEMATICS FOR ENGINEERING H860/02 Paper 2 Candidates answer on the Answer Booklet OCR Supplied Materials: 8 page Answer Booklet Insert (inserted) List of Formulae (MF1) Other Materials

### MATHEMATICS A A502/01 Unit B (Foundation Tier)

THIS IS A NEW SPECIFICATION F GENERAL CERTIFICATE OF SECONDARY EDUCATION MATHEMATICS A A502/01 Unit B (Foundation Tier) *A515440111* Candidates answer on the question paper. OCR supplied materials: None

### B294A. MATHEMATICS B (MEI) Paper 4 Section A (Higher Tier) GENERAL CERTIFICATE OF SECONDARY EDUCATION. Monday 1 June 2009 Morning WARNING

H GENERAL CERTIFICATE OF SECONDARY EDUCATION MATHEMATICS B (MEI) Paper 4 Section A (Higher Tier) B294A *OCE/T75115* Candidates answer on the question paper OCR Supplied Materials: None Other Materials

### Friday 18 January 2013 Afternoon

Friday 18 January 2013 Afternoon AS GCE MATHEMATICS (MEI) 4752/01 Concepts for Advanced Mathematics (C2) QUESTION PAPER * 4 7 3 3 9 7 0 1 1 3 * Candidates answer on the Printed Answer Book. OCR supplied

### The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72.

ADVANCED SUBSIDIARY GCE UNIT 475/0 MATHEMATICS (MEI) Introduction to Advanced Mathematics (C) THURSDAY 7JUNE 007 Additional materials: Answer booklet (8 pages) MEI Examination Formulae and Tables (MF)

### Tuesday 15 January 2013 Afternoon

H Tuesday 15 January 2013 Afternoon GCSE MATHEMATICS A A502/02 Unit B (Higher Tier) * A 5 1 6 8 8 0 1 1 3 * Candidates answer on the Question Paper. OCR supplied materials: None Other materials required:

### PhysicsAndMathsTutor.com

1 (i) both curves with positive gradients in 1 st and do not award if clearly not exponential 2 nd quadrants; ignore labels for this mark shape; condone touching negative x-axis but not crossing it consider

### MATHEMATICS (MEI) 4755 Further Concepts for Advanced Mathematics (FP1)

ADVANCED SUBSIDIARY GCE MATHEMATICS (MEI) 4755 Further Concepts for Advanced Mathematics (FP1) QUESTION PAPER Candidates answer on the printed answer book. OCR supplied materials: Printed answer book 4755

### Monday 16 January 2012 Morning

THIS IS A LEGACY SPECIFICATION H Monday 16 January 2012 Morning GCSE MATHEMATICS B (MEI) B294A Paper 4 Section A (Higher Tier) *B216710112* Candidates answer on the Question Paper. OCR supplied materials:

### physicsandmathstutor.com

ADVANCED GCE MATHEMATICS 4737 Decision Mathematics 2 Candidates answer on the Answer Booklet OCR Supplied Materials: 8 page Answer Booklet Graph paper Insert for Questions 5 and 6 (inserted) List of Formulae

### Introduction to Advanced Mathematics (C1) THURSDAY 15 MAY 2008

ADVANCED SUBSIDIARY GCE 471/01 MATHEMATICS (MEI) Introduction to Advanced Mathematics (C1) THURSDAY 1 MAY 008 Additional materials: Answer Booklet (8 pages) MEI Examination Formulae and Tables (MF) Morning

### Thursday 9 June 2016 Morning

Oxford Cambridge and RSA H Thursday 9 June 2016 Morning GCSE MATHEMATICS B J567/04 Paper 4 (Higher Tier) * 5 9 8 5 6 1 0 7 3 8 * Candidates answer on the Question Paper. OCR supplied materials: None Other

### Wednesday 8 June 2016 Afternoon

Oxford Cambridge and RSA F Wednesday 8 June 2016 Afternoon GCSE GEOGRAPHY B B563/01 Key Geographical Themes (Foundation Tier) *5818140845* Candidates answer on the Question Paper. OCR supplied materials:

### MATHEMATICS (MEI) 4776/01 Numerical Methods

ADVANCED SUBSIDIARY GCE MATHEMATICS (MEI) 4776/01 Numerical Methods *OCE/T61324* Candidates answer on the Answer Booklet OCR Supplied Materials: 8 page Answer Booklet MEI Examination Formulae and Tables

### * * MATHEMATICS (MEI) 4767 Statistics 2 ADVANCED GCE. Monday 25 January 2010 Morning. Duration: 1 hour 30 minutes. Turn over

ADVANCED GCE MATHEMATICS (MEI) 4767 Statistics 2 Candidates answer on the Answer Booklet OCR Supplied Materials: 8 page Answer Booklet Graph paper MEI Examination Formulae and Tables (MF2) Other Materials

### B293B. MATHEMATICS B (MEI) Paper 3 Section B (Higher Tier) GENERAL CERTIFICATE OF SECONDARY EDUCATION. Tuesday 12 January 2010 Morning

H GENERAL CERTIFICATE OF SECONDARY EDUCATION MATHEMATICS B (MEI) Paper 3 Section B (Higher Tier) B293B * OCE / 14186* Candidates answer on the Question Paper OCR Supplied Materials: None Other Materials

### Thursday 14 June 2012 Morning

Thursday 14 June 01 Morning A GCE MATHEMATICS (MEI) 4757 Further Applications of Advanced Mathematics (FP3) QUESTION PAPER *471575061* Candidates answer on the Printed Answer Book. OCR supplied materials:

### Tuesday 13 June 2017 Morning Time allowed: 1 hour 30 minutes

Oxford Cambridge and RSA GCSE (9 1) Mathematics J560/06 Paper 6 (Higher Tier) H Tuesday 13 June 2017 Morning Time allowed: 1 hour 30 minutes *7349142012* You may use: A scientific or graphical calculator

### Thursday 4 June 2015 Morning

Oxford Cambridge and RSA H Thursday 4 June 2015 Morning GCSE MATHEMATICS A A503/02 Unit C (Higher Tier) * 5 0 8 5 0 1 4 2 2 7 * Candidates answer on the Question Paper. OCR supplied materials: None Other

### Friday 8 November 2013 Morning

H Friday 8 November 2013 Morning GCSE MATHEMATICS B J567/04 Paper 4 (Higher Tier) *J517180313* Candidates answer on the Question Paper. OCR supplied materials: None Other materials required: Geometrical

### Monday 20 June 2016 Morning

Oxford Cambridge and RSA Monday 20 June 2016 Morning A2 GCE PHYSICS A G484/01 The Newtonian World *1320816367* Candidates answer on the Question Paper. OCR supplied materials: Data, Formulae and Relationships

### Monday 10 November 2014 Morning

F Monday 10 November 2014 Morning GCSE METHODS IN MATHEMATICS B391/01 Methods in Mathematics 1 (Foundation Tier) * 1 3 9 3 7 3 9 1 6 8 * Candidates answer on the Question Paper. OCR supplied materials:

### * * MATHEMATICS (MEI) 4757 Further Applications of Advanced Mathematics (FP3) ADVANCED GCE. Wednesday 9 June 2010 Afternoon

ADVANCED GCE MATHEMATICS (MEI) 757 Further Applications of Advanced Mathematics (FP) Candidates answer on the Answer Booklet OCR Supplied Materials: 8 page Answer Booklet MEI Examination Formulae and Tables

### * * MATHEMATICS (MEI) 4762 Mechanics 2 ADVANCED GCE. Monday 11 January 2010 Morning. Duration: 1 hour 30 minutes. Turn over

ADVANCED GCE MATHEMATICS (MEI) 4762 Mechanics 2 Candidates answer on the Answer Booklet OCR Supplied Materials: 8 page Answer Booklet Graph paper MEI Examination Formulae and Tables (MF2) Other Materials

### Monday 19 May 2014 Morning

Monday 19 May 014 Morning AS GCE MATHEMATICS (MEI) 4751/01 Introduction to Advanced Mathematics (C1) QUESTION PAPER * 3 1 8 7 0 4 0 5 4 * Candidates answer on the Printed Answer Book. OCR supplied materials:

### OXfORD CAMBRIDGE AND RSA EXAMINATIONS GCSE J567/04. MATHEMATICS B Paper 4 (Higher Tier)

OXfORD CAMBRIDGE AND RSA EXAMINATIONS GCSE J567/04 MATHEMATICS B Paper 4 (Higher Tier) friday 13 JUNE 2014: Morning DURATION: 1 hour 45 minutes plus your additional time allowance MODIfIED ENLARGED Candidate

### MATHEMATICS A A502/02 Unit B (Higher Tier)

THIS IS A NEW SPECIFICATION H GENERAL CERTIFICATE OF SECONDARY EDUCATION MATHEMATICS A A502/02 Unit B (Higher Tier) *A515450111* Candidates answer on the question paper. OCR supplied materials: None Other