Paper Reference(s) 6664/01 Edexcel GCE

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Paper Reference(s)6664/01Edexcel GCECore Mathematics C2Advanced SubsidiaryMonday 14 January 2013 MorningTime: 1 hour 30 minutesMaterials required for examinationMathematical Formulae (Pink)Items included with question papersNilCandidates may use any calculator allowed by the regulations of the Joint Councilfor Qualifications. Calculators must not have the facility for symbolic algebramanipulation, differentiation or integration, or have retrievable mathematicalformulae stored in them.Instructions to CandidatesWrite the name of the examining body (Edexcel), your centre number, candidate number, theunit title (Core Mathematics C2), the paper reference (6664), your surname, initials andsignature.Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.The marks for the parts of questions are shown in round brackets, e.g. (2).There are 9 questions in this question paper. The total mark for this paper is 75.Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You must show sufficient working to make your methods clear to the Examiner.Answers without working may not gain full credit.P41487AThis publication may only be reproduced in accordance with Edexcel Limited copyright policy. 2013 Edexcel Limited.

1.Find the first 3 terms, in ascending powers of x, in the binomial expansion of(2 5x)6.Give each term in its simplest form.(4)f(x) ax3 bx2 4x 3, where a and b are constants.2.Given that (x – 1) is a factor of f(x),(a) show that a b 7.(2)Given also that, when f(x) is divided by (x 2), the remainder is 9,(b) find the value of a and the value of b, showing each step in your working.(4)3.A company predicts a yearly profit of 120 000 in the year 2013. The company predicts that theyearly profit will rise each year by 5%. The predicted yearly profit forms a geometric sequencewith common ratio 1.05.(a) Show that the predicted profit in the year 2016 is 138 915.(1)(b) Find the first year in which the yearly predicted profit exceeds 200 000.(5)(c) Find the total predicted profit for the years 2013 to 2023 inclusive, giving your answer to thenearest pound.(3)4.Solve, for 0 x 180 ,cos (3x 10 ) 0.4,giving your answers to 1 decimal place. You should show each step in your working.(7)P41487A2

5.The circle C has equationx2 y2 20x 24y 195 0.The centre of C is at the point M.(a) Find(i) the coordinates of the point M,(ii) the radius of the circle C.(5)N is the point with coordinates (25, 32).(b) Find the length of the line MN.(2)The tangent to C at a point P on the circle passes through point N.(c) Find the length of the line NP.(2)6.Given that 2 log2 (x 15) log2 x 6,(a) show that x2 34x 225 0.(5)(b) Hence, or otherwise, solve the equation 2 log2 (x 15) log2 x 6.(2)P41487A3Turn over

7.Figure 2The triangle XYZ in Figure 1 has XY 6 cm, YZ 9 cm, ZX 4 cm and angle ZXY .The point W lies on the line XY.The circular arc ZW, in Figure 1 is a major arc of the circle with centre X and radius 4 cm.(a) Show that, to 3 significant figures, 2.22 radians.(2)(b) Find the area, in cm2, of the major sector XZWX.(3)The region enclosed by the major arc ZW of the circle and the lines WY and YZ is shown shadedin Figure 1.Calculate(c) the area of this shaded region,(3)(d) the perimeter ZWYZ of this shaded region.(4)P41487A4

8.The curve C has equation y 6 3x 4, x 0.x3(a) Use calculus to show that the curve has a turning point P when x 2.(4)(b) Find the x-coordinate of the other turning point Q on the curve.(1)(c) Findd2 y.dx 2(1)(d) Hence or otherwise, state with justification, the nature of each of these turning pointsP and Q.(3)P41487A5

9.Figure 2The finite region R, as shown in Figure 2, is bounded by the x-axis and the curve with equationy 27 2x 9 x 16,x2x 0.The curve crosses the x-axis at the points (1, 0) and (4, 0).(a) Copy and complete the table below, by giving your values of y to 3 decimal places.x11.5y05.86622.55.21033.541.8560(2)(b) Use the trapezium rule with all the values in the completed table to find an approximatevalue for the area of R, giving your answer to 2 decimal places.(4)(c) Use integration to find the exact value for the area of R.(6)TOTAL FOR PAPER: 75 MARKSENDP41487A6

EDEXCEL CORE MATHEMATICS C2 (6664) – JANUARY 2013QuestionNumber1.FINAL MARK SCHEMEScheme 2 5x 2 6Marks6Award this when first seen (not 64x0)64B1Attempt binomial expansion with correctstructure for at least one of these terms.E.g. a term of the form: 6 2 ( 5 x) 56 54 2 ( 5x)22 6 6 pp 2 ( 5 x) with p 1 or p 2 p consistently. Condone sign errors.Condone missing brackets if later workimplies correct structure and allowalternative forms for binomial coefficients6M1 6 6 or even 1 1 e.g. C1 or 960x( )6000xNot 960x2A1 (first)A1 (Second)(4)2. (a)f (1) a b 4 3 0or a b – 7 0a b 7*Attempt f( 1)M1Must be f(1) and 0 needs to be seenA1(2)(b)f ( 2) a 2 b 2 4 2 3 9Attempt f( 2) and uses f( 2) 9M1 8a 4b 8 3 9Correct equation with exponentsof (–2) removedA132(–8a 4b 4)Solves the given equation from part (a) andtheir equation in a and b from part (b) as faras a . or b .a 2 and b 5M1Both correctA1(4)[6]1

EDEXCEL CORE MATHEMATICS C2 (6664) – JANUARY 2013QuestionNumberFINAL MARK SCHEMEScheme3. (a)120000 (1.05) 138915 *(b)120000 (1.05)MarksOr 120000 1.05 1.05 1.05 138915Or 120000, 126000, 132300, 1389153 Or a 120000 and a 1.05 3 138915B1(1) 200000 5 log1.05n 1 log 3 n 1 n 1 5 3 or equivalentAllow n or n – 1 and “ ”, “ ”, or “ ” etc.M1Takes logs correctlyAllow n or n – 1 and “ ”, “ ”, or “ ” etc.M1Allow n or n – 1 and “ ”, “ ”, or “ ” etc.1.6 or awrt 1.67 for 5/3.AllowA1M1: Identifies a calendar year using theirvalue of n or n – 1M1 A1log log1.05 7 log 4 e.g n log1.052024a(1 r n ) 120000 1 1.05 1 r1 1.0511(c) 1704814(5)M1: Correct sum formula with n 10, 11 or12A1: Correct numerical expression withn 11Cao (Allow 1704814.00)M1 A1A1(3)[9]4.cos 1 0.4 113.58 ( )3x 10 x 103Awrt 114B1Uses their to find x. 10 Allow x not 1033M1x 41.2 3x 10 360 A1360 (can be implied by 246.4.)(246.4.)x 85.5M1A1360 (Can be implied by 3x 10 360 473.57. M1473.57.)x 161.2A12

EDEXCEL CORE MATHEMATICS C2 (6664) – JANUARY 2013QuestionNumberFINAL MARK SCHEMESchemeMarksB1: x 10B1: y 125. (a) (i)The centre is at (10, 12)(ii)Uses ( x 10) ( y 12) 195 100 144 r .Completes the square for both x and y in an attempt to find r.( x "10")2 a and ( y "12")2 b and 195 0, a, b 0 2B1 B12M1Allow errors in obtaining their r2 but must find square rootA correct numerical expressionfor rA1including the square root and canimplied by a correct value for rNot r 7 unless 7is rejectedA1r 102 122 195r 7(5)(b)MN (25 "10")2 (32 "12")2 Correct use of PythagorasM1 MN 625 25A1(2)(c)NP ("25"2 "7"2 ) NP (MN 2 r 2 )M1 NP 576 24A1(2)[9]6. (a)2log( x 15) log( x 15)2( x 15)2log( x 15)2 log x logxB1Correct use of log a log b logabM126 64 or log 2 64 664 used in the correct contextB1( x 15)2( x 15)2log 2 6 64xxRemoves logs correctlyM1 x 2 30 x 225 64 xMust see expansion ofscore the final mark. 1or x 30 225 x 64 x 15 2 x2 34 x 225 0 *toA1(5)(b)M1: Correct attempt to solve thegiven quadratic as far as x .A1: Both 25 and 9( x 25)( x 9) 0 x 25 or x 9M1 A1(2)[7]3

EDEXCEL CORE MATHEMATICS C2 (6664) – JANUARY 2013QuestionNumber7. (a)Scheme92 42 62 2 4 6cos cos .cos (NB 2.219516005)2 2.22( 4.06366.)12MarksCorrect use of cosine ruleleading to a value for cos M1Cso (2.22 must be seenhere)A142 62 92 29 0.604. 2 4 6 48 2.22(b)FINAL MARK SCHEME 42 "4.06"32.5(2)(2)2 2.22 or awrt 4.06Correct method for majorsector area.B1Awrt 32.5A1M1(3)(c)Area of triangle Correct expression for the area oftriangle XYZB1So area required “9.56” “32.5”Their triangle XYZ (part (b)answer or correct attempt at majorsector)M1 42.1 cm2 or 42.0 cm2Awrt 42.1 or 42.0 (Or just 42)A112 4 6 sin 2.22 9.56 (3)(d)M1: 4 their 2 2.22 Arc length 4 4.06 16.24 Or 8 4 2.22Perimeter ZY WY Arc LengthPerimeter 27.2 or 27.3Or circumference – minor arcA1: Correct ft expression9 2 Any ArcM1A1ftAwrt 27.2 or awrt 27.3A1M1(4)[12]4

EDEXCEL CORE MATHEMATICS C2 (6664) – JANUARY 2013QuestionNumber8. (a)FINAL MARK SCHEMESchemey 6 3x 4x3M1: x n x n 1dy12 3 4 or 3 12 x 4dxx1dx12 2 4or 3 12 2 4M1 A1M1Substitutes x 2 into their y So x 4 and x 2 or 40A1: Correct derivativey 0 and attempt to solve for xMay be implied bydy1212 3 4 0 4 3 x .dxxxor4 3 3( x x or x x or 6 0)dy12 0 3 4 0 x . ordxxdy12 3 4dx2dyMarksCorrect completion to answer with noerrors by solving their y 0 or 0A1substituting x 2 into their y (4)(b)x 2(c)d y 48 5 or 48 x 52dxxAwrt –1.41 B1(1)2Follow through their first derivativefrom part (a)B1ft(1)(d)An appreciation that eithery 0 a minimumor y 0 a maximumMaximum at P as y 0Minimum at Q as y 0B1B1 csoB1 cso(3)[9]5

EDEXCEL CORE MATHEMATICS C2 (6664) – JANUARY 2013QuestionNumberFINAL MARK SCHEMESchemeMarksy 27 2 x 9 x 9. (a)16x26.272 , 3.634B1, B1(2)(b)1 11 or2 24B1. (0 0) 2 5.866 "6.272" 5.210 "3.634" 1.856 Need {} orimplied later forA1ftM1A1ft1 0.5 (0 0) 2 5.866 "6.272" 5.210 "3.634" 1.856 21 45.6764 11.42A1 cao(4)M1: x n x n 1 on any termA1: 27x x 2(c) y dx 27 x x 2 6 x 2 16 x 1 c 327 4 4 6 4 2 16 4 32 27 1 1 6 1 2 16 1 23 1 13A1: 6x 2A1: 16x 1 M1A1A1A1Attempt to subtract either wayround using the limits 4 and 1. dM1Dependent on the previous M1 (48 – 36)12A1 cao(6)[12]6

2013 Edexcel Limited. Paper Reference(s) 6664/01 Edexcel GCE Core Mathematics C2 Advanced Subsidiary Monday 14 January 2013 Morning Time: 1 hour 30 minutes Materials required for examination Items included with question papers Mathematical Formulae (Pink) Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the .

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