Pearson Edexcel Level 3 GCE Mathematics

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Pearson Edexcel Level 3GCE MathematicsAdvanced LevelPaper 1 or 2: Pure MathematicsPractice Paper CTime: 2 hoursPaper Reference(s)9MA0/01 or 9MA0/02You must have:Mathematical Formulae and Statistical Tables, calculatorCandidates may use any calculator permitted by Pearson regulations. Calculators must nothave the facility for algebraic manipulation, differentiation and integration, or haveretrievable mathematical formulae stored in them.Instructions Use black ink or ball‐point pen. If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Answer all questions and ensure that your answers to parts of questions are clearlylabelled. Answer the questions in the spaces provided – there may be more space than you need. You should show sufficient working to make your methods clear. Answers withoutworking may not gain full credit. Inexact answers should be given to three significant figures unless otherwise stated.Information A booklet ‘Mathematical Formulae and Statistical Tables’ is provided. There are xx questions in this paper. The total mark is 100. The marks for each question are shown in brackets – use this as a guide as to how muchtime to spend on each question.Advice Read each question carefully before you start to answer it. Try to answer every question. Check your answers if you have time at the end. If you change your mind about an answer, cross it out and put your new answer and anyworking underneath.1

Answer ALL questions.18 x 2 98 x 78ABC , x 4223x 1( x 4) (3x 1) x 4 ( x 4)1.Find the values of the constants A, B and C.(6 marks)2.A curve C has equation 4 x 2 xy for x 0Find the exact value ofdyat the point C with coordinates (2, 4).dx(5 marks)3.(a) Show that cos7 x cos3x 2cos5 x cos 2 x by expanding cos 5 x 2 x and cos 5 x 2 x using thecompound-angle formulae.(3 marks)(b) Hence find cos 5 x cos 2 x dx .(3 marks)4.The temperature of a mug of coffee at time t can be modelled by the equationT t TR 90 TR e 1t20,where T(t ) is the temperature, in C, of the coffee at time t minutes after the coffee was poured into themug and TR is the room temperature in C.Using the equation for this model,(a) explain why the initial temperature of the coffee is independent of the initial room temperature.(2 marks)(b) Calculate the temperature of the coffee after 10 minutes if the room temperature is 20 C.(2 marks)5.Prove by contradiction that if n is odd, n3 1 is even.(5 marks)2

6.A curve C has parametric equations x sec 2 t 1 , y 2sin t , 4„ t„ 4.8 4xfor a suitable domain which should be stated.1 x(4 marks)Show that a cartesian equation of C is y 7.An infinite geometric series has first four terms 1 4 x 16 x 2 64 x3 . The series is convergent.(a) Find the set of possible values of x for which the series converges.(2 marks) Given that 4 x r 1 4,r 1(b) calculate the value of x.(3 marks)8.f (x) 2 3sin3 x cos x , where x is in radians.(a) Show that f(x) 0 has a root α between x 1.9 and x 2.0.(2 marks)Using x0 1.95 as a first approximation,(b) apply the Newton–Raphson procedure once to f(x) to find a second approximation to α, giving youranswer to 3 decimal places.(5 marks)9.Given that b a i 2abcj 2k 10i 96 j 7a 5b k , find the values of a, b and c.(6 marks)10.Use proof by contradiction to show that there are no positive integer solutions to the statement x 2 y 2 1.(5 marks)11.The function g(x) is defined by g( x) x 2 8 x 7 , x , x 4.Find g 1(x) and state its domain and range.(6 marks)3

f ( x) 12.4 x 2 x 23, x 4.( x 3)(4 x)( x 5)ABC, find the values of A, B and C. x 3 4 x x 5(6 marks)Given that f (x) can be expressed in the form13The curve C has equation y x3 6 x 2 12 x 6 .(a) Show that C is concave on the interval [–5, –3].(3 marks)(b) Find the coordinates of the point of inflection.(3 marks)14Findπ8sin 4 xπ12 1 cos 4 x 3 dx .(4 marks)BC4x 2 4x 9. A 2x 1 x 1(2x 1)(x 1)15(a) Find the values of the constants A, B and C.(6 marks)2(b) Hence, or otherwise, expand4x 4x 9in ascending powers of x, as far as the x2 term.(2x 1)(x 1)(6 marks)(c) Explain why the expansion is not valid for x 3.4(1 mark)16A large cylindrical tank has radius 40 m. Water flows into the cylinder from a pipe at a rate of4000π m3 min 1. At time t, the depth of water in the tank is h m. Water leaves the bottom of the tankthrough another pipe at a rate of 50πh m3 min 1.(a) Show that t minutes after water begins to flow out of the bottom of the cylinder, 160dh 400 5h .dt(6 marks)When t 0 min, h 50 m.(b) Find the exact value of t when h 60 m.(6 marks)TOTAL FOR PAPER IS 97 MARKS4

A level Pure Maths: Practice Paper C mark scheme1SchemeStates that:MarksAOsPearsonProgression Stepand ProgressdescriptorM12.2a7thA(x 4)(3x 1) B(3x 1) C(x 4)(x 4) 18x 2 98x 78M11.1bM12.2aMakes an attempt to manipulate the expressions in order to findA, B and C. Obtaining two different equations in the same twovariables would constitute an attempt.M11.1bFinds the correct value of any one variable:A11.1bA11.1bFurther states that:A(3x 2 11x 4) B(3x 1) C(x 2 8x 16) 18x 2 98x 78Equates the various terms.Decomposealgebraicfractions intopartial fractions repeated factors.Equating the coefficients of x2: 3A C 18Equating the coefficients of x: 11A 3B 8C 98Equating constant terms: 4 A B 16C 78either A 4, B 2 or C 6Finds the correct value of all three variables:A 4, B 2, C 6(6 marks)NotesAlternative methodUses the substitution method, having first obtained this equation:A(x 4)(3x 1) B(3x 1) C(x 4)(x 4) 18x 2 98x 78Substitutes x 4 to obtain 13B 26Substitutes x 11693381014to obtainC C 6931693Equates the coefficients of x2: 3A C 18Substitutes the found value of C to obtain 3A 12 Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free1

A level Pure Maths: Practice Paper C mark schemeMarksAOsPearsonProgression Stepand ProgressdescriptorDifferentiates 4x to obtain 4x ln 4M11.1b7thdyDifferentiates 2xy to obtain 2 x 2 ydxM12.2aA11.1bMakes an attempt to substitute (2, 4)M11.1bStates fully correct final answer: 4ln 4 2A11.1b2SchemeRearranges 4 x ln 4 2 xdydy 4 x ln 4 2 y 2 y to obtain dxdx2xDifferentiatesimple functionsdefined implicitly.Accept ln 256 2(5 marks)Notes Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free2

A level Pure Maths: Practice Paper C mark schemeMarksAOsPearsonProgression Stepand ProgressdescriptorCorrectly states cos 5 x 2 x cos 5 x cos 2 x sin 5 x sin 2 xM11.1b6thCorrectly statesM11.1bIntegrate usingtrigonometricidentities.A11.1b3(a)Schemecos 5 x 2 x cos 5 x cos 2 x sin 5 x sin 2 x or statescos 5 x 2 x cos 5 x cos 2 x sin 5 x sin 2 x Adds the two above expressions and statescos7x cos3x 2cos5x cos2x(3)(b)States that1 cos5x cos 2x dx 2 cos7 x cos3x dxMakes an attempt to integrate. Changing cos to sin constitutesan attempt.Correctly states the final answer11sin 7 x sin 3x C o.e.1462.2a6thM11.1bIntegratefunctions of theform f(ax b).A11.1bM1(3)(6 marks)Notes(b) Student does not need to state ‘ C’ to be awarded the first method mark. Must be stated in the final answer. Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free3

A level Pure Maths: Practice Paper C mark scheme4(a)SchemeMakes an attempt to substitute t 0 intoT t TR 90 TR 1 te 20MarksAOsPearsonProgression Stepand ProgressdescriptorM13.1a6th. For example,T t TR 90 TR e 0 or T t TR 90 TR is seen.Concludes that the TR terms will always cancel at t 0,therefore the room temperature does not influence the initialcoffee temperature.B13.5aSet up and useexponentialmodels of growthand decay.(2)(b)Makes an attempt to substitute TR 20 and t 10 intoT t TR 90 TR 1 t20eT 10 20 90 20 e M11.1bSet up and useexponentialmodels of growthand decay. For example,1 10 206this seen.Finds T 10 62.457. C . Accept awrt 62.5 .A11.1b(2)(4 marks)Notes Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free4

A level Pure Maths: Practice Paper C mark scheme5SchemeBegins the proof by assuming the opposite is true.MarksAOsPearsonProgression Stepand ProgressdescriptorB13.17thComplete proofsusing proof bycontradiction.‘Assumption: there exists a number n such that n is odd andn3 1 is also odd.’Defines an odd number.B12.2aM11.1bM11.1bB12.4‘Let 2k 1 be an odd number.’Successfully calculates (2k 1)3 1 (2k 1)3 1 8k 3 12k 2 6k 1 1 8k 3 12k 2 6k 2Factors the expression and concludes that this number must beeven. 8k 3 12k 2 6k 2 2 4k 3 6k 2 3k 1 2 4k 3 6k 2 3k 1 is even.Makes a valid conclusion.This contradicts the assumption that there exists a number nsuch that n is odd and n3 1 is also odd, so if n is odd, thenn3 1 is even.(5 marks)NotesAlternative methodAssume the opposite is true: there exists a number n such that n is odd and n3 1 is also odd. (B1)If n3 1 is odd, then n3 is even. (B1)So 2 is a factor of n3. (M1)This implies 2 is a factor of n. (M1)This contradicts the statement n is odd. (B1) Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free5

A level Pure Maths: Practice Paper C mark schemeSchemeMarksAOsPearsonProgression Stepand ProgressdescriptorRecognises that the identity sin 2 t cos 2 t 1 can be used to findthe cartesian equation.M12.2a6th6States sin t yy2or sin 2 t 24Also states cos 2 t 1.1bM11.1bA11.1b1x 11y2and cos 2 t into sin 2 t cos 2 t 1x 14y21y2 x 2 1 4 x 14x 1Substitutes sin 2 t Solves to find y M14x 88 4x, x 1 or x 2, accept y x 11 xConvert betweenparametricequations andcartesian formsusingtrigonometry.(4 marks)Notes Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free6

A level Pure Maths: Practice Paper C mark scheme7SchemeMarksAOsPearsonProgression Stepand Progressdescriptor(a)Understands that for the series to be convergent r 1 or statesM12.2a6th 4 x 1Correctly concludes that x 111. Accept x 444A11.1bUnderstandconvergentgeometric seriesand the sum toinfinity.(2)(b)Understands to use the sum to infinity formula. For example,1 4states1 4xMakes an attempt to solve for x. For example, 4 x States x 3163is seen.4M12.2a5thUnderstand sigmanotation.M11.1bA11.1b(3)(5 marks)Notes Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free7

A level Pure Maths: Practice Paper C mark schemeMarksAOsPearsonProgression Stepand ProgressdescriptorFinds f (1.9) 0.2188. and f (2.0) ( )0.1606.M11.1b5thChange of sign and continuous function in the interval 1.9, 2.0 rootA12.4Use a change ofsign to locateroots.6th8(a)Scheme(2)(b)Makes an attempt to differentiate f(x)M12.2aCorrectly finds f (x) 9sin 2 x cos x sin xA11.1bFinds f (1.95) 0.0348. and f (1.95) 3.8040.M11.1bAttempts to find x1M11.1bA11.1bx1 x0 Solve equationsapproximatelyusing the NewtonRaphson method.f (x0 ) 0.0348. x1 1.95 3.8040.f (x0 )Finds x1 1.959(5)(7 marks)Notes(a) Minimum required is that answer states there is a sign change in the interval and that this implies a root inthe given interval. Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free8

A level Pure Maths: Practice Paper C mark schemeMarksAOsPearsonProgression Stepand ProgressdescriptorStates a b 10 and 7 a 5b 2M12.2a6thMakes an attempt to solve the pair of simultaneous equations.Attempt could include making a substitution or multiplying thefirst equation by 5 or by 7.M11.1bFinds a 4A11.1bFind b 6A11.1bStates 2abc 96M12.2aFinds c 2A11.1b9SchemeSolve geometricproblems usingvectors in 3dimensions(6 marks)Notes Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free9

A level Pure Maths: Practice Paper C mark scheme10SchemeBegins the proof by assuming the opposite is true.MarksAOsPearsonProgression Stepand ProgressdescriptorB13.17thComplete proofsusing proof bycontradiction.‘Assumption: there exist positive integer solutions to thestatement x 2 y 2 1’Sets up the proof by factorising x 2 y 2 and stating(x y)(x y) 1M12.2aStates that there is only one way to multiply to make 1:M11.1bSolves this pair of simultaneous equations to find the values ofx and y: x 1 and y 0M11.1bMakes a valid conclusion.B12.41 1 1and concludes this means that:x y 1x y 1x 1, y 0 are not both positive integers, which is acontradiction to the opening statement. Therefore there do notexist positive integers x and y such that x 2 y 2 1(5 marks)Notes Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free10

A level Pure Maths: Practice Paper C mark scheme11MarksAOsPearsonProgression Stepand ProgressdescriptorM12.2a6thA11.1bFind the domainand range ofinverse functions.B12.2aMakes an attempt to rearrange to make y the subject. Attemptmust include taking the square root.M11.1bCorrectly states g 1 ( x) x 9 4A11.1bCorrectly states domain is x 9 and range is y 4B13.2bSchemeUnderstands the need to complete the square, and makes anattempt to do this. For example, x 4 is seen.2Correctly writes g( x) x 4 92Demonstrates an understanding of the method for finding theinverse is to switch the x and y. For example, x y 4

Pearson Edexcel Level 3 GCE Mathematics Advanced Level Paper 1 or 2: Pure Mathematics Practice Paper C Time: 2 hours Paper Reference(s) 9MA0/01 or 9MA0/02 You must have: Mathematical Formulae and Statistical Tables, calculator Candidates may use any calculator permitted by Pearson regulations.