Centre Number Candidate Number Level 3 GCE Mathematics
marksphysicshelp MPHWrite your name hereSurnameOther namesPearson EdexcelLevel 3 GCECentre NumberCandidate NumberMathematicsAdvancedPaper 1: Pure Mathematics 1Sample Assessment Material for first teaching September 2017Time: 2 hoursPaper Reference9MA0/01You must have:Mathematical Formulae and Statistical Tables, calculatorTotal MarksCandidates may use any calculator permitted by Pearson regulations.Calculators must not have the facility for algebraic manipulation,differentiation and integration, or have retrievable mathematicalformulae stored in them.Instructionsblack ink or ball-point pen. UseIf pencil is used for diagrams/sketches/graphs it must be dark (HB or B).Fill in the boxes at the top of this page with your name, centrenumber and candidate number.Answer all questions and ensure that your answers to parts of questions are clearlylabelled.Answerthe questions in the spaces provided – there maybe more space than you need.should show sufficient working to make your methods clear. Answers Youwithout working may not gain full credit. Answers should be given to three significant figures unless otherwise stated.Informationbooklet ‘Mathematical Formulae and Statistical Tables’ is provided. AThereare 15 questions in this question paper. The total mark for this paper is 100.Themarkseach question are shown in brackets – use this asfora guideas to how much time to spend on each question.Adviceeach question carefully before you start to answer it. ReadTry to answer every question.your answers if you have time at the end. CheckIf you change your mind about an answer cross it out and put your newanswer and any working out underneath.S54259A 2017 Pearson Education Ltd.1/1/1/1/1/1/1/Turn over*S54259A0132*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 20175
MPHAnswer ALL questions. Write your answers in the spaces provided.y 3x4 – 8x3 – 3(a) Find (i)(ii)dydxd2 ydx 2(3)(b) Verify that C has a stationary point when x 2(2)DO NOT WRITE IN THIS AREA1. The curve C has equation(c) Determine the nature of this stationary point, giving a reason for your answer.(2)DO NOT WRITE IN THIS AREA26*S54259A0232*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017DO NOT WRITE IN THIS AREA
MPHDO NOT WRITE IN THIS AREAQuestion 1 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 1 is 7 marks)*S54259A0332*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 201737Turn over
MPH2.CBDO NOT WRITE IN THIS AREA3cmOA0.4O12cmDFigure 1The shape ABCDOA, as shown in Figure 1, consists of a sector COD of a circle centre Ojoined to a sector AOB of a different circle, also centre O.Given that arc length CD 3 cm, COD 0.4 radians and AOD is a straight line oflength 12 cm,(a) find the length of OD,(2)(b) find the area of the shaded sector AOB.DO NOT WRITE IN THIS AREA(3)(Total for Question 2 is 5 marks)48*S54259A0432*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017DO NOT WRITE IN THIS AREA
MPHDO NOT WRITE IN THIS AREA3. A circle C has equationx2 y2 – 4x 10y kwhere k is a constant.(a) Find the coordinates of the centre of C.(2)(b) State the range of possible values for k.(2)DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 3 is 4 marks)*S54259A0532*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 201759Turn over
MPH4. Given that a is a positive constant andt 1dt ln 7tashow that a lnk, where k is a constant to be found.(4)DO NOT WRITE IN THIS AREA 2aDO NOT WRITE IN THIS AREA(Total for Question 4 is 4 marks)610*S54259A0632*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017DO NOT WRITE IN THIS AREA
MPHDO NOT WRITE IN THIS AREA5. A curve C has parametric equationsx 2t – 1,y 4t –7 3,tt 0Show that the Cartesian equation of the curve C can be written in the formy 2 x 2 ax b,x 1x –1where a and b are integers to be found.(3)DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 5 is 3 marks)*S54259A0732*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017711Turn over
MPH6. A company plans to extract oil from an oil field.DO NOT WRITE IN THIS AREAThe daily volume of oil V, measured in barrels that the company will extract from thisoil field depends upon the time, t years, after the start of drilling.The company decides to use a model to estimate the daily volume of oil that will beextracted. The model includes the following assumptions: The initial daily volume of oil extracted from the oil field will be 16 000 barrels.The daily volume of oil that will be extracted exactly 4 years after the start of drillingwill be 9000 barrels.The daily volume of oil extracted will decrease over time.The diagram below shows the graphs of two possible models.VV(0, 16 000)(0, 16 000)O(4, 9000)tOModel AtModel B(a) (i) Use model A to estimate the daily volume of oil that will be extracted exactly3 years after the start of drilling.DO NOT WRITE IN THIS AREA(4, 9000)(ii) Write down a limitation of using model A.(2)(b) (i) Using an exponential model and the information given in the question, find apossible equation for model B.(5)812*S54259A0832*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017DO NOT WRITE IN THIS AREA(ii) Using your answer to (b)(i) estimate the daily volume of oil that will be extractedexactly 3 years after the start of drilling.
MPHDO NOT WRITE IN THIS AREAQuestion 6 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 6 is 7 marks)*S54259A0932*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017913Turn over
MPH7.BDO NOT WRITE IN THIS AREAACFigure 2Figure 2 shows a sketch of a triangle ABC. Given AB 2i 3j k and BC i – 9j 3k,BAC 105.9 to one decimal place.(5)DO NOT WRITE IN THIS AREAshow that1014*S54259A01032*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017DO NOT WRITE IN THIS AREA
MPHDO NOT WRITE IN THIS AREAQuestion 7 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 7 is 5 marks)*S54259A01132*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 20171115Turn over
MPH8.f ( x ) ln(2 x 5) 2 30,x 2.5in the interval [3.5, 4](2)A student takes 4 as the first approximation to .Given f (4) 3.099 and fʹ(4) 16.67 to 4 significant figures,(b) apply the Newton-Raphson procedure once to obtain a second approximation for ,giving your answer to 3 significant figures.(2)(c) Show thatis the only root of f(x) 0(2)DO NOT WRITE IN THIS AREA(a) Show that f ( x ) 0 has a root2DO NOT WRITE IN THIS AREA1216*S54259A01232*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017DO NOT WRITE IN THIS AREA
MPHDO NOT WRITE IN THIS AREAQuestion 8 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 8 is 6 marks)*S54259A01332*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 20171317Turn over
MPH9. (a) Prove thatθ nπ,n Z2(4)(b) Hence explain why the equationtan θ cot θ 1does not have any real solutions.(1)DO NOT WRITE IN THIS AREAtan θ cot θ 2cosec2θ ,DO NOT WRITE IN THIS AREA(Total for Question 9 is 5 marks)1418*S54259A01432*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017DO NOT WRITE IN THIS AREA
DO NOT WRITE IN THIS AREAMPH10. Given that is measured in radians, prove, from first principles, that the derivativeof sin is cossin hcos h1 andYou may assume the formula for sin(A B) and that as hhh1(5)DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 10 is 5 marks)*S54259A01532*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 20171519Turn over
MPH11. An archer shoots an arrow.H 1.8 0.4d – 0.002d 2,d0where d is the horizontal distance of the arrow from the archer, measured in metres.Given that the arrow travels in a vertical plane until it hits the ground,(a) find the horizontal distance travelled by the arrow, as given by this model.(3)(b) With reference to the model, interpret the significance of the constant 1.8 in the formula.(1)DO NOT WRITE IN THIS AREAThe height, H metres, of the arrow above the ground is modelled by the formula(c) Write 1.8 0.4d – 0.002d 2 in the formA – B(d – C)2where A, B and C are constants to be found.(3)The adapted formula for this archer isH 2.1 0.4d – 0.002d 2,d0Hence or otherwise, find, for the adapted model(d) (i) the maximum height of the arrow above the ground.(ii) the horizontal distance, from the archer, of the arrow when it is at its maximum height.(2)DO NOT WRITE IN THIS AREAIt is decided that the model should be adapted for a different archer.1620*S54259A01632*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017DO NOT WRITE IN THIS AREA
MPHDO NOT WRITE IN THIS AREAQuestion 11 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 11 is 9 marks)*S54259A01732*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 20171721Turn over
MPH12. In a controlled experiment, the number of microbes, N, present in a culture T days afterthe start of the experiment were counted.N aT b,where a and b are constants(a) Show that this relationship can be expressed in the formlog10 N m log10 T cgiving m and c in terms of the constants a and/or b.(2)log10 N5.0 –DO NOT WRITE IN THIS AREAN and T are expected to satisfy a relationship of the form4.5 –4.0 –DO NOT WRITE IN THIS AREA3.5 –3.0 –2.5 –2.0 –1.5 –1.0 0.5 –1.4 log10 TFigure 3 shows the line of best fit for values of log10 N plotted against values of log10 T(b) Use the information provided to estimate the number of microbes present in theculture 3 days after the start of the experiment.(4)(c) Explain why the information provided could not reliably be used to estimate the daywhen the number of microbes in the culture first exceeds 1 000 000.(2)(d) With reference to the model, interpret the value of the constant a.(1)1822*S54259A01832*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017DO NOT WRITE IN THIS AREAFigure 3
MPHDO NOT WRITE IN THIS AREAQuestion 12 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA*S54259A01932*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 20171923Turn over
MPHQuestion 12 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA2024*S54259A02032*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017DO NOT WRITE IN THIS AREA
MPHDO NOT WRITE IN THIS AREAQuestion 12 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 12 is 9 marks)*S54259A02132*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 20172125Turn over
MPH13. The curve C has parametric equations(a) Find an expression fory 3 cos 2 t,0tdyin terms of t.dxThe point P lies on C where t (2)2π3The line l is the normal to C at P.(b) Show that an equation for l isDO NOT WRITE IN THIS AREAx 2 cos t,2x 2 3y 1 0(5)The line l intersects the curve C again at the point Q.(c) Find the exact coordinates of Q.(6)DO NOT WRITE IN THIS AREAYou must show clearly how you obtained your answers.2226*S54259A02232*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017DO NOT WRITE IN THIS AREA
MPHDO NOT WRITE IN THIS AREAQuestion 13 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA*S54259A02332*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 20172327Turn over
MPHQuestion 13 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA2428*S54259A02432*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017DO NOT WRITE IN THIS AREA
MPHDO NOT WRITE IN THIS AREAQuestion 13 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 13 is 13 marks)*S54259A02532*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 20172529Turn over
MPHy14.DO NOT WRITE IN THIS AREACSO1x3Figure 4Figure 4 shows a sketch of part of the curve C with equationy x 2 ln x 2 x 5,3x 0The table below shows corresponding values of x and y with the values of ygiven to 4 decimal places as ) Use the trapezium rule, with all the values of y in the table, to obtain an estimate forthe area of S, giving your answer to 3 decimal places.(3)DO NOT WRITE IN THIS AREAThe finite region S, shown shaded in Figure 4, is bounded by the curve C, the line withequation x 1, the x-axis and the line with equation x 3(b) Explain how the trapezium rule could be used to obtain a more accurate estimate forthe area of S.(1)(6)(In part c, solutions based entirely on graphical or numerical methods are not acceptable.)2630*S54259A02632*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017DO NOT WRITE IN THIS AREAa ln c, where a, b and c(c) Show that the exact area of S can be written in the formbare integers to be found.
MPHDO NOT WRITE IN THIS AREAQuestion 14 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA*S54259A02732*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 20172731Turn over
MPHQuestion 14 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA2832*S54259A02832*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017DO NOT WRITE IN THIS AREA
MPHDO NOT WRITE IN THIS AREAQuestion 14 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 14 is 10 marks)*S54259A02932*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 20172933Turn over
MPH15.yPDO NOT WRITE IN THIS AREAOxQFigure 5Figure 5 shows a sketch of the curve with equation y f ( x ), wheref (x) 4 sin 2 x,e 2 x 10x(a) Show that the x coordinates of point P and point Q are solutions of the equationtan 2 x 2(4)(b) Using your answer to part (a), find the x-coordinate of the minimum turning point onthe curve with equation(i) y f ( 2x ).(ii) y 3 2f ( x ).DO NOT WRITE IN THIS AREAThe curve has a maximum turning point at P and a minimum turning point at Q as shownin Figure 5.(4)3034*S54259A03032*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017DO NOT WRITE IN THIS AREA
MPHDO NOT WRITE IN THIS AREAQuestion 15 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA*S54259A03132*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 20173135Turn over
MPHQuestion 15 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 15 is 8 marks)TOTAL FOR PAPER IS 100 MARKS3236*S54259A03232*Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017DO NOT WRITE IN THIS AREA
MPHPaper 1: Pure Mathematics 1 Mark SchemeQuestion1(a)Scheme(i)dydx(ii)d2 ydx 2MarksAOs12 x3 24 x 2M1A11.1b1.1b36 x 2 48 xA1ft1.1b(3)(b)Substitutes x 2 into theirShowsdydxdydx12 23 24 220 and states ''hence there is a stationary point''M11.1bA12.1(2)(c)d2 ySubstitutes x 2 into their 2dxd2 ydx 236 22 48 248 0 and states ''hence the stationary point is a minimum''M11.1bA1ft2.2a(2)(7 marks)Notes:(a)(i)M1: Differentiates to a cubic formdyA1:12 x3 24 x 2dx(a)(ii)A1ft: Achieves a correctd2 ydyfor their2dxdx36 x 2 48 x(b)M1:A1:Substitutes x 2 into theirdydxdy 0 and states ''hence there is a stationary point'' All aspects of the proofdxmust be correctShows(c)d2 ydx 2Alternatively calculates the gradient of C either side of x 2A1ft: For a correct calculation, a valid reason and a correct conclusion.M1:Substitutes x 2 into theird2 yFollow through on an incorrect 2dxPearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 201737
MPHQuestion2(a)SchemeUses sr3 r 0.4OD 7.5 cmMarksAOsM11.2A11.1b(2)(b)Uses angle AOBUses area of sector0.4 or uses radius is (12 – ‘7.5’) cm1 2r21(12 7.5)2 (2 27.8cm20.4)M13.1aM11.1bA1ft1.1b(3)(5 marks)Notes:(a)M1:A1:0.4Attempts to use the correct formula s r with s 3 andOD 7.5 cm (An answer of 7.5cm implies the use of a correct formula and scores bothmarks)(b)M1:AOB0.4 may be implied by the use of AOB awrt 2.74 or uses radius is(12 – their ‘7.5’)M1: Follow through on their radius (12 – their OD) and their angleA1ft: Allow awrt 27.8 cm2. (Answer 27.75862562). Follow through on their (12 – their ‘7.5’)Note: Do not follow through on a radius that is negative.38Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 2017
MPHQuestion3(a)SchemeAttempts x 22y 52.Centre (2, 5)MarksAOsM11.1bA11.1b(2)(b)Sets k 22 520k29M12.2aA1ft1.1b(2)(4 marks)Notes:(a)22M1:Attempts to complete the square so allow x 2A1:States the centre is at (2, 5). Also allow written separately xy 5.2, y5(2, 5) implies both marks(b)M1:Deduces that the right hand side of their x .A1ft: k29 Also allow k2y .2. is 0 or29 Follow through on their rhs of x .Question20y .Scheme4Writest 1dtt11dt and attempts to integratett ln tc2a ln 2aa ln aalnln 77with k2722.MarksAOsM12.1M11.1bM11.1bA11.1b(4 marks)Notes:M1:Attempts to divide each term by t or alternatively multiply each term by t -1M1:Integrates each term and knows1dt ln t. The c is not required for this markM1:tSubstitutes in both limits, subtracts and sets equal to ln7A1:Proceeds to aln7and states k272or exact equivalent such as 3.5Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –Issue 1 – April 2017 Pearson Education Limited 201739
MPHQuestion5SchemeAttempts to substitutex 1into y2y4x 1276( x 1)Attempts to write as a single fraction(2 x 5)( x 1) 6y( x 1)y2 x 2 3x 1x 1a3, b 1MarksAOsM12.1M12.1A11.1b(3 marks)Notes:M1:M1:3x 1or equivalent into y 4t 7 t2Award this for an attempt at a single fraction with a correc
Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials – . .
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Centre Number Candidate Number Write your name here Surname Other names Total Marks Paper Reference Turn over Sample assessment material for first teaching September 2017 Time: 1 hour 15 minutes Instructions Use black ink or ball-point pen. Fill in the boxes at the top of this page with your name, centre number and candidate number.
Edexcel GCSE You do not need any other materials. Centre Number Candidate Number Write your name here Surname Other names Total Marks Sample Assessment Material Paper Reference Time: 1 hour Turn over Instructions tt Use black ink or ball-point pen. Fill in the boxes at the top of this page with your name, centre number and candidate number.t t .
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5PH1H/01 Additional Sample Assessment Material Time: 1 hour Edexcel GCSE Centre Number Candidate Number Surname Other names Write your name here Physics/Science Unit P1: Universal Physics Higher Tier Instructions Use black ink or ball-point pen. Fill in the boxes at the top of this page with your name, centre number and candidate number.
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