# Mathematics - Mick Macve

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can be written as(I)5θ 2 – 15θ 1 » 0The solutions of the equation(3)5θ 2 – 15θ 1 0are 0.068 and 2.932, correct to 3 decimal places.(b) Comment on the validity of each of these values as approximate solutions toequation (I).DO NOT WRITE IN THIS AREA11 1 cos θ sin θ 2 tan θ 2 10DO NOT WRITE IN THIS AREA1. (a) Given that θ is small and in radians, show that the equation(1)DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA2*S63449A0248*

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 1 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 1 is 4 marks)*S63449A0348*3Turn over

y 5 43tt 0Show that the Cartesian equation of the curve can be expressed in the formy ax bx 1x kwhere a, b and k are constants to be found.(3)DO NOT WRITE IN THIS AREAx 6t 1DO NOT WRITE IN THIS AREA2. A curve has parametric equationsDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA4*S63449A0448*

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 2 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 2 is 3 marks)*S63449A0548*5Turn over

8( x 5)where k is a constant.Given that the curve has a stationary point P, where x 3(a) show that k – 8(4)(b) Determine the nature of the stationary point P, giving a reason for your answer.(c) Show that the curve has a point of inflection where x 7DO NOT WRITE IN THIS AREAy x 2 kx 14 DO NOT WRITE IN THIS AREA3. A curve has equation(2)(2)(1)DO NOT WRITE IN THIS AREA(d) Explain the error in Jane’s reasoning.DO NOT WRITE IN THIS AREAAs there is a change of sign, the curve cuts the x-axis in the interval (4.5, 5.5)DO NOT WRITE IN THIS AREAJane uses this information to write down the followingDO NOT WRITE IN THIS AREAThe curve passes through the points (4.5, 14.25) and (5.5, –15.75)6*S63449A0648*

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 3 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA*S63449A0748*7Turn over

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 3 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA8*S63449A0848*

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4.DO NOT WRITE IN THIS AREAQPROxDO NOT WRITE IN THIS AREAyy f(x)Figure 1(a) Write f(x) as a product of two algebraic factors.(2)(b) Find, giving your answer in simplest form,(i) the exact x coordinate of P,(ii) the exact x coordinate of R.(2)(c) Deduce the number of real solutions, for –π θ 12π, to the equationsin3 θ – 6 sin2 θ 7 sin θ 2 0justifying your answer.(2)10DO NOT WRITE IN THIS AREAThe coordinates of Q are (2, 0)DO NOT WRITE IN THIS AREAThe curve cuts the x-axis at the points P, Q and R, as shown in Figure 1.DO NOT WRITE IN THIS AREAf(x) x3 – 6x2 7x 2    x DO NOT WRITE IN THIS AREAFigure 1 shows a sketch of part of the curve with equation y f(x), where*S63449A01048*

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 4 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 4 is 6 marks)*S63449A01148*11Turn over

yDO NOT WRITE IN THIS AREADiagram notdrawn to scaleRODO NOT WRITE IN THIS AREA5.xThe finite region R, shown shaded in Figure 2, is bounded by the graph with equationy f(x) and the x-axis.(a) Find the area of R, giving your answer in simplest form.The equation(4)DO NOT WRITE IN THIS AREAf(x) 7 – ½3x – 5½     x DO NOT WRITE IN THIS AREAFigure 2 shows part of a graph with equation y f(x), whereDO NOT WRITE IN THIS AREAFigure 2DO NOT WRITE IN THIS AREAy f(x)7 – ½3x – 5½ kwhere k is a constant, has two distinct real solutions.(b) Write down the range of possible values for k.(1)12*S63449A01248*

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 5 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 5 is 5 marks)*S63449A01348*13Turn over

1125 2 Ax x1632where A is a constant.(a) Find the value of A, giving your answer in simplest form.(5)(b) Determine, giving a reason for your answer, whether the binomial expansion for f(x)is valid when x 110DO NOT WRITE IN THIS AREAThe binomial expansion of f(x), in ascending powers of x, up to and including the termin x2, isDO NOT WRITE IN THIS AREAf(x) (2 kx)– 4    where k is a positive constant6.(1)DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA14*S63449A01448*

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 6 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 6 is 6 marks)*S63449A01548*15Turn over

7.1.5OBDO NOT WRITE IN THIS AREAAxDO NOT WRITE IN THIS AREAy2 ex 2x2xx , x 0The curve cuts the x-axis at the point A, where x α, and at the point B, where x β, asshown in Figure 3.(a) Show that α lies between –1.5 and –1DO NOT WRITE IN THIS AREAf ( x) DO NOT WRITE IN THIS AREAFigure 3 shows a plot of part of the curve with equation y f(x), whereDO NOT WRITE IN THIS AREAFigure 3DO NOT WRITE IN THIS AREAy f(x)(2)(b) The iterative formula 11 xn 1 e xn xn 2n with x1 –1 can be used to estimate the value of α.(i) Find the value of x3 to 4 decimal places.(ii) Find the value of α correct to 2 decimal places.16*S63449A01648*(2)

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAThe value of β lies in the interval [1.5, 3]A student takes 3 as her first approximation to β.Given f(3) –1.4189 and fʹ(3) – 8.3078 to 4 decimal places,(c) apply the Newton-Raphson method once to f(x) to obtain a second approximation to β.Give your answer to 2 decimal places.(2)A different student takes a starting value of 1.5 as his first approximation to β.(d) Use Figure 3 to explain whether or not the Newton-Raphson method with this startingvalue gives a good second approximation to β.(2)[If you need to rework your answer to part (d) turn over for a spare copy of Figure 3]DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA*S63449A01748*17Turn over

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 7 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA18*S63449A01848*

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 7 continuedOnly use this spare copy of Figure 3 if you have to rework your answer to part (d).yDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAAO1.5Bxy f(x)Spare copy of Figure 3DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 7 is 8 marks)*S63449A01948*19Turn over

8.DO NOT WRITE IN THIS AREAOxDO NOT WRITE IN THIS AREAyFigure 4The line l cuts the circle at points P and Q.Given that the distance PQ is 8(b) find the two possible equations for l.(4)DO NOT WRITE IN THIS AREAA line l is parallel to the x-axis.DO NOT WRITE IN THIS AREA(3)DO NOT WRITE IN THIS AREA(a) Write down an equation for the circle.DO NOT WRITE IN THIS AREAA circle with centre (9, –6) touches the x-axis as shown in Figure 4.20*S63449A02048*

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 8 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 8 is 7 marks)*S63449A02148*21Turn over

where a and b are constants.(a) Show that this equation can be written in the formlog10 y t log10 b cexpressing the constant c in terms of a.(2)DO NOT WRITE IN THIS AREAy abtDO NOT WRITE IN THIS AREA9. The amount of antibiotic, y milligrams, in a patient’s bloodstream, t hours after theantibiotic was first given, is modelled by the equationA doctor measures the amount of antibiotic in the patient’s bloodstream at regularintervals for the first 5 hours after the antibiotic was first given.(c) (i) give a practical interpretation of the value of the constant a,(ii) give a practical interpretation of the value of the constant b.(d) Use the model to estimate the time taken, after the antibiotic was first given, for theamount of antibiotic in the patient’s bloodstream to fall to 30 milligrams. Give youranswer, in hours, correct to one decimal place.(e) Comment on the reliability of your estimate in part (d).(2)(2)(1)22*S63449A02248*DO NOT WRITE IN THIS AREAWith reference to this model,DO NOT WRITE IN THIS AREA(2)DO NOT WRITE IN THIS AREA(b) Estimate, to 2 significant figures, the value of a and the value of b.DO NOT WRITE IN THIS AREAShe plots a graph of log10 y against t and finds that the points on the graph lie close to astraight line passing through the point (0, 2.23) with gradient – 0.076

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 9 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA*S63449A02348*23Turn over

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DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 9 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 9 is 9 marks)*S63449A02548*25Turn over

Initially 9.00 cm2 of the bread was covered by mould and 6 days later, 56.25 cm2 of thebread was covered by mould.In the biologist’s model, the rate of increase of the surface area of bread covered bymould, at any time t days, is proportional to the square root of that area.By forming and solving a differential equation,(a) show that the biologist’s model leads to the equation 3 A t 3 4 DO NOT WRITE IN THIS AREAThe biologist measured the surface area of bread, A cm2, covered by mould at times, t days,after the start of the experiment.DO NOT WRITE IN THIS AREA10. A biologist conducted an experiment to investigate the growth of mould on a slice of bread.2612182430A (cm2)9.0056.25143.78271.19334.81337.33Table 1Use the last four measurements from Table 1 to(b) (i) evaluate the biologist’s model,(ii) suggest a possible explanation of the results.DO NOT WRITE IN THIS AREA0DO NOT WRITE IN THIS AREAt (days)DO NOT WRITE IN THIS AREAThe biologist’s full set of results are shown in the table below.DO NOT WRITE IN THIS AREA(6)(3)26*S63449A02648*

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 10 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA*S63449A02748*27Turn over

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DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 10 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 10 is 9 marks)*S63449A02948*29Turn over

yDO NOT WRITE IN THIS AREAQy f(x)OxDO NOT WRITE IN THIS AREA11.P0 x πThe curve has a minimum turning point at P and a maximum turning point at Q,as shown in Figure 5.(a) Show that the x coordinate of P and the x coordinate of Q are solutions of the equationcos 2 x 13DO NOT WRITE IN THIS AREAsin 2 x 3 cos 2 xDO NOT WRITE IN THIS AREAf ( x) DO NOT WRITE IN THIS AREAFigure 5 shows a sketch of the curve with equation y f(x), whereDO NOT WRITE IN THIS AREAFigure 5(4)(b) Hence find, to 2 decimal places, the x coordinate of the maximum turning point onthe curve with equationπ3(i) y f (3 x) 50 x 1 (ii) y f x 4 0 x 4π(4)30*S63449A03048*

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 11 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA*S63449A03148*31Turn over

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 11 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA32*S63449A03248*

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 11 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 11 is 8 marks)*S63449A03348*33Turn over

(a) Find the total mass of the mineral which the company expects to extract from 2018to 2040 inclusive, giving your answer to 3 significant figures.(b) Find the mass of the mineral which the company expects to extract during 2040,giving your answer to 3 significant figures.(2)(2)The costs of extracting the mineral each year are assumed to be:DO NOT WRITE IN THIS AREAThe mass of the mineral which the company expects to extract in each subsequent year ismodelled to decrease by 2% each year.DO NOT WRITE IN THIS AREA12. A company extracted 4500 tonnes of a mineral from a mine during 2018. 800 per tonne for the first 1500 tonnes 600 per tonne for any amount in excess of 1500 tonnesDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(3)DO NOT WRITE IN THIS AREA(c) Find the value of x, giving your answer to 3 significant figures.DO NOT WRITE IN THIS AREAThe expected cost of extracting the mineral from 2018 to 2040 inclusive is x million.34*S63449A03448*

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 12 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA*S63449A03548*35Turn over

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 12 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA36*S63449A03648*

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 12 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 12 is 7 marks)*S63449A03748*37Turn over

13.DO NOT WRITE IN THIS AREAROx3DO NOT WRITE IN THIS AREAyFigure 6π2The finite region R, shown shaded in Figure 6, is bounded by the curve, the x-axis, andthe line with equation x 3Use calculus to show that the area of R is 20 1532(7)DO NOT WRITE IN THIS AREA0 tDO NOT WRITE IN THIS AREAy 5sin 2tDO NOT WRITE IN THIS AREAx 6cos tDO NOT WRITE IN THIS AREAFigure 6 shows the curve with parametric equations38*S63449A03848*

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DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 13 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA40*S63449A04048*

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 13 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 13 is 7 marks)*S63449A04148*41Turn over

y DO NOT WRITE IN THIS AREAy kx2kxRODO NOT WRITE IN THIS AREAyy kx 2x 0y kxx 0where k is a positive constant.The finite region R, shown shaded in Figure 7, is bounded by the two curves.Show that, for all values of k, the area of R is13DO NOT WRITE IN THIS AREAFigure 7 shows the curves with equationsDO NOT WRITE IN THIS AREAFigure 7DO NOT WRITE IN THIS AREAxDO NOT WRITE IN THIS AREA14.(5)42*S63449A04248*

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3kann where k is a constant.The sequence is periodic of order 3Given that a2 2(a) show that k2 k – 12 0(3)DO NOT WRITE IN THIS AREAan 1 k DO NOT WRITE IN THIS AREA15. A sequence of numbers a1, a2, a3, is defined byGiven that a1 ¹ a2121(b) find the value of arr 1(4)DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA46*S63449A04648*

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Question 15 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 15 is 7 marks)TOTAL FOR PAPER IS 100 MARKS48*S63449A04848*DO NOT WRITE IN THIS AREA

10 *S63449A01048* T TE T AEA T TE T AEA The coordinates of T TE T AEA T TE T AEA T TE T AEA T TE T AEA 4. y P O x y f(x) Q R Figure 1 Figure 1 shows a sketch of part of the curve with equation y f(x), where f(x) x3 – 6x2 7x 2 x The curve cuts the x-axis at the points PQ, and R, as shown in Figure 1. Q are (2, 0) (a) Write f(x) as a product of two algebraic factors.

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