WMA01_01_que_20160112

www.dynamicpapers.com Write your name here Surname Other names Centre Number Candidate Number Pearson Edexcel International Advanced Level Core Mathematics C12 Advanced Subsidiary Tuesday 12 January 2016 – Morning Paper Reference Time: 2 hours 30 minutes WMA01/01 You must have: Total Marks Mathematical Formulae and Statistical Tables (Blue) Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable 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). Coloured pencils and highlighter pens must not be used. • centre Fill in the boxes at the top of this page with your name, number and candidate number. • clearly labelled. Answer all questions and ensure that your answers to parts of questions are • Answer the questions in the spaces provided – there may be more space than you need. • without You should show sufficient working to make your methods clear. Answers working may not gain full credit. • degree When a calculator is used, the answer should be given to an appropriate of accuracy. Information •• The The total mark for this paper is 125. marks for each question are shown in brackets – use this as a guide as to how much time 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. Turn over P46957A ©2016 Pearson Education Ltd. *P46957A0152* 1/1/1/1/ www.dynamicpapers.com Leave blank 1. A sequence of numbers u1, u2, u3, ... satisfies un+1 = 2un – 6, n.1 DO NOT WRITE IN THIS AREA Given that u1 = 2 (a) find the value of u3 (2) 4 (b) evaluate ∑u i i =1 (3) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 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___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ Q1 (Total 5 marks) 3 *P46957A0352* Turn over www.dynamicpapers.com Leave blank 49 2. (i) Given that = 7 a, find the value of a. 7 (2) DO NOT WRITE IN THIS AREA 10 (ii) Show that = 15 2 + 20 18 − 4 You must show all stages of your working. 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___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ Q2 (Total 5 marks) 5 *P46957A0552* Turn over Find. using calculus and showing each step of your working.dynamicpapers. www.com Leave blank 3. DO NOT WRITE IN THIS AREA 4 ∫ ⎛ 2 ⎞ ⎜⎝ 6 x − 3 − ⎟ dx 1 x⎠ (5) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 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The 4th term of an arithmetic sequence is 3 and the sum of the first 6 terms is 27 Find the first term and the common difference of this sequence.dynamicpapers. 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(2) The table below gives corresponding values of x and y.dynamicpapers.5 0. for y = sin2x.866 1 (b) Use the trapezium rule with all the values of y from the table to find an approximate value for π DO NOT WRITE IN THIS AREA ∫ 4 sin 2 x dx 0 (3) DO NOT WRITE IN THIS AREA 10 *P46957A01052* .com Leave blank 3π 5. www. π π π x 0 12 6 4 y 0 0. The values of y are rounded to 3 decimal places where necessary. 0 -
x - 2 DO NOT WRITE IN THIS AREA Show the coordinates of the points where your graph crosses the x-axis. 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(2) (b) Factorise f(x). f( x) = x3 + x 2 − 12 x − 18 DO NOT WRITE IN THIS AREA (a) Use the factor theorem to show that (x + 3) is a factor of f (x). (2) (c) Hence find exact values for all the solutions of the equation f (x) = 0 (3) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 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___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ Q6 (Total 7 marks) 15 *P46957A01552* Turn over . in ascending powers of x. 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DO NOT WRITE IN THIS AREA (4) Given that the coefficient of x3 in this expansion is 1512 (b) find the value of k. (a) Find the first 4 terms. where k is a non-zero constant. of the binomial expansion of (1 + kx)8. dynamicpapers.com Leave blank Question 7 continued ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 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ș< 360° 7 sin(2ș+ 30°) = 3cos(2ș + 30°) giving your answers to one decimal place. (1) DO NOT WRITE IN THIS AREA (b) Hence solve for 0 .) 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(Solutions based entirely on graphical or numerical methods are not acceptable.dynamicpapers. (a) Given that 7 sin x = 3cosx. www.dynamicpapers.com Leave blank Question 8 continued ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ Q8 (Total 6 marks) 21 *P46957A02152* Turn over . 02 (4) DO NOT WRITE IN THIS AREA (d) Find the value of N.com Leave blank 9. with the populations at the end of each year forming a geometric sequence.dynamicpapers. 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(a) Show that the predicted resident population at the end of Year 2 is 132 600 (1) (b) Write down the value of the common ratio of the geometric sequence. The resident population of a city is 130 000 at the end of Year 1 A model predicts that the resident population of the city will increase by 2% DO NOT WRITE IN THIS AREA each year. (1) The model predicts that Year N will be the first year which will end with the resident population of the city exceeding 260 000 (c) Show that log10 2 N +1 log10 1. dynamicpapers.com Leave blank Question 9 continued ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 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___________________________________________________________________________ Q9 (Total 7 marks) 25 *P46957A02552* Turn over . www. The curve C has equation 5 5 2 DO NOT WRITE IN THIS AREA y = 12 x − 4 x − 1000 . x 
0 18 dy (a) Find dx (2) (b) Hence find the coordinates of the stationary point on C. dx 2 (3) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 26 *P46957A02652* .dynamicpapers.com Leave blank 10. (5) d2 y (c) Use to determine the nature of this stationary point. com Leave blank Question 10 continued ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 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The shaded region S. is bounded by the lines BZ. ZY.com Leave blank 11. shown in Figure 1. (4) (c) the area of the shaded region to 3 significant figures. (3) The point A lies on the line XY and the point B lies on the line XZ and AX = BX = 5 cm. YA and the arc AB. (a) Find the size of the angle YXZ. YZ = 16 cm and ZX =12 cm.dynamicpapers. (4) ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 30 *P46957A03052* . Find (b) the perimeter of the shaded region to 3 significant figures. X DO NOT WRITE IN THIS AREA 12 B cm cm A 10 S Y Z 16cm Figure 1 Figure 1 shows a triangle XYZ with XY = 10 cm. giving your answer in radians to DO NOT WRITE IN THIS AREA 3 significant figures. AB is the arc of a circle with centre X. dynamicpapers. www.com Leave blank Question 11 continued ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 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C and k are constants to be determined. www. where A. (4 + 3  x)2 f (x) = .dynamicpapers. B.com Leave blank 12. (4)  E. 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___________________________________________________________________________ ___________________________________________________________________________ Q12 (Total 10 marks) 37 *P46957A03752* Turn over . www. (a) Show that k satisfies the inequality DO NOT WRITE IN THIS AREA 11k2 – 30k – 9 
0 (4) (b) Find the range of possible values for k. The equation k(3x 2 + 8x + 9) = 2 – 6x. has no real roots. where k is a real constant. www. 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___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 39 *P46957A03952* Turn over .dynamicpapers. www. com Leave blank Question 13 continued ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 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___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 40 *P46957A04052* .dynamicpapers. www. com Leave blank Question 13 continued ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 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___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ Q13 (Total 8 marks) 41 *P46957A04152* Turn over . www.dynamicpapers. an expression for x in terms of a. (4) (ii) Solve the equation (log5 y)2íORJ5 y) + 12 = 0 showing each step of your working.dynamicpapers. (4) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 42 *P46957A04252* . in its simplest form.com Leave blank 14. (i) Given that log a x + log a 3 = log aíZKHUHa is a positive constant DO NOT WRITE IN THIS AREA find. www. dynamicpapers. www.com Leave blank Question 14 continued ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ Q14 (Total 8 marks) 43 *P46957A04352* Turn over . www.com Leave blank 15. The points A and BKDYHFRRUGLQDWHVíí.dynamicpapers. DQG. 8) also lies on C. giving your answer in the form ax + by + c = 0 (4) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 44 *P46957A04452* .UHVSHFWLYHO\ AB is the diameter of a circle C. 8). (b) Find an equation of the tangent to C at the point (4. (6) The point (4. DO NOT WRITE IN THIS AREA (a) Find an equation for the circle C. dynamicpapers.com Leave blank Question 15 continued ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 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(5) The curve C crosses the x-axis at the points T and S. at the points P and Q. This region R is bounded by the line segment PQ. (8) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 48 *P46957A04852* . www. the line segment TS. y DO NOT WRITE IN THIS AREA C Q l P R O T S x DO NOT WRITE IN THIS AREA Figure 2 1 The straight line l with equation y = x + 1 cuts the curve C. (b) Write down the coordinates of the points T and S. DO NOT WRITE IN THIS AREA (c) Use integration to find the exact area of the shaded region R. (2) The finite region R is shown shaded in Figure 2. with equation 2 y = x 2 í 4x + 3. and the arcs PT and SQ of the curve. as shown in Figure 2 (a) Use algebra to find the coordinates of the points P and Q.com Leave blank 16.dynamicpapers. dynamicpapers. www.com Leave blank Question 16 continued ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 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___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 49 *P46957A04952* Turn over . com Leave blank Question 16 continued ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 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___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 50 *P46957A05052* . www.dynamicpapers. www.dynamicpapers.com Leave blank Question 16 continued ___________________________________________________________________________ DO NOT WRITE IN THIS AREA ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 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___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 51 *P46957A05152* Turn over . dynamicpapers. www.com Leave blank Question 16 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 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___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ Q16 (Total 15 marks) TOTAL FOR PAPER: 125 MARKS END 52 *P46957A05252* .