Full name: Teacher name: U6 A Level Maths PURE MOCK Tuesday 5 th February 2019 PM Time: 2 hours Total Marks: 100 You must have: Mathematical Formulae and Statistical Tables, Calculator Instructions Use black ink or ball-point pen. If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Fill in your name at the top of this page and the name of your teacher Answer all questions and ensure that your answers to parts of questions are clearly labelled. You should show sufficient working to make your methods clear. Answers without working may not gain full credit. Answers found from the calculator without working may not gain full credit. Answers should be given to three significant figures unless otherwise stated. Information A booklet Mathematical Formulae and Statistical Tables is provided. There are 14 questions. The total mark for this paper is 100. The 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. Q1 Q2 Q3 Q4 Q5 Q6 Q7 6 5 4 10 4 6 5 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Total 5 10 7 7 10 11 10 100 Grade
1. x The diagram shows a sketch of the curve with equation y =, x 0. 1 x The finite region R, shown shaded the diagram, is bounded by the curve, the line with equation x = 1, the x-axis and the line with equation x = 3. x The table below shows corresponding values of x and y for y =. 1 x (a) (b) x 1 1.5 2 2.5 3 y 0.5 0.6742 0.8284 0.9686 1.0981 Use the trapezium rule, with all the values of y in the table, to find an estimate for the area of R, giving your answer to 3 decimal places. Explain how the trapezium rule can be used to give a better approximation for the area of R. (3) (1) (c) Giving your answer to 3 decimal places in each case, use your answer to part (a) to deduce an estimate for 3 (i) 5x dx, (ii) x 6 dx. 1 x 1 x 1 3 1 (2) (6 marks)
2. (a) Given that θ is small, use the small angle approximation for cos θ to show that 1 + 4 cos θ + 3 cos 2 θ 8 5θ 2. (3) Adele uses θ = 5 to test the approximation in part (a). Adele s working is shown below. Using my calculator, 1 + 4 cos (5 ) + 3 cos 2 (5 ) = 7.962, to 3 decimal places. Using the approximation 8 5θ 2 gives 8 5(5) 2 = 117 Therefore, 1 + 4 cos θ + 3 cos 2 θ 8 5θ 2 is not true for θ = 5. (b) (i) Identify the mistake made by Adele in her working. (ii) Show that 8 5θ 2 can be used to give a good approximation to 1 + 4 cosθ + 3 cos 2 θ for an angle of size 5. (5 marks) (2)
3. The diagram shows a sector AOB of a circle with centre O and radius r cm. The angle AOB is θ radians. The area of the sector AOB is 11 cm 2. Given that the perimeter of the sector is 4 times the length of the arc AB, find the exact value of r. (4 marks)
4. f(x) = x 3 + ax 2 ax + 48, where a is a constant. Given that f( 6) = 0, (a) (i) show that a = 4. (ii) express f(x) as a product of two algebraic factors. (4) Given that 2 log2 (x + 2) + log2 x log2 (x 6) = 3, (b) show that x 3 + 4x 2 4x + 48 = 0. (4) (c) Hence explain why 2 log2 (x + 2) + log2 x log2 (x 6) = 3 has no real roots. (2) (10 marks)
5. A cup of hot tea was placed on a table. At time t minutes after the cup was placed on the table, the temperature of the tea in the cup, θ C, is modelled by the equation where A is a constant. θ = 25 + Ae 0.03t The temperature of the tea was 75 C when the cup was placed on the table. (a) Find a complete equation for the model. (1) (b) Use the model to find the time taken for the tea to cool from 75 C to 60 C, giving your answer in minutes to one decimal place. (2) Two hours after the cup was placed on the table, the temperature of the tea was measured as 20.3 C. Using this information, (c) evaluate the model, explaining your reasoning. (4 marks) (1)
6. Complete the table below. The first one has been done for you. For each statement below you must state if it is always true, sometimes true or never true, giving a reason in each case. Statement Always True Sometimes True Never True Reason The quadratic equation ax 2 + bx + c = 0 (a 0) has 2 real roots. It only has 2 real roots when b 2 4ac > 0 When b 2 4ac = 0 it has 1 real root and when b 2 4ac < 0 it has 0 real roots. (i) When a real value of x is substituted into x 2 6x + 10 the result is positive. (ii) If ax > b then x b a (2) (2) (iii) The difference between consecutive square numbers is odd. (2) (6 marks)
7. Given that θ is measured in radians, prove, from first principles, that d d (cos ) = sin sin h cos h 1 You may assume the formula for cos (A ± B) and that as h 0, 1 and 0. h h (5 marks)
8. The second, third and fourth terms of an arithmetic sequence are 2k, 5k 10 and 7k 14 respectively, where k is a constant. Show that the sum of the first n terms of the sequence is a square number. (5 marks)
1 4x 5 5 9. (a) Use binomial expansions to show that 1 + x x 2. 1 x 2 8 (6) 1 A student substitutes x = into both sides of the approximation shown in part (a) in an attempt 2 to find an approximation to 6. 1 (b) Give a reason why the student should not use x =. 2 (1) 1 (c) Substitute x = into 11 1 4x 1 x 5 5 = 1 + x x 2 2 8 to obtain an approximation to 6. Give your answer as a fraction in its simplest form. (3) (10 marks)
10. Show that 2 sin 2 0 1 cos d = 2 2 ln 2. (7 marks)
2 1 11x 6x B C 11. A + +. ( x 3)(1 2x) ( x 3) ( 1 2x) (a) Find the values of the constants A, B and C. (4) 2 1 11x 6x f(x) =, x > 3. ( x 3)(1 2x) (b) Prove that f (x) is a decreasing function. (3) (7 marks)
12. (a) Show that cosec 2x + cot 2x cot x, x 90n, n Z. (5) (b) Hence, or otherwise, solve, for 0 θ < 180, cosec(4θ + 10 ) + cot(4θ + 10 ) = 3. You must show your working. (Solutions based entirely on graphical or numerical methods are not acceptable.) (5) (10 marks)
13. (a) Express 2 sin θ 1.5 cos θ in the form R sin (θ α), where R > 0 and 0 < α <. 2 State the value of R and give the value of α to 4 decimal places. (3) Tom models the depth of water, D metres, at Southview harbour on 18th October 2017 by the formula 4 t 4 t D = 6 + 2 sin 1.5 cos, 0 t 24, 25 25 where t is the time, in hours, after 00:00 hours on 18th October 2017. Use Tom s model to (b) find the depth of water at 00:00 hours on 18th October 2017, (1) (c) (d) find the maximum depth of water, find the time, in the afternoon, when the maximum depth of water occurs. Give your answer to the nearest minute. Tom s model is supported by measurements of D taken at regular intervals on 18th October 2017. Jolene attempts to use a similar model in order to model the depth of water at Southview harbour on 19th October 2017. Jolene models the depth of water, H metres, at Southview harbour on 19th October 2017 by the formula 4 x 4 x H = 6 + 2 sin 1.5 cos, 0 x 24, 25 25 where x is the time, in hours, after 00:00 hours on 19th October 2017. By considering the depth of water at 00:00 hours on 19th October 2017 for both models, (e) (i) explain why Jolene s model is not correct, (ii) hence find a suitable model for H in terms of x. (1) (3) (3) (11 marks)
14. The diagram shows a sketch of the curve C with equation y = 5x 9x + 11, x 0 The point P with coordinates (4, 15) lies on C. The line l is the tangent to C at the point P. The region R, shown shaded in the diagram, is bounded by the curve C, the line l and the y-axis. Show that the area of R is 24, making your method clear. 3 2 (Solutions based entirely on graphical or numerical methods are not acceptable.) (10 marks)