Mathematics AS/P2/M18 AS PAPER 2

Size: px

Start display at page:

Download "Mathematics AS/P2/M18 AS PAPER 2"

Harvey Benson
5 years ago
Views:

1 Surname Other Names Candidate Signature Centre Number Candidate Number Examiner Comments Total Marks Mathematics AS PAPER 2 March Mock Exam (OCR Version) CM Time allowed: 1 hour and 30 minutes Instructions to candidates: In the boxes above, write your centre number, candidate number, your surname, other names and signature. Answer ALL of the questions. You must write your answer for each question in the spaces provided. You may use a calculator. Information to candidates: Full marks may only be obtained for answers to ALL of the questions. The marks for individual questions and parts of the questions are shown in square brackets. There are 12 questions in this question paper. The total mark for this paper is 75. Advice to candidates: You should ensure your answers to parts of the question are clearly labelled. You should show sufficient working to make your workings clear to the Examiner. Answers without working may not gain full credit. AS/P2/M crashmaths Ltd.

2 2 Section A: Pure Mathematics Answer all the questions 1 Find (2x 3) 4, giving each term in its simplest form. [3] Show, in clear stages, that dx = k(3 3), where k is a constant to be found. [5] 1 x 3 3 The function f is defined such that f(x) = 2x 3 x 2 25x 12. (i) Show that (x + 3) is a factor of f(x). [2] (ii) Solve the equation f(x) = 0. [3] 4 Using differentiation from first principles, show that (i) the gradient of the line y = mx + c is m, [2] (ii) the derivative of 6x 3 is 18x 2. [3] 5 Given that the solution to the inequality x 2 + ax 3 < 2 is { x : x < c} { x : 1 < x < 7}, where c < 7, find the values of the constants a and c. [3]

3 3 6 The diagram below shows the circle C which passes through the points A( 3, 2) and B(0, 1). l y A B x C The straight line l has the equation 2x y + 4 = 0 and is a tangent to C at A. (i) Find (a) the gradient of the perpendicular bisector of the line segment AB [3] (b) the equation of the perpendicular bisector of the line segment AB [3] (ii) Show that the coordinates of the centre of C are ( 1, 3). [5] (iii) Calculate the radius of C. [2] (iv) Express the equation of the circle C in the form ( x a) 2 + ( y b) 2 = k where a, b and k are constants to be found. [2] 7 The binomial coefficient n C r is defined such that n C r = n r = n! r! ( n r)!, (i) By starting with the right-hand side, or otherwise, prove that (ii) Deduce that n r r m = n n m m r m n r = n n 1 r r 1 [3] [2] Turn over

4 4 8 In this question you must show detailed reasoning. r cm h cm The diagram above shows a solid. The solid is formed by attaching a hemisphere with radius r cm to each end of a cylinder, which has radius r cm and height h cm. The centres of cylinder and the hemispheres lie on the same line. The solid has a volume of 120 cm 3. (i) (a) Show that h = 120. [2] πr 4r 2 3 (b) Hence, obtain an expression for the surface area, A cm 2, of the solid in terms of r. [1] (ii) Show that the surface area of the solid is minimised when r = [4] π (iii) Justify, by further calculus, that A is a minimum for the value of r in part (b). [3] 1

5 5 Section B: Mechanics Answer all the questions 9 A ball, with mass 0.5 kg, is thrown vertically upwards from a point P at 22 m s 1. The point P is 10 m above a large water reservoir. The ball is modelled as a particle that moves freely under the influence of gravity until it reaches the reservoir. (i) Find the speed of the ball as it hits the reservoir. [2] After the ball hits the reservoir, it decelerates uniformly and comes to rest in 3 s. (ii) Calculate the deceleration of the ball in the reservoir. [1] (iii) Hence, find the magnitude of the resistive forces acting on the ball in the reservoir. [2] 10 A particle P moves on the x-axis. At time t s, P is moving with a velocity v m s 1, where v = a bt 2 0 t 5 0 otherwise and a and b are positive constants. The magnitude of the acceleration of P at t = 2 is 4 m s 2. (i) Find the values of the constants a and b. [3] (ii) Find the total distance travelled by the particle P. [4] 11 In this question, i and j are perpendicular unit vectors. A particle P has a position vector (xi + yj) m relative to a fixed origin O. Two variable forces, F 1 N and F 2 N, act on the particle as it moves, where F 1 = (4ysin 2 x + x)i + e y j F 2 = (4ycos 2 x)i 6j The particle passes through the point Q, which has position vector (ai + bj) m relative to O. When the particle passes through Q, it is moving at constant speed. Find the exact value of a and b. [4] Turn over

6 6 12 A P B The mass A is held at rest on a rough horizontal table and is attached to one end of a string. The mass of A is 2 kg. The string passes over a pulley P, which is fixed at the edge of the table. The other end of the string is attached to the mass B, which has mass 4.5 kg and hangs freely, vertically below P. The magnitude of the frictional force between A and the table is modelled as having a constant value of 0.4 R N, where R is the magnitude of the normal reaction force exerted by the table on A. The string is released from rest, with the string taut, as shown in the diagram above. The masses are modelled as particle, the string is modelled as light and inextensible, the pulley is modelled as small and the acceleration due to gravity, g, is modelled as being 9.8 m s 2. The pulley is not modelled as a smooth pulley and the difference in tension between the two sides, T N, is modelled as T = α + βa where α = 3 N, β = 0.3 kg and a is the acceleration of the masses. Given that the tension in the string at B is greater than the tension in the string at A, (i) find the acceleration of the masses. [4] (ii) Find the magnitude of the resultant force acting on the pulley. [3] (iii) Suggest one improvement that can be made to the model. [1] END OF QUESTION PAPER Copyright 2018 crashmaths Ltd.

MATHEMATICS AS/P2/M18 AS PAPER 2

MATHEMATICS AS/P2/M18 AS PAPER 2 Surname Other Names Candidate Signature Centre Number Candidate Number Examiner Comments Total Marks MATHEMATICS AS PAPER 2 March Mock Exam (Edexcel Version) CM Time allowed: 1 hour and 15 minutes Instructions