Wednesday 6 October 2021 – Afternoon AS Level Mathematics A H230/01 Pure Mathematics and Statistics. Graded A+. Download to score.

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AS Level Mathematics A (H230) Binomial series a b n a a n nC C n n b an n b a C b b r n r r n 1 1 2 2 2 ^ + = h + + - - + + f f - + + ^n ! Nh, where C ! ! n ! r r n r n nC r n r = = =... c m ^ - h Differentiation from first principles x lim h x h x f f f h 0 = + - " l^ h ^ ^ h h Standard deviation n x x xn x 2 2 - 2 = - /^ h / or f f x x f fx x 2 2 - 2 = - ^ h / / // The binomial distribution If X n + B^ , ph then X x n x P^ = = h c mp p x^1- hn x - , mean of X is np, variance of X is np^1-ph Kinematics v u = +at s ut at 12 2 = + s u v t 12 = + ^ h v u 2 2 = +2as s vt at 12 2 = -3 © OCR 2021 H230/01 Oct21 Turn over Section A: Pure Mathematics Answer all the questions. 1 In the triangle ABC, AB = 3, BC = 4 and angle ABC = 30°. Find the following. (a) The area of the triangle. [2] (b) The length AC. [2] (c) The angle ACB. [3] 2 The number of people, n, living in a small town is changing over time. In an attempt to predict the future growth of the town, a researcher uses the following model for n in terms of t, where t is the time in years from the start of the research. n = + 12 500 5000 t , for t H 1 Find the rate of change of n when t = 5. [4] 3 The diagram shows the curve y x = f( ), where f( ) x is a cubic polynomial in x. This diagram is repeated in the Printed Answer Booklet. x y 0 1 1 2 2 –3 –2 3 –2 –1 –1 (a) State the values of x for which f( ) x 1 2 1, giving your answer in set notation. [3] (b) On the diagram in the Printed Answer Booklet, draw the graph of y x = - f( ). [2] (c) Explain how you can tell that f( ) x cannot be expressed as the product of three real linear factors. [1]4 © OCR 2021 H230/01 Oct21 4 (a) Simplify 2 6 3 3 12 - - c c m m - . [2] (b) The vector a is defined by a r s 6 3 12 = - + c c m m - , where r and s are constants. Determine two pairs of values of r and s such that a is parallel to the y-axis and a = 3. [5] 5 The fuel consumption of a car, C miles per gallon, varies with the speed, v miles per hour. Jamal models the fuel consumption of his car by the formula C v v 5 12 125 3 2 = - , for 0 8 G G v 0. (a) Suggest a reason why Jamal has included an upper limit in his model. [1] (b) Determine the speed that gives the maximum fuel consumption. [4] Amaya’s car does more miles per gallon than Jamal’s car. She proposes to model the fuel consumption of her car using a formula of the form C v v k 5 12 125 3 2 = - + , for 0 8 G G v 0, where k is a positive constant. (c) Give a reason why this model is not suitable. [1] (d) Suggest a different change to Jamal’s formula which would give a more suitable model. [2] 6 The power output, P watts, of a certain wind turbine is proportional to the cube of the wind speed vms-1. When v = 3.6, P = 50. Determine the wind speed that will give a power output of 225 watts. [3] 7 The relationship between the variables P and Q is modelled by the formula Q a = Pn where a and n are constants. Some values of P and Q are obtained from an experiment. (a) State appropriate quantities to plot so that the resulting points lie approximately in a straight line. [1] (b) Explain how to use such a graph to estimate the value of n. [2]5 © OCR 2021 H230/01 Oct21 Turn over 8 (a) Prove that the following statement is not true. p is a positive integer & 2 p H p2 [1] (b) Prove that the following statement is true. m and n are consecutive positive odd numbers & mn+1 is the square of an even number [4] 9 In this question you must show detailed reasoning. Find the equation of the straight line with positive gradient that passes through (0, 2) and is a tangent to the curve y x = - 2 x+6 [Show More]

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