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Monday 18 October 2021 – Afternoon A Level Mathematics A H240/03 Pure Mathematics and Mechanics. Latest Revision Guide. Graded A+

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Formulae A Level Mathematics A (H240) Arithmetic series S n a l n a 2 1 n d n 2 1 12 = + ^ ^ h h = + " - , Geometric series S r a r 11 n n = - - ^ h S r a 3 = 1- for r 1 1 Binomi... al series a b n a a n nC C n n b an n b a C b b n N r n r r n 1 1 2 2 2 ^ ^ + = h h + + - - + + f f - + + ! , where ! ! n ! r r n r n nC C r n r = = = c m ^ - h x nx ! ! , n n x r n n n r 1 1 x x n 2 1 1 1 ^ + = hn + + ^ ^ - h h 2 + + f - - f^ + h r +f ^ 1 1 ! Rh Differentiation tankx k k sec2 x sec t x x an -cosec2x -cosec c x x ot sec x cot x cosec x f^xh f l^xh Quotient rule y uv = , yx v v ux u vx dd dd dd = 2 - Differentiation from first principles x lim h x h x f f f h 0 = + - " l^ h ^ ^ h h Integration ln xx x x c ff d f = + c l dde ^^ ^ hh h x x x n x c 1 1 f f d f n n 1 = + + + ; l^ h h a ^ k a ^ hk Integration by parts u vx x uv v ux x dd d dd ; ; = - d Small angle approximations sin i i . , Section A: Pure Mathematics Answer all the questions. 1 Show in a sketch the region of the x-y plane within which all three of the following inequalities hold. y x 2 H , x y + G 2, x H 0. You should indicate the region for which the inequalities hold by labelling the region R. [3] 2 A B C 60° (4 +h) cm (4 –h) cm The diagram shows triangle ABC in which angle A is 60° and the lengths of AB and AC are ( ) 4 c +h m and ( ) 4 c -h m respectively. (a) Show that the length of BC is pcm where p h 2 2 = + 16 3 . [2] (b) Hence show that, when h is small, p h . 4+ + m n 2 4 h , where m and n are rational numbers whose values are to be determined. [4] 3 An arithmetic progression has first term 2 and common difference d, where d ! 0. The first, third and thirteenth terms of this progression are also the first, second and third terms, respectively, of a geometric progression. By determining d, show that the arithmetic progression is an increasing sequence. [5]5 © OCR 2021 H240/03 Oct21 Turn over 4 (a) Sketch, on a single diagram, the following graphs. • y x = -1 • y kx = , where k is a negative constant [2] (b) Hence explain why the equation x x- = 1 k has exactly one real root for any negative value of k. [1] (c) Determine the real root of the equation x x- = 1 6 - . [2] 5 A particle P moves along a straight line in such a way that at time t seconds P has velocity vm s-1, where v t = + 12 cos s 5 in t. (a) Express v in the form R t cos( ) - a , where R 2 0 and 0 1 1 a r 2 1 . Give the value of a correct to 4 significant figures. [3] (b) Hence find the two smallest positive values of t for which P is moving, in either direction, with a speed of 3m s-1. [3] 6 The equation 6 2 arcsin( ) x x - - 1 0 2 = has exactly one real root. (a) Show by calculation that the root lies between 0.5 and 0.6. [2] In order to find the root, the iterative formula x p n n +1 = +q r sin( ) x2 , with initial value x 0 5 . 0 = , is to be usedcos 1 i i . - 2 1 2, tan i i . where i is meas [Show More]

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