OCR GCE A LEVEL 2022 MATHEMATICS A QUESTION PAPER H240-03 PAPER 3-H240/03 Pure Mathematics and Mechanics

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Arithmetic series S n a l n a2 1 n d n 2 1 2 1 = + ^ ^ h h = + " - , Geometric series S r a r 1 1 n n = - ^ - h S r a 1 = 3 - for r 1 1 Binomial series a b a a C C b a b a C b b n N... n n n n n n n r n r r n 1 1 2 + = + + 2 2 + + f f+ + ! - - - ^ ^ h h, where ! ! n ! r r n r n C C n 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 1 R + =n + + 2 f f r f 1 ! - + + - - + ^ + ^ ^ ^ h ^ h h h h Differentiation tan kx k k sec x 2 sec t x x an cosec x 2 - -cosec c x x ot sec x cot x cosec x f^xh fl^xh Quotient rule y v u = , x y v v x u u x v d d d d d d = 2 - Differentiation from first principles x lim h x h x f f f h 0 = + - " l^ ^ ^ h h h Integration ln x x x x c f f d f = + c l e dd ^ ^ ^ h h 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 x v x uv v x u x d d d d d ; ; = - d Small angle approximations sin i i . , cos 1 2 1 2 i i . - , tan i i . where i is measured in radians 3 © OCR 2022 H240/03 Jun22 Turn over Trigonometric identities sin s ^A B ! ! h = inA B cos cosA B sin cos c ^A B ! " h = osA B cos sinA B sin tan tan tan tan tan A B A B A B 1 ! " ! ^ h = A B k 2 1 a ! ! ^ + hrk Numerical methods Trapezium rule: y xd h y y y 2 y y a b 2 n n 1 . 0 1 + + + +2 1 f+ ^ ^ - y " h h,, where h n b a = - The Newton-Raphson iteration for solving f^xh = 0: x x x x f f n n n n 1 = - + l ^ ^ h h Probability P P ^ ^ A B, + h h = + A B P P ^ ^ h h - A B P P ^ ^ A B+ h h = = A B P P ^ A B h ^ hP^A Bh or A B B A B P P P + ^ = ^ ^ h h h Standard deviation n x x n x 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 P X x n x p p 1 x n x = = - - ^ h c m ^ h , mean of X is np, variance of X is np^1-ph Hypothesis test for the mean of a normal distribution If X N , 2 + ^n v h then X , n N 2 + n v c m and , n X + N 0 1 n v - ^ h Percentage points of the normal distribution If Z has a normal distribution with mean 0 and variance 1 then, for each value of p, the table gives the value of z such that P^Z z G h = p. p 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 0.9995 z 0.674 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291 Kinematics Motion in a straight line Motion in two dimensions v u = +at v u = +at s ut at 2 1 2 = + s ut t a2 1 2 = + s u v t 2 1 = + ^ h s u v t 2 1 = + ^ h v u 2as 2 2 = + s vt at 2 1 2 = - s vt t a2 1 2 = - 4 © OCR 2022 H240/03 Jun22 Section A: Pure Mathematics Answer all the questions. 1 Solve the equation 2 3 x- = 9. [3] 2 (a) Give full details of the single transformation that transforms the graph of y x3 = to the graph of y x 8 3 = - . [2] The function f is defined by f( )x x 8 3 = - . (b) Find an expression for f ( )x -1 . [2] (c) State how the graphs of y x = f( ) and y x f ( ) 1 = - are related geometrically. [1] 3 The points P and Q have coordinates (2, -5) and (3, 1) respectively. Determine the equation of the circle that has PQ as a diameter. Give your answer in the form xyax by c 0, 2 2 + +++ = where a, b and c are integers. [4] 4 The positive integers x, y and z are the first, second and third terms, respectively, of an arithmetic progression with common difference -4. Also, x, y 15 and z are the first, second and third terms, respectively, of a geometric progression. (a) Show that y satisfies the equation y y 16 225 0. 4 2 - - = [4] (b) Hence determine the sum to infinity of the geometric progression. [4] 5 © OCR 2022 H240/03 Jun22 Turn over 5 In this question you must show detailed reasoning. y x O P The diagram shows the curve with equation y x x 4 1 2 3 = 2 + - . The tangent to the curve at the point P has gradient 2. (a) Show that the x-coordinate of P satisfies the equation 4 3 x x 3 0. 3 + - = [5] (b) Show by calculation that the x-coordinate of P lies between 0.5 and 1. [2] (c) Show that the iteration x x 3 3 4 n n 1 3 = - + cannot converge to the x-coordinate of P whatever starting value is used. [2] (d) Use the Newton-Raphson method, with initial value 0.5, to determine the coordinates of P correct to 5 decimal places. [Show More]

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