Wednesday 6 October 2021 – Afternoon A Level Mathematics A H240/01 Pure Mathematics. 100% predictor paper. Download to score. 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 ! ! ! C nr r n r n nC r n r = = = - JKKL ^ NOOP 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 . , cos 1 i i . - 2 1 2, tan i i . where i is measured in radians3 © OCR 2021 H240/01 Oct21 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 ^ ! h = 1" ! aA B ! ! ^k+ 2 1hrk Numerical methods Trapezium rule: y x d h y y y 2 y y b a 2 n n 1 y . "^ ^ 0 1 + + h h + + 2 1 f+ - ,, where h = b a -n The Newton-Raphson iteration for solving f^xh = 0: x x x x f f n n n n +1 = - l ^^ hh 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 xn x 2 2 - 2 = - - /^ h / - or f f x x f fx x 2 2 - 2 = - - ^ h - / / // The binomial distribution If X + B^n p , h then P X x n x = = p p x 1- n x - JKKL ^ ^ NOOP h 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^n v , 2h then X , n N 2 + n v JKKL NOOP and , n X + N 0 1 v - n ^ 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. 0.75 0.90 0.95 0.674 1.282 1.645 0.975 0.99 0.995 1.960 2.326 2.576 0.9975 0.999 0.9995 2.807 3.090 3.291 pz Kinematics Motion in a straight line Motion in two dimensions v u = +at v u = +at s ut at 12 2 = + s ut t a 12 2 = + s u v t 12 = + ^ h s u = + 2 1^ vht v u 2 2 = +2as s vt at 12 2 = - s vt t a 12 2 = -4 © OCR 2021 H240/01 Oct21 Answer all the questions. 1 Determine the set of values of k such that the equation x x 2 + + 4 3 ( ) k+ = 0 has two distinct real roots. [4] 2 Alex is comparing the cost of mobile phone contracts. Contract A has a set-up cost of £40 and then costs 4p per minute. Contract B has no set-up cost, does not charge for the first 100 minutes and then costs 6p per minute. (a) Find an expression for the cost of each of the contracts in terms of m, where m is the number of minutes for which the phone is used and m 2 100. [2] (b) Hence find the value of m for which both contracts would cost the same. [2] 3 It is given that x is proportional to the product of the square of y and the positive square root of z. When y = 2 and z = 9, x = 30. (a) Write an equation for x in terms of y and z. [2] (b) Find the value of x when y = 3 and z = 25. [2] 4 In this question you must show detailed reasoning. The cubic polynomial f( ) x is defined by f( ) x = - 2 3 1 x x 3 2 - + 1 6 x . (a) Use the factor theorem to show that ( ) 2 1 x- is a factor of f( ) x . [1] (b) Express f( ) x in fully [Show More]

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