2 Find ( ) d x x x 1 1 1 1 2 3 5 - - - e o y , giving your answer in the form ln a b + , where a and b are rational numbers. [5]

2 > x B1 2 ( ) 5 3 1 ln 1 1   − +   −   x x 2 B1 for ( ) ln 1 − x B1 for 1 1 + − x 1 1 ln 4 ln 2 4 2     + − +         M1 Dep on at least one B mark, for correct use of limits 1 ln2 4 − 2 A1 for ln2 A1 for 1 4 − oe

2 A particle moves in a straight line such that its velocity, v ms 1 - , at time t seconds after passing through a fixed point O, is given by v 25 e t3 = - . Find the speed of the particle when t 1 = . [2]

2 4.91 =− v soi B1 Speed = 4.91 B1

4 A curve is such that - ( ) d d x y x 3 2 2 2 3 1 = + . The curve has gradient 4 at the point (2, 6.2). Find the equation of the curve. [6]

4 ( ) ( ) 2 3 d 1 3 2 d 2   = + +     y x c x M1 For ( ) 2 3 1 3 2 + k x where 1k a constant. 4 2 = + c M1 Dep for use of 4 and 2 = x in their d d y x to obtain c ( ) 2 3 d 1 3 2 2 d 2   = + +     y x x A1 May be implied by subsequent integration or by 2 = c ( ) ( ) 5 3 1 3 2 2 10 = + + + y x x d M1 For ( ) 5 3 2 3 2 + k x where 2k is a constant. ( ) 1 6.2 32 4 10 = + + d M1 Dep on previous M1 for use of 2 = x and 6.2 = y in their y ( ) 5 3 1 3 2 2 1 10 = + + − y x x A1 Must be an equation The first mark is for correctly integrating the given derivative to get the x term with a constant k. The second mark is for using the given gradient and x-value to find the constant of integration for the derivative. The third mark is for the correct derivative with the constant. The fourth mark is for correctly integrating the derivative to get the y term with a constant d. The fifth mark is for using the given point (x, y) to find the constant of integration for the curve. The final mark is for the complete and correct equation of the curve.

6 DO NOT USE A CALCULATOR IN THIS QUESTION. A curve has equation y x x 3 5 8 5 60 2 = + - + ` j . (a) Find the x-coordinate of the stationary point on the curve, giving your answer in the form a b 5 + , where a and b are integers. [4] (b) Hence find the y-coordinate of this stationary point, giving your answer in the form c 5, where c is an integer. [3]

6(a) ( ) d 2 3 5 8 5 d = + − y x x or ( ) 8 5 2 3 5 = + x M1 Either For differentiation must have one correct term. or for use of 2 ' 4 0' − = b ac , so 2 = −b x a at the stationary point. ( ) ( ) 3 5 4 5 3 5 3 5 − = × + − x oe leading to 12 5 20 4 − oe, this is the minimum acceptable working for this method. M1 Dep for equating their d d y x to zero with attempt to solve and rationalisation using a two term factor, or rationalisation of 2 = −b x a , using a two term factor with sufficient detail to imply no use of a calculator. Allow multiple equivalents. Allow one numerical slip or sign error. 5 3 5 = −+ x 2 A1 for 5 − A1 for 3 5 6(b) ( )( ) 2 3 5 3 5 5 = + − y ( ) 8 5 3 5 5 60 − − + ( )( ) 3 5 45 25 30 5 120 40 5 60 = + + − − + + 210 70 5 90 5 150 120 40 5 60 = + − − − + + M1 For substitution of their x and simplification with sufficient detail to imply no use of a calculator. Allow one numerical slip or sign error in the expansion of ( )( ) 2 3 5 3 5 5 + − or one sign error in the other terms. 20 5 = 2 A1 for all non surd terms = 0 A1 for 20 5

7 (a) In this question, all lengths are in metres and time, t, is in seconds. –10 0 10 20 30 40 50 displacement time 10 20 30 40 The diagram shows the displacement–time graph for a runner, for t 0 40 G G . (i) Find the distance the runner has travelled when t 40 = . [1] (ii) On the axes, draw the corresponding velocity–time graph for the runner, for t 0 40 G G . [2] –1 0 2 4 6 velocity time 1 –2 –3 – 4 –5 –6 3 5 10 20 30 40 (b) A particle, P, moves in a straight line such that its displacement from a fixed point at time t is s. The acceleration of P is given by t2 4 2 1 + - ` j , for t 0 2 . (i) Given that P has a velocity of 9 when t 6 = , find the velocity of P at time t. [3] (ii) Given that s 3 1 = when t 6 = , find the displacement of P at time t. [3]

7(a)(i) 110 (m) B1 7(a)(ii) B2 B1 for a line joining ( ) 0,5 and ( ) 10,5 B1 for a line joining ( ) 10, 2 − and ( ) 40, 2 − 7(b)(i) ( ) ( ) 1 2 2 4 = + + v t c M1 For ( ) 1 2 2 4 + k t 9 4 = + c M1 Dep for attempt to find c using 9 = v and 6 = t in their v ( ) 1 2 2 4 5 + + t A1 7(b)(ii) ( ) ( ) 3 2 1 2 4 5 3 = + + + s t t d M1 For ( ) 3 2 2 4 + k t 1 64 30 3 3 = + + d M1 Dep for attempt to find d using 1 3 = s and 6 = t in their s ( ) 3 2 1 2 4 5 51 3 + + − t t A1

Cambridge O Level Additional Mathematics · 4037

O-Level Additional Mathematics (4037) Calculus past paper questions

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