IJC_JC2_H2_Maths_2012_Questions_Paper_2

2 IJC/2012/JC2 9740/02/S/12 Section A: Pure Mathematics [40 marks] 1 The graph of ( )fy x = has a turning point at ( )0,2 and passes through the points ( )4,0 − and ( )4,0 . The graph of ( )fy x ′= has a point of inflexion at the origin. On separate diagrams, sketch the graphs of (i) ( )f 2 y x ′= − , [2] (ii) ( )fy x = , [3] stating the equations of any asymptotes, the coordi nates of any stationary points and points of intersection with the axes. 2 (i) By using the substitution 1x u= , show that 4 2 2 2 1 d 24 4 x x x π= −∫ . [4] (ii) O is the origin and A is a point on the curve 2 1 4 y x x = − where 2 2 x = . The region R is enclosed by the curve 2 1 4 y x x = − , the line OA , the line 4x = and the x-axis. Find the exact area of R. [4] 3 A sequence of positive numbers 1 2 3 , , , ... x x x satisfies the recurrence relation 1 10 3 , n n x x + = − for 1, 2, 3, ... n = . (i) Given that the sequence converges to l, find the value of l. [2] (ii) Prove that ( ) ( ) 2 2 1 3n n x l l x + − = − . [2] (iii) Use the result in part (ii) to show that if nx l < , then 1nx l + > . [2] 4− 4 y x O ( )fy x = 1y = 2x = 2x = − 2 ( )fy x ′= 2x = 2x = − O y x

3 IJC/2012/JC2 9740/02/S/12 4 A sector of angle θ radians and radius 4 cm π is used to form the curved surface of a right circular cone such that there is no overlapping. Prove that the volume of the cone is 2 4 6 3 8 4 cm 3 π π θ θ − . [3] Show that the maximum volume of the cone is 43p q π cm 3 as θ varies, where p and q are integers to be determined. [5] [The formula for the arc length of a sector of a ci rcle is s r θ= .] 5 The diagram shows a solid with a horizontal rectang ular base OABC in which the lengths of OA and OC are 12 m and 8 m respectively. M and N are the mid-points of OC and AB respectively, and D and E are 7 m and 5 m vertically above M and N respectively. The point O is taken as the origin for position vectors. Perpe ndicular vectors i, j, k are such that i and j are parallel to OA and OC respectively, and k is perpendicular to the plane OABC . (i) Find a vector equation of the line DB . Hence find the foot of perpendicular from E to the line DB . [5] (ii) Find the acute angle between line DB and the plane OBE . [4] (iii) Find the length of projection of DE onto DB . [3] (iv) State, giving a reason, whether lines DB and AC are coplanar. [1] M D O B A E C N 7 5 8 12 i j k [Turn over

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