HCI_9758_2025_Prelim Paper 2

Section A: Pure Mathematics [40 marks] 1 The region A is bounded by the curves 1yx=+ , 72yx=− , the x-axis and the y-axis. (a) Find the exact area of A. [4] (b) Find the volume of the solid obtained when A is rotated through 2 radians about the y-axis. [3] 2 It is given that ln sin 4yx  =+   , where 3 44 x−   . (a) Show that 22 2 dd 10dd yy xx + + = . Hence find the first four non -zero terms of the Maclaurin expansion of y, leaving your answer in exact form. [6] (b) Verify the result obtained in part (a) using standard series from the List of Formulae (MF27). [5] 3 The parametric equations of the curve C are 1 3cosec and 2cot 3, where 0xy    = − = −   . (a) Show d2 secd3 y x =− . Hence f ind the equation of the normal to C at the point where 4  = . Give the equation in the form y Ax B=+ , where A and B are exact constants to be found. [4] (b) Show that the normal found in part (a) will cut C again. [2] (c) Find the Cartesian equation of C. [2] (d) Sketch C, indicating clearly its key features. [3] (e) Find the range of values of m such that there is no intersection between the line ( )13y m x= − − and C. [2]

2 4 Let A, B and C be the points on the same plane with position vectors a, b and c respectively and + + =a b c 0 . It is given that vectors a, b and c are unit vectors. (a) (i) By considering cc , find the value of ab . [3] (ii) Find the angle AOB . [2] (iii) Draw the position vectors a, b and c on a single diagram. Using your diagram, identify the type of triangle ABC. [2] (b) The point D has position vector +ab . Find the area of the quadrilateral ACBD. [2] Section B: Probability and Statistics [60 marks] 5 The eleven letters in the word INSPIRATION are each printed on separate, identical cards. (a) Find the number of ways in which the cards can be arranged in a row if, (i) there are no restrictions, [1] (ii) the letters N are together or the letters I must all be separated, but not both. [3] (b) Three of the eleven cards are removed at random. Find the probability that the letters on the eight cards left behind are all distinct. [2] 6 A basketball free throw game involves a team of two players, Ben and John. The game consists of at most three throws and the moment a player makes a successful shot, the team wins, and the game will end. The game uses the following rules. • Only one player is selected for each game and the probability that Ben is selected in a game is 0.7. • The probabilities that Ben and John make a successful shot in any single attempt are 0.1 and 0.07 respectively. • The shots are independent of each other. (a) Find the probability that the team wins the game. [3] (b) The team did not win the game. Find the proba