HCI 9758 2023 Prelim P2

Section A: Pure Mathematics [40 marks] 1 A curve G has equation 1 coscosyx x= + . The region bounded by G , the line π 3x= , the x-axis and the y-axis is rotated through 4 right angles about the x-axis. Find the exact volume generated, giving your answer in terms of π . [4] 2 (a) Use the substitution 31ux= − to find 5 3 d 1 x x x−∫ . [4] (b) Hence find 8 3 d 1 x x x−∫ . [4] 3 A complex number 1z is given by 1 ( 3) izaa= +− , where a is a positive real constant. It is given that 1arg z θ= , where π 02 θ−<< . (a) Find, leaving your answers in terms of θ, (i) 1arg( 2 )z− , [2] (ii) 1arg( 2 )za− . [2] Another complex number 2z is given by 2 1 3iz = + . (b) Without using a calculator, find the range of values of a such that 2 12 12 10.Im ( ) zz zz ≤ [5] 4 A curve C has parametric equations 2 3,1x t yt= = + for t∈  . (a) Sketch the graph of C , giving the coordinates of any axial intercepts. [2] (b) Without using a calculator, find the equation of the tangent to C at the point P where 1t = . [2] (c) The tangent to C at P meets the curve again at the point Q . Find the exact coordinates of Q . [3] (d) Hence find the exact area of the region bounded by C and the tangent to C at P . [3]

2 © HCI 2023 5 The equations of two planes 1p and 2p are 2 3, 2 3 1, x sy z xyz ++= − −+= respectively, where s is a negative real constant. (a) Find a vector parallel to both 1p and 2p , giving your answer in terms of .s [2] (b) It is given that 1p and 2p intersect in a line l , and l meets the xz -plane at a point A . Find the coordinates of A . [3] (c) The shortest distance between the origin O and 1p is 3 6 . Find the value of s . [2] It is now given that 2,s=− with the normal vectors to 1p and 2p denoted as 2 2 1  = − 1n and 1 2 3  = − 2n respectively. It is given that 1p and 2p divide the real space into four regions 1R , 2R , 3R and 4R as shown in the following two -dimensional diagram (not drawn to scale). (d) Relative to the origin O , the point B has position vector k . Determine whether B lies in 1R , 2R , 3R or 4R , justifying your answer. [2]

3 © HCI 2023 Section B: Probability and Statistics [60 marks] 6 Eight students are shortlisted to be prize recipients for a competition. Two students are awarded the first prize and another two students are awarded the second prize. The eight shortlisted students are to stand in a single row for a photo- taking session with a guest of honour. Find the number of different possible arrangements if the two first prize awardees must stand together while the two second prize awardees also must stand together during the session. [2] After the photo-taking session, the eight students proceed for a tea reception and are to be seated at a round table with 10 identical seats. If 4 students are to be seated on each s