SAJC_9758_2022_Prelim_P2

1 Section A: Pure Mathematics (40 marks) 1 A sequence 1 2 3, , ,...u u u is such that 1 ! nu n= for n + and 1 1 ! nn nuu n − −=+ for all 2n . (i) Find 2 1 ! N n n n= −  . [3] (ii) Give a reason why the series in (i) is convergent and state the sum to infinity. [2] (iii) Hence find ( ) 5 8 2 1! N n n n + = − − in terms of N. [3] 2 The equations of three planes 1 , 2 and 3 are 9, 3 2 10 and 5x py z x y z x ay z− + = − − = − − = respectively, where a and p are constants. The line 1l has equation 2 4 3 ( 3 ) = − + + − +r i j k i j k , where  . (i) Given that 1l does not intersect with 1 , show that 4p=− and find the shortest distance between the line 1l and 1 . [3] (ii) The line 2l is the reflection of the line 1l in 1 . Hence, or otherwise, find the Cartesian equation of the plane that contains the line l2 and is parallel to 1 . [2] (iii) Given that the line 3l lies on both 2 and 3 , find a vector equation of 3l , leaving your answer in terms of a. [3] (iv) Let  be the acute angle between l3 and 1 . Find the value(s) of a if 3sin 18 = . [3]

2 3 In this question you may use expansions from the List of Formulae (MF26). (i) Find the Maclaurin expansion of ln(1 cos3 )x+ in ascending powers of x, up to and including the term in 4x , for 0 π 3x . [4] (ii) Use your expansion from part (i) and to find an approximate value for 0.5 0 ln(1 cos3 ) dx x x + , giving your answer to 5 decimal places. [2] (iii) Use your calculator to find the value of 0.5 0 ln(1 cos3 ) dx x x + up to 5 decimal places. [1] (iv) With the aid of a suitable diagram, comparing your answers in (ii) and (iii), comment on the accuracy of your approximations. [2] 4 A curve C has parametric equations 23 9 , 12xy ttt =−−= , where  3, 3t− . (i) Sketch the graph of C, indicating the coordinates of the end points. [2] The normal to the curve C at the point P is given by 9 2 83yx+= . (ii) By finding the gradient of the curve C at the point with parameter t , calculate the value of t at the point P. [3] Given that the normal to curve C at P intersects the curve C at another point Q with parameter q. (iii) Show that 329 2 108 101 0q q q+ − − = at Q. Hence, find the coordinates of Q. [4] (iv) Find the area bounded by the curve C and the normal to the curve at point P, giving your answers correct to 3 significant figures. [3]

3 Section B: Probability and Statistics (60 marks) 5 A group of 10 people consists of 9 men and 1 woman. Find the number of ways which the group can be seated at a round table with 10 identical chairs if (i) 2 particular men, Caleb and James are not seated beside the woman, but are seated next to each other. [3] The 10 identical chairs at the table are replaced with 10 chairs of different colours. (ii) Find t