TJC_H2_MATHS_P2_Solutions

1 © TJC 2015 TJC/MA 9740/Preliminary Exam 2015 TEMASEK JUNIOR COLLEGE, SINGAPORE Preliminary Examination 2015 Higher 2 Mathematics Paper 2 Section A: Pure Mathematics [40 marks] 1 Sketch, on an Argand diagram, the locus of the point representing the complex number z such that 5πarg( 3 i) . 6z   [2] Give a geometrical description of the locus of the point representing the complex number w such that i,wk where k is real. [1] (i) Given that the two loci intersect at exactly one point, show that k = a or kb where a and b are real constants to be determined. [3] (ii) In the case when k takes the value of a, find the complex number representing the point of intersection, in the form x + iy, where x and y are exact. [3] [Solution] Locus – half line with correct argument and excluded point, passes through origin The locus of w is a circle centered at (0, -1) and radius k units. (i) Two loci intersect exactly at one point: k CD or k AC In triangle ACD: π 3sin 62 3 CD CD   and 3AC , 3 2k or k 3 i.e. 3 2a , 3b (ii) Consider triangle BCD:  BCD = π π π 2 6 3 π 3 1 3cos 3 2 2 4BC k     , π 3 3 3sin 3 2 2 4BD k     At point D : 3 3 1, (1 )4 4 4xy    ,  complex number representing point D = 31 i44 B D –1 π 6 Re Im locus of z 5π 6 k A( 3, 1 ) C O locus of w

2 © TJC 2015 TJC/MA 9740/Preliminary Exam 2015 Alternatively, π11cos 32OD  complex number representing point D = 1 π π 1 3 1cos isin i2 6 6 2 2 2 31 i44                    2 (a) Given that the sequence 5, 11, 17, , x is arithmetic, solve the equation 5 11 17 2760 x     . [4] (b) Mr Tan set aside $80,000 for his two sons. On the first day of the year that his sons turned 7 and 17 years old, he deposited $ x into the younger son’s bank account and the remaining sum of money into the elder son’s bank account. Mr Tan adds a further $1000 into the younger son’s account on the first day of each subsequent year. The bank pays a compound interest at a rate of 2% per annum on the last day of each year. Each son will withdraw the full sum of money from his account (