PJC_H2_MATH_P2_SOLUTIONS

PJC2011 [Turn Over] 1 PJC 2011 JC2 H2 Mathematics End of Year Paper 2 Solution 1 32Let sequence, then 1 8 4 2 2 27 9 3 4 64 16 4 8 11 4Using GC: , , , 062 3 rTa rb rc r d abcd abc d abc d ab c d ab cd             32114 623 rTrrr 2(i) 3 21 14 x x   21 104 5 04 4 5 ---------------(1) x x x x x      (2 e ) e 1(1 4e ) e 2e 1 1e4 xx xx x x       1 2 Let lim i.e as , and then 3 30 11 1 2 2 11 3 11 3 or (NA since >0)22 nn nn n Xn X X x               Note: Graph is not a straight line 3y xx  1 1 from graph of 3 for 0 , 0 30 0 nn nn nn y xx xy xx xx xx           – 4 5 + + –

PJC2011 [Turn Over] 2 Replace in (1) with e then 4 e 5 since e 0 for all 0e 5 ln 5 ln 5 x x x x x x x x             4(i)  04 2 11 OL 4 2 322 06 3 OC OD                   4(ii)  11area sin 22 24 1 302 30 0 1 122 12 1 288 6 22 OAL OL OA AOL OL OA            

4(iii) 0 using dot 22 1 1 14 14 cos 33 6cos 14 64.6 ML OD                      5(i) 2 22 22 22 2 12 1 1 4422 4 1 2 44 44 24 xxx x x xx xx x         

PJC2011 [Turn Over] 3 5(ii) 22 1 0 0 21 11 1 4d 4 4 s i n22 2 2 1 4 4sin 42 4 kk xxx x x kkkk a                   5(iii) 2244yx 2 2 2 12 xy  2 0 1 4d2 k R xx  5(iv) 1 Required area 4 with 1 1 3 4sin 2 3 4 6 2 3 3 Rk          6(i) 31 zz 6(ii) arg 3 i arg66 arg66 6 arg 03 0a r g 3 31 lies on excluding point ' ' z z z z zz zO A C O                 6(iv)  23 3 3 tan 3 1 3 Hence arg 2 3i23 AB AB z            2 1 k R 3  1 A B  23 O C Locus of Q Locus of P Locus of z Im(z) Re(z) 3 −1

PJC2011 [Turn Over] 4 7(i) N = 90, n = 10, so k = 9. List all the faculty staff by their names in alphabetical order and ch oose a random number from 1 to 9. Say we have chosen the number 4, then we select the 4 th, 13th, 22nd, 31st, 40th,49th, 58th, 67 th,76th, 85th names from the list to form the sample of 10. 7(ii) Choose the number of Professor, Seni or Lecturers and Lecturers in the following: Professor Senior Lect urer Lecturer Total Population 9 18 63 90 Sample 1 2 7 10 Compile 3 lists of names – Professo r, Senior Lecturers and Lecturers. Perform simple random sampling from the separate lists of Professor, Senior Lecturers and Lecturers to select 1 Professor, 2 Senior Lecturers and 7 lecturers to form a sample of 10. Stratified