Paper 2

1 2018 H2 Mathematics 9758 Paper 2 Suggested Solutions Section A (Pure Math) 1(i) 1 3d1 15d3 y yx =− 1 31 15 d 1 d3 y y x −  −= 2 31 153 21 33 y xc  − =+       , where c is an arbitrary constant 2 391 1523 y x c − = + When x = 0, y = 69 ( ) 2 391 69 1523 c −= c = 18 2 391 15 1823 yx − = + 2 3 3 2 3 2 3 2 12 15 439 12 15 439 12 4 1539 23 4 459 yx yx yx yx  − = + − = + = + + = + + 1(ii) Since d 4d y x = , 1 31 15 43 1 15 643 237 y y y  −= −= = 3 223 4 45 2379 x + + =

2 From GC, x = 54 Coordinates of the point is (54, 237) 2(a) Since the coefficients of the equation are real, the complex roots occur in conjugate pairs. Therefore since 2 – 3i is a root, then 2 + 3i is also a root. So we have ( )( )( ) ( )( ) 4 3 2 2 22 4 20 56 2 3 2 3 4 4 13 4 , x x sx x t x i x i x ax b x x x ax b − + − + = − + − − + + = − + + + where ,ab  Comparing coefficients of 3x , 20 16 4 a a − = − =− Comparing coefficients of x, 56 4 13 4 56 13( 4) 1 ba b b − =− + = + − = Comparing constants, 13 13 tb= = Comparing coefficients of 2x , 4 52 1 4( 4) 52 69 s b a= − + = − − + =  we have ( )( ) ( )( ) 4 3 2 2 2 22 4 20 69 56 13 4 13 4 4 1 4 13 2 1 x x x x x x x x x x x − + − + = − + − + = − + − The other roots are 2 + 3i and 1 2 . Note: 1 2x= is a repeated real root. (bi) 3 3 27 27 0 w w = −= Since 3w= , ( )( ) 32 27 3w w w aw b− = − + + , where ,ab 

3 Comparing constants, 27 3 9 b b − =− = Comparing coefficients of w, 03 3 3 ba ba =− = =  we have ( )( ) 32 27 3 3 9w w w w− = − + + To find the other roots, let 2 3 9 0ww+ + = 3 9 4(1)(9) 2 3 27 2 3 3 3 2 w i −  −= −  −= −= The other roots are 3 3 3 22 i− + and 3 3 3 22 i− − (bii) Let 1w = 3, 2 3 3 3 22 iw −=+ and 3 3 3 3 22 iw −=− 1 3w = 22 2 3 3 3 9 27 32 2 4 4w −= + = + =   22 3 3 3 3 9 27 32 2 4 4w −−= + = + =   1arg 0w = 1 2 33 22arg tan 3 3 2 w  −= − = − 1 3 33 22arg tan 3 3 2 w  − − =− + =− −

4 ( )1 3 cos0 sin 0wi = + , 2 223 cos sin33wi =+ and 3 223 cos sin33 223 cos sin33 wi i   −−=+ =− (biii) 22 33 1 2 3 3 3 3 27 ii w w w e e  −  ==      1 2 3 2 2 2 23 3 cos sin 3 cos sin3 3 3 3 1 3 1 33 3 3 2 2 2 2 0 w w w i i ii       + + = + + + −          −−= + + + −          = 3(i) 5 5 5 4 4 4 2 0 1 5 4 3 AD BC OD OA OC OB OD OC OB OA OD = − = − = − + −           = − + −                − =− ( )5, 4,3D − − 3 Re(w) Im(w) 3 3

5 (ii) 5 0 5 4 0 4 2 10 8 EC −−           = − =           −      5 0 5 4 0 4 0 10 10 EB            = − =           −      5 5 8 4 4 90 8 10 40 EC EB −−            =  = −          − − −     