RVHS_9758_2025_Prelim_P1_Solutions

River Valley High School 1 2025 H2MA Prelim Paper 1 1 Solution [4] Complex Number (a) Method 1 ( ) ( ) ( ) ( ) 32 f 1 0 1 1 1 0 0 a b c d a b c d −= − + − + − + = − + − + = 0b c d a− + − = ------------------- (1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 32 f 4 0 4 4 4 0 52 47 15 8 4 0 i a i b i c i d a i b i c i d += + + + + + + = + + + + + + = Comparing real parts, 15 4 52 0b c d a+ + + = ----------------- (2) Comparing imaginary parts, 8 47 0b c a+ + = -------------------(3) Solving (1), (2) and (3) using the GC, 7ba=− , 9ca= and 17da= Method 2 Since all the coefficients of ( )f x are real, 4i+ and 4i− are conjugate roots of ( )f0 x = . ( ) ( ) ( )     32 2 32 32 f 4 i 4 i 1 8 17 1 7 9 17 7 9 17 x ax bx cx d a x x x a x x x a x x x ax ax ax a = + + + = − + − − +     = − + + = − + + = − + + 7ba=− , 9ca= and 17da=

River Valley High School 2 2 Solution [6] Inequality ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) 2 2 2 2 95 11 9 5 1 011 9 4 5 011 44 011 2 011 x xx xx xx xx xx xx xx x xx +−+ − + − +− − − − + +− ++ +− + +− 1x− or 1x  , 2x− (Alternatively, 2x− or 21 x−  − or 1x  ) 2 9 e 5 .1 e e 1 x xx +−+ Replace x with ex in the previous inequalitry, e1x − or e1x  (no solution since e0x  for x ) 0x (Note e2x − is always true) Thus 0x . Alternative presentation: e2x − or 2 e 1x−  − or e1x  (no solution since e0x  for x ) 0x Thus 0x .

River Valley High School 3 3 Solution [6] Complex Numbers ( ) 2 1 2 1 7 0z i z i− + + + = In the above equation, 1 , 1 2i, 1 7ia b c= =− − = + ( ) ( )( ) ( ) 2 2 4 2 1 2i 1 2i 4 1 1 7i 21 1 2i 4i 3 4 28i 2 1 2i 7 24i= 2 b b acz a −  −= +  − − − += +  − − −= +  − − We want to find 7 24i−− . Let 7 24i i , ,x y x y− − = +  ( ) ( ) ( ) 2 22 7 24i i ---(*) 7 24i 2 i xy x y xy − − = + − − = − + Comparing real and imaginary parts, 22 7 ---(1) 2 24 12 ---(2) xy xy x y − =− =− −= Substitute (2) into (1): ( )( ) 2 2 2 2 42 42 22 12 7 144 7 144 7 7 144 0 16 9 0 yy yy yy yy yy − − =− − =− − =− − − = − + = 22 22 16 0 or 9 0 16 9 4 (reject as ) yy yy yy − = + = = =− =  When 4, 3yx= =− . When 4, 3yx=− =

River Valley High School 4 The square roots of 7 24i−− are 3 4i− and 3 4i−+ . Hence, ( )1 2i 3 4i 2z +  −= 1 2i 3 4i 1 2i 3 4i or 22 2 i or 1 3i zz zz + + − + − +== = − =− +

River Valley High School 5 4 Solution [6] Summation (a) ( )