RVHS 9758 2023 Prelim P1 Solution

River Valley High School 1 2023 H2MA Prelim Paper 1 1 Solution [6] Inequality P1 Q1 (i) ( ) ( )( ) 2 2 2 4 41 012 21 01 21 xx xx x xx −+ <+− − <−+ Method 1: ( ) ( )( ) ( ) ( )( ) 2 2 21 01 21 Since 2 1 0 for all real values of , 1 210 x xx xx xx − <−+ −≥ − +< 1 or 12xx∴ <− > Method 2: ( ) ( )( ) 2 21 01 21 x xx − <−+ 1 or 12xx∴ <− > (ii) ( ) ( )( ) ( ) ( ) 21 21 2 2 21 012 2 22 1 0 1 2 22 x xx x xx + + − ≤+− − ≤ +− Replace with 2 ,xx 1 2x=− 1x= 1 2x= + + − − o ο ο

River Valley High School 2 112 OR 2 1 OR 2 22 (rejected, since 0 1 2 is positive for all real values of .) x xx x xx x <− > = >= −

River Valley High School 3 2 Solution [6] Complex Numbers P1 Q2 (i) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 22 22 22 22 22 11 *1 i 2 i1 1 i*1 i 2 1i 1 i1i 2 1i 1 2i 1i 11 1 211 1 2 2 11 2 1 4 222 1 2 44 1 z z a a a a a a a a aa a aa a aa a − =+ +− =++ −+ =−+ −+ =−+ −+ = ++ − ++= + −+= + − += + 2 4 30 1 or 3 (rejected since 2) aa aa − += = < (ii) Consider 1arg * n z zw −  ( ) ( ) ( ) 1arg * arg 1 arg * arg arg i arg 1 i 4 2 44 2 zn zw nz z w n n n π π ππ π −=  = −− − = − −−  = −− −  = If 1 * n z zw −  is purely imaginary and negative, then:

River Valley High School 4 1 3 7 11arg , , ,...* 22 2 n z zw ππ π− = or 3 22 kπ π+ , k +∈  Thus, 3 7 11, , , ...2 22 2 3, 7, 11, ... n n π ππ π= = 3 smallest positive values of 3,7,11n=

River Valley High School 5 3 Solution [7] Summation P1 Q3 (i) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) 11 2 22 2 3 2 where 2 121 11 12 1 1 1 11 22 11 2 1 2 n nnu uu n n nn nn n n n n n nn nnn nn nn n nn nn −+ −+ ≥ = −+−+ +− − ++−= −+ +− ++ −= −+ = − = − 2A= (ii) 3 2 1N n nn= −∑ ( ) 3 2 11 2 1 23 12 2 1 22 2 1 2 N n N n nn n nn u uu u uu = −+ = = − = −+ −+ = ∑ ∑ 23 2uu+− 4 3 u u + + 45 3 21 2 ... ... ... 2N NN uu u uu− −− −+ +−+ 21 2NNuu−−+− 1 N N u u − + + ( ) 1 12 1 2 1 2 NN NN uu uuu u + +            −+ = −−+ 1 11 1122 1 11 1 1 22 1 NN NN = −− + + = −+ +

River Valley High School 6 (iii) 3 2 1 11 1 1lim 22 1 11Since 0 as , 0 as 1 It converges. Nn nn NN NNNN ∞ →∞= = −+ −+  → →∞ → →∞ + ∴ ∑ (iv) ( )( )( )3 1 21 N n n nn= −−∑ ( )( )( ) ( ) 1 13 1 Replace with 1.12 11 1 nN n nnn nn += += = ++− +− +∑ ( )( )( ) 1 2 1 3 2 1 11 1 N n N n n nn nn − = − = = −+ = − ∑ ∑ 11 1 1 22 1 NN =−+ − (Alternative, but not recommended) ( )( ) ( )( )( ) ( )( )( ) 3 22 1 12 1 3 11 11 1 ( replaced by 1)12 1 12 NN nn nN n N