RVHS_8865_2022_Prelim_Solutions_vetted

1 ©RIVER VALLEY HIGH SCHOOL 8865/01/2022 Section A: Pure Mathematics [40 marks] 1 Solution [4] Inequalities For ( ) 21 24 0k x x k− − −  , Discriminant < 0 ( ) ( )( ) 2 2 2 1 0 & 24 4 1 0 1 & 24 4 4 0 1 & 6 0 1 & ( 3)( 2) 0 1 & 2 or 3 k k k k k k k k k k k k k k k −  − − − −   + −   − −   − +   −  taking intersection, 3k  2 Solution [7] Differentiation & Integration Techniques (i) Let ( ) 3 212yx=− ( )( ) ( ) 11 22 d3 2 1 2 3 1 2d2 y xxx = − − =− − (ii) ( ) ( ) ( ) 1 2 1 2 1 d 1 2 d 12 12 1 22 12 x x x x x c xc − =− − −=+ − =− − +  (iii) ( )6 5 3 1 2 2xx− =− − − ( ) ( ) ( ) ( ) 3 2 3 2 3 1 2 265 dd 1 2 1 2 23 1 2 d 12 13 1 2 d 2 d 12 1 2 2 1 2 1 2 2 1 2 xx xx xx xx x x x x x x x c x x c − − −− = −− = − − − − = − − − − = − − − − + = − + − +   

2 ©RIVER VALLEY HIGH SCHOOL 8865/01/2022 3 Solution [8] Solving Equations using GC (i) 22 2 2 11e 8 e4 4 2 for stationary points, let 0 11 e042 1e 2 1ln( )22 1ln or ln 44 xx x x x dyy dx dy dx x x = − +  = − = −= = = =−  (ii) (iii) 2 2 e ln( )4 e 8 8 ln( )4 Sketch =8 ln( ) from GC, 8.51 or 2.48 x x x x x x yx x − =− − − + = − − −− =− − 4 Solution [8] Graphing + Appln of Differentiation & Integration (i)

3 ©RIVER VALLEY HIGH SCHOOL 8865/01/2022 (ii) ( ) 12 1 2 1 2 3 2e d 2 2 e 4ed x xx y y x − −− =+ = − =− When 2x= , ( ) ( ) 1 2 2 3 1 2 2 3 3 2e 3 2e d 4e 4ed y y x − − − − = + = + =− =− Equation of tangent at x = 2: ( ) ( ) ( ) 33 3 3 3 33 3 2e 4e 2 4e 8e 3 2e 4e 3 10e yx yx x −− − − − −− − + =− − =− + + + =− + + (iii) Required area ( ) ( ) 2 1 2 3 3 0 2 3 2e 4e 3 10e d 2.07105 2.07 units x xx− − −= + − − + + =   5 Solution [13] Graphing + Application of Differentiation (i) 32 2 30 585 1980 12000 d 90 1170 1980d x m m m x mmm = − + + = − + At stationary point, ( ) ( )( ) ( ) ( ) 2 2 d 0d 90 1170 1980 0 1170 1170 4 90 1980 2 90 1170 810 2 90 2 or 11 x m mm m = − + = −= = = When 2m= , 13860x= When 11m= , 2925x= 2 2 2 d 90 1170 1980d d 180 1170d x mmm x mm = − + =−

4 ©RIVER VALLEY HIGH SCHOOL 8865/01/2022 When 2m= , ( ) 2 2 d 810 0d x m =−  ( )2,13860 is a maximum point When 11m= , ( ) 2 2 d 810 0d x m = ( )11,2925 is a minimum point (ii) (iii) Required area ( ) 12 32 0 30 585 1980 12000 d 105120 m m m m= − + + =  The total production of fishballs by Todo Fishball Company for the fiscal year 2021 is 105120 kg. (iv) When 0d = , 40 5 35y= − = Therefore, an employee produces 35 kg of fishball immediately after the training programme. (v) The model is suitable as the employee shows gradual improvement in the efficiency over time after the training and the improvement tapers off which is realistic. d y 35 40

5 ©RIVER VAL