2022 HS AMath Prelims P1 Ans

2022 Prelim A Maths Paper 1 Marking Scheme Qn Solution Marks Remarks 1 2 38y x x= + − ----- (1) 37 22yx=− ----- (2) (1) = (2) 2 38xx+− = 37 22x− ( )( ) 22 3 9 0 2 3 3 0 3 or 32 5 or 84 xx xx xx yy + − = − + = = =− =− =− Coord of A and B are ( )35, & 3, 824  − − − . Distance btw A and B ( ) ( ) 221.5 3 1.25 8 8.11 units = + + − + = [M1] [M1] [M1] [A1] Accept exact surd 1053 4 2 ( )( ) ( ) 22 78 2 1 12 1 1 1 x A B C xxx x x + = + ++−+ − − ( ) ( )( ) ( ) 27 8 1 2 1 1 2 1x A x B x x C x− = − + + − + + Let x = 1, 15 = 3C , C = 5 Let 1 2x=− , 293 22 A=−  , A = 2 Let x = 0, 28 2( 1) (1)( 1) 5(1)B= − + − + , B = ─1 ( )( ) ( ) 22 7 8 2 1 5 2 1 12 1 1 1 x xxx x x + = − ++−+ − − [M1] [M1] [A3]

3 ( ) 2 2 43 3 23 19 8 3 2 3 = 2 3 2 3 38 19 3 16 3 24 = 23 = 62 35 3 62, 35 ab ab − += + −−  +− −−+ − − = =− [M1] [M1] [A1] Accept alternative method ( )( ) ( ) 2 3 2 3 4 3ab+ + = − 4 ( ) 2 2 2 At ( 1, 4), 4 ( 1 2) 4 y a x k ak ak = + + − = − + + += Since a > 0 & k > 0 as curve lies above x-axis, let a = 1 , then k = 3 A possible equation for the curve is ( ) 223yx= + + [M1] [M1] [M1] [A1] (Any other suitable equation satisfying the relevant conditions accepted) 5(a) (b) (c) max min 1 ( 3) 1(shown)22r + + −= = =− [B1] Period of curve = 2 8 , 41 q q  == Amplitude = max min 1 ( 3) 222 − − −== , 2p=− [B1] [B1] Equation of the curve is 2sin 14 xy=− − [B1] 6(a) ( ) ( ) 2 f ' 6 2 32 x x dx x x c =+ = + +  At '1, f ( ) 11xx=− = ( ) 211 3 1 2( 1) 10 c c = − + − + = ( ) 2f ' 3 2 10x x x= + + [M1] [M1] [A1]

(b) ( ) ( ) 2 32 f 3 2 10 10 x x x dx x x x d = + + = + + +  At ( ) ( ) ( ) 321,10 , 10 1 1 10( 1) 20 d d − = − + − + − + = ( )f x = 32 10 20x x x+ + + [M1] [A1] (c) For y = f(x) to have stationary points, set ( )f ' 0x = . 2 22 3 2 10 0 4 2 4(3)(10) 116 0 xx b ac + + = − = − =−  ( )f ' 0x = has n o real solution, ie no stationary points. [M1] [M1] [A1] Solve equation to show no real roots accepted No marks if no real roots is not mentioned 7(a) ( ) ( ) 214.5, 4.5 18 4.53 729 8 xV   = = − = 729 8 81 98 dV dt  == cm/s ( or 110 8 cm/s) [M1] [M1.A1] Accept 10.125 31.8or cm/s (b) ( ) ( ) 23 2 183 36 33 V x x dV xxdx   =− =− ( ) ( )( ) 236 4.5 3 4.53 15 4.5,  3 4 Ax dV dx t   =− = = 81 8 135 4 Using 0.3 dV dV dx dt dx dt dx dt   = == The water level is rising at a rate of 0.3 cm/s [M1] [M1] [M1. A1]

8(a) ( )( ) 33 22 sin cos sin cos sin sin cos cos xx x x x x x x + = + − + [B1] (b) ( )( ) ( ) 33 22 22 sin cos sin cos sin cos sin sin cos cos sin cos 1sin 2sin cos cos2 11 s