2025 Prelim AM Bedok View P1 MS

Bedok View Secondary School Department of Mathematics Mark Scheme 1 Year 2025 Level & Stream 4E5N Type of Assessment Preliminary Exam Subject & Paper Additional Math P1 Qns Working 1 24 6 6 4x xy y− − = ----(1) 4yx−= ------(2) Sub (2) into (1): ( )( ) 2 22 2 2 4 6 (4 ) 6(4 ) 4 4 24 6 24 6 4 2 30 28 0 15 14 0 1 14 0 14 or 1 10 or 3 x x x x x x x x xx xx xx xx yy − + − + = − − − − = + + = + + = + + = =− =− =− = 2 9sin 2sec 29sin cos 2sin cos 9 42sin cos 9 4sin 2 9 xx x x xx xx x =− =− =− =− =− Basic angle = 1 4sin 9 −   = 0.46055 ( )2 0.46055, π 0.46055 0.46055 or 2.68104 0.230 or 1.34 x x =− − − =− − =− −

Bedok View Secondary School Department of Mathematics Mark Scheme 2 Qns Working 3a 3b 11 1 Principal range of cos is 0 cos π, 23 πhence principal value of cos is 24 πinstead of . 4 xx−− −  − − L.H.S: 1cot tan tan tanA A A A+ = + = cos sin sin cos AA AA+ = 22cos sin sin cos AA AA + = 1 sin cosAA = sec cosecAA = RHS (shown) OR RHS: 11sec cos ec cos sinAA AA= = 22 22 1 cos sin sin cos cos sin sin cos cos sin cos sin sin cos cos sin tan cot LHS (shown) AA AA AA AA A A A A AA AA AA += =+ =+ = + =

Bedok View Secondary School Department of Mathematics Mark Scheme 3 Qns Working 4 By long division, ( ) ( ) ( ) ( ) ( ) ( ) 33 2 32 2 2 2 22 22 2 4 2 4 442 8 8 42 2 8 8 4 222 8 8 4 2 2 xx x x xxx xx xx x x A B C xxx x x x x A x Bx x Cx ++ = +++ − − +=+ + − − + = + + +++ − − + = + + + + Let 0x= : 44 1 A A = = Let 2x=− : 12 2 6 C C − =− = Let 1x= : 8 8 4 9 3 6 9 B B − − + = + + =− ( ) ( ) 3 22 2 4 1 9 6 2 222 x xxx x x + = + − + +++ 5a ( ) ( ) ( ) ( ) 2 22 2 22 32 f '( ) 3 23 3 x x x a x x x ax x + − + = + − − += + ( ) ( ) 2 22 22 2 2 2 23 0, for 1 3 Since 3 0, 2 3 0 2 3 0 Given that 1 ( )( 1) 0 0 By comparing: 3 21 2 3 1 1 x ax bx x x x ax x ax bx x b x x bx x b b ab a a − − +    + + − − +  + −   − −  − − +  =− =− − =− = 5b

Bedok View Secondary School Department of Mathematics Mark Scheme 4 Qns Working 6 ( ) 2 2 2 68 Discriminant ( 8) 4( 6)( ) 64 4 24 y k x x k kk kk = − − + = − − − = − + For curve that does not intersect the x-axis, Discriminant < 0 2 6 16 0 ( 2)( 8) 0 2 or 8 kk kk kk − −  + −  −  The curve has a minimum point: 60 6 8 k k k −   7a ( ) ( )d 1 e e 1 ed 2e e x x x xx xxx x − − − −−  − = − − =− 7b ( ) ( ) ( ) ( ) 2e e d 1 e 2e d e d 1 e e d 2e d 1 e e d 2e 1 e 2e e e ee x x x x x x x x x x x x x x x xx x x x c x x x x c x x x x c x x x c xc xc − − − − − − − − − − − − − − − −− − = − + − = − + − =− + − + =− − − + =− − + + =− − +    

Bedok View Secondary School Department of Mathematics Mark Scheme 5 Qns Working 8a ( ) ( ) ( ) ( ) 2 2 2 1 d2 d 1 2 d 1 2 1 d 21 2 1 A