AM4 4049 BSSS 2023 P1 MS

Marking Scheme BSSS AM4 4049 P1 2023 1 ( ) ( )( ) 2 2 2 321 4 32 1 4 4 2 3 ... eqn A 2 3 40 2 3 4 ... eqn B Sub eqn B into A, 4( 4) 16 ... eqn C Sub eqn C into B, 1623 4 2 48 4 2 4 48 0 2 24 0 6 40 6 or 4 22 or 3 xy yx xy x y xy xy xy xy y x x x xx xx xx xx xx yy += + = += + += += − −= =− +− = − −= − +−= +−= + −= = −= = = ( ) 4 2 intersections are 6, 2 and 4, 43 − ∴ −−  M1 – form eq in terms of x/y M1 - factorization M1 – solve for x & y A1, A1 2a ( ) ( ) 30 18 2 22 2230 18 2 22 22 60 36 2 30 2 36= 42 24 6 2= 2 =12+3 2 + + −+= × + − +−− − + M1 A1 2b ( )( ) ( )( ) ( )( ) Consider area of XYZ, 1 4 2 2 2 sin135 15 9 22 22 2 2 15 9 2 2 2 1 2 15 9 2 1 59 2 212 21 21 15 2 9 2 15 9 22 21 2 3 62 3, 6 oab ab ab ab ab ab ab ++ = + ++ = + ++ = + +−+= × +− +×− −+= − += + ∴= =  M1 M1 A1 3a ( ) ( ) 2 2 22 ' 22 33 22 22 22 22 (5 )( ) , 0.( 3) 3 ( 2 ) (5 ) 2 () ( 3) 2 6 10 2 = ( 3) 16 = ( 3) Since ( 3) 0, for 0, 16 0( 3) Therefore, is a decreasing function. xfx x x x x xx fx x x x xx x x x xx x x f −= > + + − −− = + −−− + + − + +> > − <+ M1 – Quotient Rule M1 A1 3b ( ) ( ) ( ) ' 22 2 2 2 16( )= ( 3) 64 = 19 64 when 4, 0.2 0.2 = 1.1281 = 1.13 (to 3 1 s.f 9 19 64 .) xfx x dy dy dx dt dx dt dx dt dx t x d − + − − − = = × = × = × M1 M1 A1 4a ( ) ( ) ( ) 2 2 2 2 2 At intersections, 5 24 20 5 100 0 Discriminant = 5 400 = 25 400 = 25 16 = 2 hx x x hx h h h − += −+= −− − − ( )( ) ( )( ) 54 4 For no intersections, 25 4 4 0 44 hh hh h −+ − +< −<< M1 M1 A1 4b ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 47 2 47 = 2 3 Minimun point is 2,3 4 7 3. Alternatively, Since 2 0 2 3 3 xx x x xx x x x −+ = − −+ −+ ∴ − +≥ −≥ − +≥ ∴ 4 7 3.x− +≥ B1 – complete sq only B1

5a ( )( ) ( ) ( ) 2 2 22 22 2 2 2 22 2 22 22 sin( )sin( ) sin cos cos sin sin cos cos sin sin cos cos sin 1 cos cos cos 1 cos cos cos cos cos cos cos cos cos AB AB A B AB A B AB A B AB AB A B B AB A AB BA +− = +− = − = − −− =− −+ = − M1 M1 A1 5b ( ) ( ) oo oo oo oo 2o 2o 22 Let 45 and 30 , sin 75 sin15 sin 45 30 sin 45 30 cos 30 cos 45 32 22 32 44 1 4 AB= = = +− = −   = −       = − = M1 A1 6a 8cos 20sin RQM RSM d θ θθ ∠= ∠= = + M1 A1 6b 22 2 1 sin 8 cos 20 88 20 =tan 20 464 = 21.54 o RR R αα α − = = = +  = B1 – r B1– α 6c Max 21.54 = 21.5 (to 3 sig. fig