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.) Corresponding = 90 21.54 = 68.45 = 68.5 (to 1 dec. pl.) oo o o d θ = − B1 – maximum d B1– corresponding θ 7a ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 3 32 52 152 3 52 532 2 52 1 11 52 6 5 3252 2 52 30 12 15 10 25 2 15 22 25 2 xy x x xxdy dx x xxx x xx x x x − += − −− − += − = × ×× − − + − − −− −= − −= − M1 Qoutient Rule A1 7b b = 1 there is only 1 cycle in for oo0 360θ≤≤ , ( ) ( ) o At 0, 3 , 3 sin 0 30 1 12 2 4 a a a − −= − − −= −  = B1 M1 A1 8ai 0when 0, 195(0.8) 195 tm m = = = B1 8aii ( ) 6when 6, 195(0.8) 51.118 51.1 to 3 sig. fig. tm m m = = = = B1 8aiii ( ) ( ) 1when 195 , 4 1 195 195(0.8)4 1 (0 .8)4 1 ln 0.8 ln 4 6.2125 t t m t t = = = = = ( ) 6.21 t o 3 sig. fig.t = M1 M1 A1 8b Since 0 0.8 1, 195(0.8) 195.tt≤≤ ≤ B1 8c G1- Correct Shape G1 - Labels 9a ( ) [ ] 4 0 4 0 4 0 3 21 13 21 13 ln 2 12 3 ln 9 ln12 3ln 3 dxx dxx x + = + = + = − = ∫ ∫ M1 A1 9bi ( ) ( ) 53 05 53 5 30 3 2 ( ) ( ) 2 ( ) ( ) ( ) 263 15 f x dx f x dx f x dx f x dx f x dx − = + −− =×+ = ∫∫ ∫∫ ∫ M1, M1 A1 9bii 5 23 55 233 5 3 5 23 () ( ) 13 113 53 23 15 () 0 230 15 45 2 kf x dx x kf x dx dx x k x k k kf x dx x k k  − = − = −−   =− −+  = −  −=  ⇒− = ⇒= ∫ ∫∫ ∫ M1 M1 A1 M

10a ( ) ( ) ( ) ( ) ( ) 22 22 22 2 8 12 27 0 4 16 6 36 27 0 4 65 Centre is 4,6 Radius = 5 xy x y xy xy ++− += ⇒+ −+− −+= ⇒+ +− = − M, A1, A1 10b ( )( ) ( ) ( ) ( ) 1 2 2 222 2 Consider C , at -axis, 0, 12 27 0 9 30 3 or 9 Consider C , 39Centre = 0, 2 = 0, 6 93Radius = 2 = 3 C is 6 3 yx yy yy yy xy = − += − −= = = +  − +− = M1 M1 M1 A1 10c ( ) ( ) ( ) ( ) ( ) ( ) ( ) 22 1 1 22 2 2 Distance from 3,3 to C 3 4 3 6 = 10 5, radius C Distance from 3,3 to C 3 0 3 6 = 18 3, radius C 3, 3 l − = −+ + − < − = −− + − > ∴− 12ies in C but not in C . M1 A1 11ai ( ) ( ) Horizontal distance from A to B = 6 0 6 6Horizontal distance from P to B = 23 Vertical distance from A to B = 10 1 9 9Vertical distance from P to B = 33 P is 2, 4 Alternatively, 2 2 AP PB pa b p pa −= = −= = = − ∴ = − −=     22 32 0632 1 10 63 12 2 4 bp p ba x y x y x y − = +   = +      =   = B1 11aii ( ) ( ) ( ) 10 1 3Gradient AB = 60 2 Gradient PQ = tan 180 = tan = tan = Gradient AB 3 = 2 Equation PQ is o PQC PQC PCQ − =− −∠ −∠ −∠ − − 43 22 28 36 2 3 14 y x yx yx − =−− −= − + += M1 M1 – find gradient M1 – valid attempt to find equation (ecf) A1 11aiii When 0, 14 3 14 is ,0 .3 y x Q = = ∴  B1 11b ( ) ( ) ( ) Note 2 1 14Area =2x 2 4 10423 104 34 8 4 39 35 is 2, 35 DP D D D xx y y y D = =   −−=  ×−= = − =− ∴− M1 – find xD M1 – find area A1 12 ( ) ( ) ( )( ) 7 22 2log 3 log 49 2 lg 2 lg 73lg 7 lg 2 lg 3lg 7(lg ) 2 lg 7 0 2 lg lg 7 lg 2 lg 7 0 1lg lg 7 or lg 2 lg 72 1 or 49 7 pp p p pp pp pp pp = + = + − −= + −= =−= = = M1 – Change of Base M1 – Quadratic Eqn M1- Solve for lg p A1 - Solve for p 12bi 3log 3a xa x = = 9 2 log 3 b yb y = = 99 9 3 3 log 9 loglog 9 log 1 log log 9 22 x yy x b x b a += +=    += M1 – change of base M1 – Add/Subtract Law A1 12bii ( ) ( ) 332 32 33 = 3 ab ab xy + = M1 – equivalent exponential A1 G F C1 C2 A P B D

13a ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 11 11 1 211 2 3 9 9 311 3 4 8 113 ,T 3 11When 2, T = 3 2 = 55 9 = 495 11When 3, T = 3 3 = 165 27 nn nFor x xn nx x x nx x − + − − −= −  = − = − − ( ) ( ) ( ) ( ) 8 9 11 2 311 10 11 2 11 3 11 10 9 = 4455 Considering coefficient of , 495 4455 1386 1 5 Alternatively, 11 113 11 3 3 3 ...23 33 495 4455 x x k k x xx x x xx x x −− − −= =  − = + −+ − + − +    = −+ − ( )( ) ( )( ) 8 11 11 10 9 8 9 99 ... 1 3 1 33 495 4455 ... terms with = 4455 495 4455 495 1386