2023 JC1 Promo Practice Paper B (solutions) VJC

2023 PROMO PRACTICE PAPER B Solutions Qn Solution Notes 1i 5 3yx= has a stationary point at ( )0,0 hence gradient of it’s graph at the origin should be 0. Graph of 3ln 3yx=+ extends down to infinity as x approaches 0. Equation of asymptote: 0x= and coordinates of x- intercept: ( )0.368,0 need to be stated (required by question) 1ii Graphs of 3ln 3yx=+ and 5 3yx= intersect at 0.395x= and 3.02x= . 5 33ln 3 0.395 3.02 xx x +    2 ( ) ( ) 22 d3 2 1 d yx y xy x− = −−− Differentiating w.r.t. x ( ) 2 22 2 d d d d3 6 2 2 2 d d d d y y y yx y x y y xx x x x − + − = + When 0x= , 1y= ( ) ( )( )dd0 1 2 0 1 0dd yy xx− =  = ( ) ( )( )( ) ( )( ) 22 22 dd0 1 0 2 0 0 2 2 0 0 2dd yy xx− + − = +  =−  the Maclaurin’s series for y is ( ) 2 2 210 2! 1 y x x x −= + + + = − + Differentiate (1) immediately using product rule. No need to make d d y x the subject before differentiation.

2023 PROMO PRACTICE PAPER B Solutions 3i ( ) ( ) 2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 e sin d 11e sin e cos d22 1 1 1 1e sin e cos e sin d2 2 2 2 1 1 1e sin e cos e sin d2 4 4 5 1 1e sin d e sin e cos4 2 4 21e sin d e sin e cos55 1 e 2sin cos5 x xx x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x D x x x x C x x C =− = − − −  = − − = − + = − + = − +       3ii Let ( ) 2f ( ) e 2sin cosxx x x=− . Observe that ( ) ( )( ) 22e 2sin 1 cos 1 f ( 1)xy x x x+= + − + = + is a translation of f ( )yx= by 1 unit in the negative x- direction. Hence, the gradient of the curve f ( 1)yx=+ at 12x =− is the gradient of the curve f ( )yx= at 2x = , which is given by f 2y =  . From part (i), 2f ( ) 5e sin xxx = . Hence, the required gradient is f 5e2  = .

2023 PROMO PRACTICE PAPER B Solutions 4i Asymptotes: ( ) ( ) ( ) 22 23 94 23 32 33 22 3 5 3 13 or 2 2 2 2 yx yx xy y x y x −− = −− = −= + = − =− + 4ii ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 22 22 22 22 22 12 48 48 6 3 0 12 4 6 3 48 0 12 4 4 48 6 9 9 3 48 0 12 2 3 12 23 112 y y ax ax a y y a x x a y y a x x a a y a x a yx a − + + − − = − + − − + = − + − + − + − − + = − + − = −− += which is an ellipse with centre ( )3,2 and vertices at ( )3,2 a+ and ( )3,2 a− For curve C and D to not intersect, 39aa   Since a is a positive constant,  :0 9aa    Modified mark allocation from 2 to 4.

2023 PROMO PRACTICE PAPER B Solutions 5i ( )( ) ( ) ( ) ( ) ( )( ) ( ) 2 122 2 12 21 222 2 24 2 244 2 24 1 122 1 2 1 2 121 1 1 12 2 2! 2 1 112 2 4 1 12 2 2 4 1 3 3 2 4 8 x xxx xx xxx xxx xxxx xx − − − + = + −− = + −   −−   = + + − − + − +        = + + + +  = + + + + +  = + + + Refer to MF26 and apply the standard series ( ) ( ) 2111 2! n nnx nx x −+ = + + + 5ii For expansion to be valid, 22 1122 xx−    ( ) ( )( )   2 2 2 2 2 2 2 since 0 2 2 0 : 2 2 x x x x x xx xx   =  − +   −   Note that the standard series ( ) ( ) 2