2024 TJC JC2 Prelim Exam H2 Maths Paper 2 (Solutions) For Studens (1)

1 2024 TJC Preliminary Exam H2 Mathematics Paper 2 (Suggested solutions) Section A: Pure Mathematics [40 marks] 1 A curve C has equation 3 1 xy x −= − . (a) Sketch C, stating the equations of the asymptotes. [2] (b) Find the exact volume of the solid generated when the region bounded by C, the x-axis and the line x = 9 is rotated through 2 radians about the x-axis. Leave your answer in the form ( )ln2ab + , where a and b are constants to be determined. [5] [Solutions] Remarks 1(a) 32 111 xy xx −= =− +−− Use dashed lines for asymptotes. When sketching additional lines to solve other parts of the problem, it is recommended that a side sketch be redrawn to avoid damaging the solution. 1(b) Required volume ( ) ( ) ( )( ) ( )( ) ( ) 2 2 9 3 9 3 9 23 91 3 11 3 3 d1 21d 1 441 + d 1 1 4ln 1 4 1 9 4ln 9 1 4 9 1 3 4ln 2 4 2 15 4ln 2 4ln 22 15 8ln 22 x xx xx xx x x x x        − −− −=  − = − + − =− − − = − − − − = − − − − − − − = − + =−     An additional diagram drawn by the side is useful in helping identify the solid of revolution. It is more efficient to perform a long division first then squaring. Squaring first will result in a more complicated algebraic fraction that would still have to be dealt with by long division (or further splitting), which would increase the complexity of the working. x y 3 −3

2 2 (a) Find the series expansion of ( ) 1 38 cos 2 x x + in ascending powers of x up to and including the term in 2x . [5] (b) Find the range of validity of x for the expansion to be valid. [2] [Solutions] Remarks (a) ( ) ( ) ( ) ( ) ( ) 11 1 33 3 2 2 2 12 2 2 2 2 2 12 33 818 8 cos 2 cos 2 121 3 2! 1 2 2 1 1 224 576 2 1 1 224 576 2 1 2 24 576 1 11512 1 8 8 2 2 2 8 8 x x xx x xx x xx x xx x xx x x − ++ =  −   + + +       = −+ = + − + − + = + − + + + = + + − + = + + + • Do not differentiate and use Maclaurin’s theorem. • Use binomial expansion/standard series in MF26: ( ) ( ) 211 1 , 1 2! n x x x n xnn −+ = + + +  21cos 1 2x x= − + , all x • Expand until x2 term. Do not waste time expanding more than what is required. • Rewrite ( ) 2 1 12 x− as ( ) 1212 x − − so that we can further expand it using binomial expansion again (b) For ( ) 1212 x − − : 2 2 1x  1 2 x  11 22 x−   For 1 3 1 8 x− : 18 x   88 x−   Taking intersection, the range of validity is 11 22 x−   Note that there are two main binomial expansions, 1 3 1 8 x+ and ( ) 1212 x − − . We need to find the validity range for each expansion and then find the range of values which satisfy both validity range by taking intersection.

3 3 A sequence  na is defined by 0 2a = and 2 1 21 where 1.33 n nna a n − − = −   (a) By