TMJC H2 Mathematics Prelims Paper 2 (A)

TMJC/2024 JC2 Preliminary Examination Suggested Solutions/H2 Math (9758/02)/Math Dept Page 1 of 16 2024 H2 MATH (9758/02) JC 2 PRELIMINARY EXAMINATION – SUGGESTED SOLUTIONS Qn Solution 1 Maclaurin’s Series (i) ( ) ( ) 2 2 2 ln 1 3 2 d 3 4 d 1 3 2 d1 3 2 3 4 d y x x yx x x x yx x x x = + + += ++ + + = + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 3 2 2 2 3 2 2 32 2 32 diff wrt , dd1 3 2 3 4 4 showndd diff wrt , d d d d1 3 2 3 4 4 3 4 0d d d d d d d1 3 2 6 8 4 0 d d d x yyx x x xx x y y y yx x x x x x x x y y yx x x x x x + + + + = + + + + + + + = + + + + + = 23 23 23 23 d d dwhen 0, 0, 3, 5, 18,d d d 5 183 ... 2! 3! 5 3 3 ... 2 y y yxy x x x y x x x x x x = = = =− = = − + + = − + + (ii) ( ) ( ) ( ) ( ) ( ) 2 2322 2 2 2 3 4 3 23 ln 1 3 2 3 2 3 2 3 2 ... 23 11 3 2 9 12 4 27 ...23 5 3 3 ... (verified)2 y x x x x x x xx x x x x x x x x x = + + ++ = + − + + = + − + + + + = − + + Alternative: ( ) ( )( ) 2ln 1 3 2 ln 1 1 2y x x x x= + + = + +  ( ) ( )ln 1 ln 1 2y x x= + + + Using the standard series in MF26, ( ) ( ) 2323 23 22... 22 3 2 3 5 3 3 ... (verified)2 xxxxy x x x x x  = − + + + − +  = − + + Useful technique: Multiply both sides by the denominator before further differentiation. AVOID using quotient rule for further differentiation as much as possible. Use implicit differentiation Use implicit differentiation Find the values of f(0), f’(0), f”(0) and f ’” (0). Substitute into the general form of the Maclaurin Series that can be found in MF26 Use the standard series in MF26 for ( )ln 1 x+ up to and including the term in 3x and show clearly that it is the same as the Maclaurin Series found in (i).

TMJC/2024 JC2 Preliminary Examination Suggested Solutions/H2 Math (9758/02)/Math Dept Page 2 of 16 (iii) ( ) 2 2 3 5ln 1 3 2 3 3 ... 2x x x x x+ + = − + + Sub 1 ,2x= 2 2 3 1 1 1 5 1 1ln 1 3 2 3 3 ... 2 2 2 2 2 2           + + = − + +                     5ln 3 4 Substitute 1 2x= into both sides of the Maclurin Series to get the approximate value for ln 3

TMJC/2024 JC2 Preliminary Examination Suggested Solutions/H2 Math (9758/02)/Math Dept Page 3 of 16 Qn Solution 2 Vectors (i) 3 2 By ratio theorem, 41 55 OC OD =− =+ a ab Area of triangle 1 2 1 3 4 1 2 2 5 5 3 4 1 4 5 5 3 4 1 4 5 5 31 45 3 (shown)20 OCD OC OD= = −  +  =  + =  +  = +  = a a b a a b a a a b 0 a b ab (ii) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 22 22 2 22 , 2 0 AO AB ABAP AB AB  = − − −= −−  −+=− −−  −+= − = = −+ = − = + a b a b a b a b a a b a a bab a b a a b a b a a a a b b b b a b a a b a a b ab 2 22 = + a ab and OA OC are in opposite directions, thus the negative sign is important This is a SHOW question, there is a need to apply the distributive law and show the expansion clearly Must explain that =a a 0 For the projection vector formula, there is NO modulus for AO