ACJC 2025 JC1 H2 Math Promos Solution and Markers Report

ACJC Mathematics Department 2025 H2 Mathematics Promotional Examinations Solutions Qn Solutions Remarks General comments: • If additional pages are used, students should alert the marker by indicating clearly in the original space for the question in the answer booklet. • If the additional pages at the end of this booklet are used, the question number must be clearly indicated. • Please do not change the order of the question number indicated in the answer booklet by cancelling off the original indicated question number. Your answers must be written only on the correct allocated space indicated by the question number in the answer booklet. 1 cos sin 2xxyx = cos sin 2ln( ) ln( )xxyx = (cos )ln (sin 2 )lnx y x x= Differentiate both sides w.r.t x 1 d sin 2( sin )(ln ) (cos )( ) (2cos 2 )(ln )d yxx y x x x y x x− + = + ---- (1) Substitute x = into cos sin 2xxyx = , cos sin 2 1 1 yx y  − = = Hence y = 1 From (1) d0 (cos )(1) 2ln 0d y x+ = + d 2lnd y x =− As the base is not a constant, students should not apply the formula f ( ) f ( ) f '( ) lnxxd a x a adx = Students are expected to leave answers in exact form and not 3s.f.

ACJC Mathematics Department 2025 H2 Mathematics Promotional Examinations Solutions Qn Solutions Remarks 2 ( ) ( ) 1 21 2 12 2 2 22 2 22 2 2 2 111 tan 2 22 1 tan 2 Differentiate w.r.t d22 d 1 4 d1 14d 1 4 d d d 1 4 8d d d d d 8 0 (shown)d d (1 4 ) y x x yx x yy xx yyx xx y y yy x xx x x y y xy x x x − − − − = + −   =+ = + = = ++ + =− + + + = + Students should consider implicit differentiation instead of differentiating it explicity and then trying to substitute in the expressions to prove LHS = RHS. 3 2 2 2 3 2 2 3 2 2 1 2 2 3 3 Differentiate w.r.t x, d d d d d 2 8 2(1 4 ) 8 8(1 4 ) 0d d d d d when 0, 1 tan 0 1 d 1d d 1d d 8 2 1 5d y y y y yy x x x xx x x x x x y y x y x y x −− −  + + + − + + + =  = = + = = =− =− + + =− 2 3 2 3 Maclaurin series for y: 1 5 1 511 2! 3! 2 6y x x x x x x + − − = + − − Most students were able to differentiate to find the third derivative but made careless mistakes.

ACJC Mathematics Department 2025 H2 Mathematics Promotional Examinations Solutions Qn Solutions Remarks 3(a) Given that ( )1 1 11 n r n r r n= =++ , ( ) ( ) 2 1 1 1 [Note: replace by 1]1 1 1 1 11 1 11 (shown) n r n r rrrr rr n n n n n = − = +− = + −= −+ −= =−   Some students who have identified the correct replacement fail to write down the correct start and end values for r. (b) ( ) ( ) 2 22 2 11 21 11 21 1 12 lim 111 2 1since as , 0, 1 12 3 2 r r r rr n rr rr n n n  =  == →  + − =+   −  = + − − → → =+ =   Many students wrongly wrote 1  instead of “ as 1,0n n→ → . ” Students should list 2 1 1 1 1 ...2 4 8 16r r  =  = + + + to observe that it is the sum to infinity of a GP.

ACJC Mathematics Department 2025 H2 Mathematics