YTSS A Math 2024 Prelim P2 QP

NAME: ( ) CLASS: YISHUN TOWN SECONDARY SCHOOL PRELIMINARY EXAMINATION 2024 SECONDARY 4 EXPRESS / 5 NORMAL ACADEMIC ADDITIONAL MATHEMATICS PAPER 2 (4049/02) DATE : 23 August 2024 DAY : Friday DURATION : 2h 15min MARKS : 90 READ THESE INSTRUCTIONS FIRST Do not turn over the cover page until you are told to do so. Write your name, class and class index number in the spaces at the top of this page. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an approved scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total marks for this paper is 90. MARKS OBTAINED FULL 1 6 2 8 3 7 4 10 5 10 6 8 7 9 8 10 9 10 10 12 TOTAL 90 This question paper consists of 21 printed pages and 1 blank page. 60

2 2024 YTSS Preliminary Examination Secondary 4E5N Additional Mathematics EXP (4049/02) Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation , Binomial Expansion where is a positive integer and 2. TRIGONOMETRY Identities Formulae for !" =++ !"#$# ! !"##$ ! "!!±!= ( ) !""!!!!! #$$$#%" !$$$#%!#%!%#% ++!! " # $$ % &++!! " # $$ % &+!! " # $$ % &+=+ ''' !!" !" !( ) ( )( ) ! "###" !$! ! ! !""" !"! " ! " +!!=!="" # $ %% & ' !! !! !! !! !! !! "#$%&""#' $()%'&" %"#''*) += += =+ ( ) ( ) ( ) ! !! !!!!! !!! "! "!"! "!"!"! "!"!"! ! !!!! "#$% "#$!!"#$&&&&&&&&&&&&&&&&&&& '($!%%)*'!'($)*'!)*' )*''($!!'($&&&&&&&& "#$"#$% "#$"#$"#$&&&&&&&& '($'($)*')*')*'&&&&&&&& '($)*')*''($'($&&&&&&&& ! = !=!=!= = ±=± =± ±=± ! ! !"#! !"# A#BB#" ! B C # A " !"#$ % &'!$ !"#!"#!"# $$$ =! "+= ==

3 2024 YTSS Preliminary Examination Secondary 4E5N Additional Mathematics EXP (4049/02) [Turn over Answer all questions. 1 (a) Given that the curve has a minimum value, find the range of values of k for which the line meets the curve. [5] M1 For line to meet curve, . M1 M1 OR M1 Since curve has a maximum value, OR A1 (b) State the largest possible integer value of k. [1] Largest possible integer value of k =3 A1 !"# $ % #!" # " # "=! + ! !" #= ! "# $ % #!" ! " ! ! "!+ ! = ! "# $ % # &!" ! " !!+ ! = !!!!"# $ %"% $" % $ &!! !!! !" !!!" #$ $% &!! !!+ " !"# $ %!!+! !" #$% &!!+! !" #!!" !!!!"!>!!<!" #!!" # !"!!<

4 2024 YTSS Preliminary Examination Secondary 4E5N Additional Mathematics EXP (4049/02) 2 Given the following equation , (a) Using the substitution , show that the equation can be written as, where k is a constant. [2] M1 (equivalent form) A1 (b) Using the value of k found in (a), show that is a factor of . [1] Let . A1 By factor theorem, is a factor of . ( )!"#$ % !& "#' % " '!! ! +!=+ + !!"= !"#$ "$ % &!!! "!! + = ( )!"#$ % !& "#' % " '!! ! +!=+ + ! "#$% & "' #$% % & # % !! ! !"=+ # + $%&' !"#$! % &' ! &" ! #!! !=!+! + !"#$! % &" ! &' ! # (!! !!" !"! = !"#$ "$ % # &!!!!! ! = !"!+!"#$ "$ %!!! "!! + !"#$ % & " & ' %!" " " "=! !! !"## # #$% & $% # " $% # ' $% &"" " "!!= ! !! !!! !"# $%!!= !"!+!"!"

5 2024 YTSS Preliminary Examination Secondary 4E5N Additional Mathematics EXP (4049/02) [Turn over (c) Hence, find the value(s) of x. [5] From (ii), . By division algorithm/ comparing coefficients, M1 M1 When , M1 OR M1 (N.A.) (3s.f.) A1 2 A1 !"# " ! $ # " #!" " # " $ " %=+ + + !"# " ! $ # " ! % & #!" " " "=+ ! ! !"# " ! $ # " % #!" " "=+ ! !" #!"= !"! # $ " %$ &!!+! = ! "!=! !!=!" # !=! !"!= !" # !" $! = !"#$!=

6 2024 YTSS Preliminary Examination Secondary 4E5N Additional Mathematics EXP (4049/02) 3 It is given that the equation of a curve is . (a) Find and . [3] A1 M1 A1 (b) Hence, find . [2] = M1 = A1 (or equivalent) !"#!"# != ! ! ! " ! ! " " ! " !"# #$%!!"#$! $ !"!=+ ! ! "#$ $%& $%& "#$!! ! !"# $! $ ! $ ! $!"!=! + + ! ! !" # $!"# $!"!= !" # $ % &! !!! !" # $ % &!"! !! !"# #$%!!"! " ! #++ !"# $%!"! !! ( )! "#$ $%&' !!"! " ! #++

7 2024 YTSS Preliminary Examination Secondary 4E5N Additional Mathematics EXP (4049/02) [Turn over (c) Using the results in (a) and (b), find . [2] M1 A1 M1 A1 !" # $ % &! !!! !"# $%! & !"#!! !"! " ! # ! "! $+= +! !"# $ %&! $ !"#!! !"! # ! " ! # ! "! $+= +!! ( ) ! "#$% &'# #$% #$%! !! ! !"! # ! ! " "! "! $=! + + +" ! ""#$% &'# #$% #$%!! !! ! !"! # ! ! " "! "! $=! ! + +" ! ""#$% &'# #$%!! !! !"! # ! ! " "! $=! + +" ! ! " " ##$%& '($ $%&)) !! !"! ! " "! !! "#=$ +%&'() () ()!!" "" ! ## # #$%& '($ $%& '($ ! $%& ! )" ) " ) ) ! "! " " " " ! !! !!"# $% $% " #=& + && +'( )* )* '(+, +, - .-./ ! " #$%& ' !"! ! ="

8 2024 YTSS Preliminary Examination Secondary 4E5N Additional Mathematics EXP (4049/02) 4 (a) A box with length 35 cm, breadth 20 cm and height y cm is being compressed from the top at a constant rate of 0.005 cm per second. The initial height of the box is 30 cm. (i) Find the height of the box after one minute. [2] After one minute, the height of the box is . (ii) The length of the diagonal BC is L cm. Show that , where k is an integer to be found. [1] (shown) !" #" $"%""&' ()%*+,-!= !!" #=+ !! ! "# !$ %&!# !"=+ = ! "#!$!"=+ 35 cm 20 cm y cm L cm A B C M1 , A1

9 2024 YTSS Preliminary Examination Secondary 4E5N Additional Mathematics EXP (4049/02) [Turn over (iii) Assuming that the shape of the box does not change under the compression, find the rate at which BC is decreasing after one minute. [3] After one minute, , The rate of decrease of AC is 0.00297 cm per second. (b) The density D of the box is related to its volume V by the formula , where m is a constant. Given that when , , find the rate at which the density is changing with respect to V when . [4] When , ! "#!$!"=+ ! " " " ## # ## # !#$ !%"&' $" '"# # #!%"& !! " #" # """ # "" #" ! =" =+ " =" + !"#$!= ! "#""$! ! "=! !! " !#$% & '$''()" !#$% *+!( '$''!#%,,&-./01) ! "=! + =! !" #=!"#!=!"!#$%!=!"!#$%!=!"# $#%#"&'( ""%'& ! ! = = !!"#$ !!"#$%%% !" ! " != = ! "! ! # $ % " ! ""=! !"!#$%!=! " !!#$% " &#&'() $*)+,#%+ $*+&&--.,-/#0#1 ! "=! "! =! M1 for ! ! ! " M1 for chain rule with substitution A1 M1 M1 M1 A1 *Deduct 1m if write ! ! ! "

10 2024 YTSS Preliminary Examination Secondary 4E5N Additional Mathematics EXP (4049/02) 5 (a) Explain why there are only odd powers of x in the expansion , where . [3] M1 M1 Since is always even, is always odd since the addition of an even number and an odd number, 27, is always odd. A1 Hence, only has odd powers in this expansion. (b) Given that the coefficient of in the expansion of is , show that the value of k is 2. [4] Using the general term formula, to find term, Let M1 M1 M1 (rej. -2 as ) A1 (with rejection) ! " #"! "! !" #$%&' !!>()( )!!" # #" ! ! !!"# $# ! + "#=! $%&' ()! ! "# $ % %$ ! !! ! !!"# # $ !! ! + "#=$ ! $ %&'( ()! ! "# $ % %& ! !! !!"# $ !! + "#=! $ %&'( !!!" #!!!!!! " #"! "! !" #$%&' !"# $" !!!" # $!!=!!=() ! "#