AHS 2025 S4 AM Prelim P2 MS

NAME: _______________________________ ( ) CLASS: 4 ( ) 9o ADDITIONAL MATHEMATICS 4049/02 Paper 2 MARKING SCHEME READ THESE INSTRUCTIONS FIRST Write your name, index number and class in the spaces at the top of this page. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, 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 number of marks for this paper is 90. ---------------------------------------------------------------------------------------------------------------------- For Examiners’ Use Question Marks Question Marks 1 8 2 9 Units 3 10 Clarity / Logic 4 11 Precision / Accuracy 5 Total: 6 90 7 Parent’s Name & Signature: Date: This paper consists of 18 printed pages. S4 ANGLICAN HIGH SCHOOL SECONDARY FOUR PRELIMINARY EXAMINATIONS 2025

Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation , Binomial expansion , where n is a positive integer and 2. TRIGONOMETRY Identities Formulae for Area of ! "!" #" $++ = !" ! !! " #$ " !± != ( ) !" " ### ###!" ! !! ! ! " " !!! !#$ # # $ # $ # $ $ " !! !"# "# "#+= + + + + + +$% $% $%&' &' &' ( ) ( )( )! """ !# ## # ! !! ! "! " "!" " !! +"#==$% !&' !"#$$%& '' =+ !! !! !! "#$%&'( += !!"#$%" & "#'!!=+ !"!"!" !"#$%!$%!!"#&!"#' ±=± !"#$ % !"# !"# #&' #&'!" ! " ! "±= ! !"# !"#!"#$ % & !"# !"# !"!" !" ±±= ! !"# $ $!"# %&!!! !=!!!!! !!!! "#$!%%&'"!"#$&'"!&'" !=!=!= ! !! !"#$% "#$!!"#$ ! = !"#!! " # A B C !"#!"#!"# == !"##"A !"#$$$$ !+= !"#!"# $ %=!

3 © Anglican High School 2025 4049/02/Prelim/25 [Turn over 1 (a) Show that . [4] (a) M1 for using changing tan and cot M1 M1 A1 (b) Hence, solve the equation for 0 < x < p. [3] (b) M1 M1 A1 ! " #$% "&'# #$% !!! =" !! ! !! ! ! "#$%&'( ()* "HHHHHHH&', ,-* ,-* &', "HHHHHHH&', ,-* &', ,-* " &', ,-* &',HHHHHHH&', ,-* &', "( ) * .( ) * !( ) *! !! !! !! !! !! !! ! !! ! ! ! ! = " = " = " =÷ " =" = ! "#$% & %'( &!! =! ! " #$% & ! !#$% & " '$()*+$%,-. /0/123 & /0/1234 /0/1234 " /0/1234 3 /0/12 20"154 +/0254 +/056+4 +" 3 07 ! ! !" " " ! = = = =+ + + =

4 © Anglican High School 2025 4049/02/Prelim/25 [Turn over 2 The expression , where a and b are constants, has a factor of , and a remainder of -9 when divided by . (a) Find the value of a and of b. [4] M1 for forming first equation M1 for forming second equation M1 for solving A1 for both answers !"!" ! #++ !!!!!+() () () () () ( ) ( ) ( ) () ()() () ! ! ! "#$%& ' &( ) '( ( ) ') '( &* + '* * + !* * + ** ! * (* , !* - + .L0% +%12$3 ( +' 4 +%%52S%% 4 !! " ! # "# "# "# "# "# "# " " " # # "# =+ + = ++ = ++= += ! ! ! ! ! ! != ! !+ ! + = ! !! + = ! !+ = ! ! ! ! ! ! =! =! =! !+ = ! = "= ! =

5 © Anglican High School 2025 4049/02/Prelim/25 [Turn over (b) Hence solve the equation expressing the non-integer roots in the form of , where c and d are constants . [4] M1 for getting quadratic factor M1 for using quadratic formula to solve the quadratic equation A2 for all answers !"#!" ! #++ = ! !"+ () ()( )( ) ()( )( ) ( )( ) ()() () ! " " ! " " " #$ B & #' $ & ()*+,-./0123*+,4##1+14256*,#* 7 B& $ #' $ $ & $B & 8 '$ $ & 8 $ $ & 8**********,0********** ' $$ $ $ & "$ $ B9 : $$ 9 : '9 **********,0********** '" !! ! !! ! " ! ! " " !! ! ! !! !! ! !! ! ! ! ! !! =! + =! +! != !! = =! +! !+ = !+ ! = +! = = !± ! != !±= !±= !±==

6 © Anglican High School 2025 4049/02/Prelim/25 [Turn over 3 The diagram shows an isosceles triangle ABC, where AB = BC and the line segment BC is parallel to the y-axis. Point A lies on the y-axis and the equation of line AC is . Point B is , where h is a positive constant. (a) Find the coordinates of A, B and C. [4] (a) Since A is on the y-axis, Since BC is parallel to the y-axis, and x-coordinate of B is 4, x-coordinate of C = 4. B1 B1 M1: form equation using length formula A1 !"!"=+ ( )!"! ( )!" #! ()!" # $$ !=+ = ( )!"##! ( ) ( ) ( ) !! !! !! "# $ $ $% # $$ $% % & $!$ !! $% &% % !! !! !! ! ! ! ! +! =! +! = ! +!+ = ! + = = ( )!" #! A B C O x y

7 © Anglican High School 2025 4049/02/Prelim/25 [Turn over Given that the point D is such that ABCD is a kite, and the line segment CD is parallel to the x-axis. (b) Find the coordinates of D. [4] (b) Gradient of BD = Equation of BD: Since CD is parallel to the x-axis, y-coordinate of D = 11. M1: for finding gradient A1 M1: for subst y = 11 A1 (c) Find the area of the kite ABCD. [2] (c) M1 in anticlockwise A1 ! " ! () ! " !#$ " % !" # # # =! + =! + = ! " # !"=! + !!! " # !$ # % ! ! ! =! + =! =! ( )!"##!! ( )( ) ! "# # A"%&'() * A %% %% *! %" ## ## %+ %! !# AA " ! ,"-./01 != =+ + ! ! + ! +"#$% =

8 © Anglican High School 2025 4049/02/Prelim/25 [Turn over 4 In the diagram, it is given that AB = 15 cm, BC = 40 cm and . The vertical line OC is perpendicular to OA and AB is perpendicular to BC. (a) Show that . [2] (i) B1 B1 (b) Express OC in the form , where R is a positive constant and is an acute angle in degrees. [3] (ii) M1 M1 A1 (c) Given that OC is at the maximum, determine the shortest distance of B to OC. [3] (iii) M1 M1 A1 !="!"# OC=15sinθ+40cosθ!"# $% &'! () !"# &'! $%!"# () &'! !" "# A# !" "# ! ! !! !! = = =+ =+ ( )!"+!"#! !!!"# $% $&!%' "#()* ' $% +,-"""---'' "!-. /0* +,-" '12 ! "# ! ! " =+ = = = =+ ° !"# $ $%&'($)!$*+,- ./'%%%''' /0 &0'11. 2+345,25$6725"-), %027-8&0'11. 9 :%'0)! !" ! ! =+ ° = ° =° =° = C O A B 40 cm 15 cm

9 © Anglican High School 2025 4049/02/Prelim/25 [Turn over 5 (i) Show , stating the value of the constant k. [4] (i) M1 – differentiate correctly M1 – simplify trigo functions M1 – correct application of sine double angle formula A1 (ii) Differentiate with respect to x. [3] (ii) M1 – use logarithm laws to simplify M1 – differentiate and A1 ( )!"# $%# &! '(# ) !""" =!"#$ ( ) ( )() () ! ! "#$% &'% ( )*+ ( (" &'% ( (+,) ( )-% ( +,) ( ( )-% ( +,) ( (! !)-% ( +,) ( . )-% . . !!!! ! !! !! !! ! " =!"#$ = = = = = ! "#$ % ! !! ( ) ( ) ( ) !! ! ! !! ! ! ! "# " $ #%& %&"$ " ! $ "$ %& # %& $"! $# $ ! !# ! $ $$ ! ! $ $ !$ !! !! ! ! !!! ! !! !! !! ! !! !" !"=#$ #$#$ %% &'&' !"()=% %#$*+&' !" ! "=%#$ # $ %&' & ' !"%%#$=#$ %&' +=% % !" #!( )!"# $!!

10 © Anglican High School 2025 4049/02/Prelim/25 [Turn over 6 The curve is such that . The y-axis is a normal to the curve at the point Q, where y = 1. (a) Show that Q is a stationary point. [2] (a) The x-axis is perpendicular to the y-axis. Since the y-axis is a normal, the x-axis is parallel to a tangent at y = 1. Since the gradient of the x-axis is 0, at y = 1. Therefore, there is a stationary point. B1: finding B1: stating why gradient = 0 (b) Hence, find an expression for . [6] (b) At x = 0, = 0, At x = 0, y = 1, M1: for correct integration M1 A1 M1 M1 A1 ()!!"= () !"# # ! !!!" " !=+ !!" !# = !" !# ()!! () ! ! ! "# ! ! ! $! ! !! ! ! ! ! !" " # ! ""$ "$ " ! ! =