AHS 2025 S4 AM Prelin P1 MS

1 © Anglican High School 2025 4049/01/Prelim/25 [Turn over NAME: _______________________________ ( ) CLASS: 4 ( ) 9o ADDITIONAL MATHEMATICS 4049/01 Paper 1 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 12 Total: 6 13 90 7 14 Parent’s Name & Signature: Date: S4 ANGLICAN HIGH SCHOOL SECONDARY FOUR PRELIMINARY EXAMINATIONS 2025

© Anglican High School 2025 4049/01/Prelim/25 NOTE: If the marks allocation is italicised, it has been modified from version 2 according to the suggestions given by the tester. 1 (a) Find the range of values of p for which lies entirely above the line.[4] Þ discriminant < 0 M1 M1 for discriminant < 0 M1 A1 (b) Hence, deduce, without finding the discriminant, the number of intersection points when p = 3. [1] Line does not intersect curve when p is in the range . For p = 3, line intersects curve at 2 points. B1 2 (a) By using substitution or otherwise, find the values of x for which , giving your answer where appropriate, to one decimal place. [5] M1 applying Laws of Indices to break them into product of two numbers M1 for getting quadratic equation M1 for using lg to solve. A2 for both answers ! !!" "=! ! !" #!" #=+ ! ! !! "# $ ! ! % !! " !" !" ! " !!= + +!! ! ! = ( ) !" #$!$%"% #"!# #&&'! #$''$%!%(! ' ' ' !<<!" <++" <++" <++++" <!!!!!" ! !! !! !!! !! !"#+= !"# !"!#$# !+= !!" !" !<<! ! !" #!! ! $!!!+=! () () ( )( ) !" # ! # ! # ! ! ! ! ! ! !$ "! ! ! !$! %&'( ! " ! !$! " ) !$! "* $) "* $) + "! $ + "!(((((,-((((( $ ! "!(((((,-(((((! $ .L ! .L"!(((((,-(((((! ! .L"!(((((,-((((( !.L ! #01)$2* #0* !! !! ! !! !! " "" "" "" "" "" "" !! ! ! !+=! ="! = =" ! =! =! !+ = !! = == == == == # #

© Anglican High School 2025 4049/01/Prelim/25 [Turn over 3 A closed circular cylinder has a volume of cm3, a radius of cm and a height h cm. Express h in the form of where p and q are integers. [4] M1 – expansion of r2 M1 – rationalisation M1 – multiplication and simplification A1 for answer 4 Express in partial fractions. [6] Marking Scheme modified from Version 2 B1 for long division working M1 for breaking into partial fractions A1 for value of A ( )!! " " # !+ ( )!! "+ !"!"+ ( ) ( ) ( ) ( ) ( ) ( ) ! ! "#$%&'(#)(*+$,-.'/(0( 11 ! ! 2 11 ! ! 2 ! ! 2 11 ! ! 2 ! 3 1 4! 11 ! ! 2 11 ! ! 2 43 3 1 11 ! ! 2 43 3 1 43 3 1 43 3 1 5!3 ! !13 4! !6 2 6 46 451 51 5!3 ! 7!6 2 !6 2 !3 ! 488 588 ! 788 2 488 5! 72 !" " " " " " " " " ! !! + =+ += + ++ = + += + +"=# +" "+ "= " "+ "= "= =" !" ! ""# $ " % !!! !! !+ ! + ( ) !! " ! " !" " !! ############################" $ " " % &" ####### " ########## ' ################# " &" ######################################## " " % &" " &" " $$ !! !!! !! !! !!! ! ! !! !! +! + ! !+ !! ! !+ ! ! ! !"= + ++ ( ) ( ) ( ) ! "! " !! !# ! $$ %&'()*'+,-+, $ ,(./M&1.M&(2 !# ! $ !! " # ! A !!! ! !! !! " ! ! # ! A !! ! + =+ ++ + !! ! = + + + () !"#$ %& '( ) % * ! " " = != + =!

© Anglican High School 2025 4049/01/Prelim/25 M1 for forming simultaneous equations to solve for value of B and of C. A1 for value of B and of C. A1 for final answer or B1 for long division working M1 for breaking into partial fractions M1 for expansion and grouping A1 for value of A ( )()( ) () ( )()( ) () ()() () ! ! "#$% &' ! & &! ( & ) & &* &* +& "#$% &' ! & &! ( & ) & &( &* !! &! , ! ! %%%%%%%%%%%%%%% & "#$% &%-./0% & , &+ &% ! "# "# "# ! "# "# "# " " " # # = !!! = ! + + + != !+ + += ! ! ! ! ! ! ! ! =! !+! = ! + +! ! + != !+ ! != ! ! ! ! ! ! ! ! += = = += =! !" !" ""# $ " $ ! " %% !!! ! !!! ! !+ ! ! =+ ! ++ ( ) !! " ! " !" " !! ############################" $ " " % &" ####### " ########## ' ################# " &" ######################################## " " % &" " &" " $$ !! !!! !! !! !!! ! ! !! !! +! + ! !+ !! ! !+ ! ! ! !"= + ++ ( ) ( ) ( ) ( ) ! "! " !! !! ! ! #! $$ %&'()*'+,-+, $ ,(./M&1.M&(2 ! #! $ ,,,,,,,,,,,,,,,,,,,,, $ ,,,,,,,,,,,,,,,,,,,,, $ ! ! " #! A !!! ! !! ! ! " ! ! #! A "! " #! A! "# ! A ! " !! ! + =+ ++ + !! ! = + + + =+ ++ =+ ++ !"#$B&'()*+,#$B+-.(+.-/###### 0 12 3 ! ! =! =! !"#$B&'()*+,#$B-..*$*-+/0#B.# 1# 2 ! "=!

© Anglican High School 2025 4049/01/Prelim/25 [Turn over A1 for value of B and of C. A1 for final answer 5 Given the curve, where k is a constant, (i) By expressing y in the form , determine the value of k and b. [3] (i) Marking Scheme modified from Version 2 M1 for A1 for value of b A1 for value of k (ii) Hence, show that the curve is always below the x-axis. [2] (ii) Hence, the graph of y is always below the x-axis. M1 A1 or Discriminant = Since discriminant < 0, the curve does not intersect the x-axis. Coefficient of x2 is also negative, hence it is a quadratic curve with a maximum point. Hence, the curve is always above the x-axis. M1 discriminant A1 for negative coefficient of x2 or The turning point is and is a maximum point. Since the y-coordinate is less than zero, the curve is always above the x-axis. M1 for maximum turning point. A1 for y-coor < 0 !"#$B&'()*+,-$B&.//+B+.,01$&/$ 2$ ! 3! 4 ! "# # # += ! !+ = ! = !" !" ""# $ " $ ! " %% !!! ! !!! ! !+ ! ! =+ ! ++ !!"!" # " #=+! !"# $!" #+! ! ! ! ! !" #! $ " #! % % $ " #% $ " % "% ! !" # " # !" # # !" # # !" # " $ " " =+! =+ ! =+ + ! ! =+ ! ! = !! = ! =! !"# $ %!" # "=+ ! ! ! ! ! ! !" #$ # "# $ % !" #$ % !" #$ # # % !" " " " =! + ! +" !+# !+! # ! < !"# $ # "! $ "% $ & '!! ! ! = ! < !" # " $!!

© Anglican High School 2025 4049/01/Prelim/25 (iii) State the turning point of the curve and determine the nature of this point. [2] (iii) B1 B1 6 The binomial expansion of where n > 0, in ascending powers of x is Find the value of p, of n and of q. [6] By comparing coefficients of x2, By comparing coefficients of x, Marking Scheme modified from Version 2 B1 for applying the formula for Binomial Coefficient M1 for forming equation. is not acceptable A1 for both answers, must reject -7. M1 for forming equation. is not acceptable A1 for value of p A1 for value of q !"#$%$&'()%$! * + , +- .$/' %0'.'1.2%1"1'()%$! =! ! ( )! ! "#+ !! "# #! !$ %%%! " ! #!!+ ++ ( ) ( ) ( ) ( ) ( ) ( )( ) !" !! "" !! " # # $$$#! " ## !# $$$ !% # #! !& $$$ ! "# !! !"# "# "# !! !! !!"# " # " # # " # $# + !" !" !"=+ + + +#$ #$ #$%& %& %& '' '=+ + + + =' + + + ( ) ( )( ) ! ! " !#! A% A% & #' & #((((()*(((( '(+,-.-/ !! !! !! !! !! != != !! = !+ = == ! !" !" # !!" !" " " =! != =! ()() !"# $ %%$& %"' ! ! !"=# $%&' =# ! !"! !"!" =#$%& !"# !"!" =#$%&'

© Anglican High School 2025 4049/01/Prelim/25 [Turn over 7 (a) The surface area