MGS 2025 Sec 4 Prelim AM P1 (Solutions)

Class Index Number Name : ___________________________________ This question paper consists of 23 printed pages and 1 blank page. METHODIST GIRLS’ SCHOOL Founded in 1887 PRELIMINARY EXAMINATION 2025 Secondary 4 Tuesday ADDITIONAL MATHEMATICS 4049/01 26 August 2025 Paper 1 2 h 15 min Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your class, index number and name in the spaces at the top of this page. Write in dark blue or black pen You may use a HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all 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. 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. 90

Page 2 of 24 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2025 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the quadratic equation , Binomial Expansion , where n is a positive integer and . 2. TRIGONOMETRY Identities Formulae for DABC a2 = b2 + c2 - 2bc cos A D = bc sin A !" =++ !"#$# ! !"##$ ! "!!±!= ( ) !""!!!!! ##$" !#$!#$!$#$ ++!! " # $$ % &++!! " # $$ % &+!! " # $$ % &+=+ ''' !!!!" !" ( ) ! " #$%%%" #$ !! ! ! !! ! ! " " "!" " !" ## +==$% #&' !!"#$ %&" '!!+= !!"#$ % &'(!!=+ !!"#$%" & "#'!!=+ !"!"!" !"#$%!$%!!"#&!"#' ±=± !"!"!" !"#!"#$%!$%!&$%!' !=± !" !"!" !"#!"#$ !"#!"#%!"#& ! ±=± !!!!"##$%&&#$%=cos2A=cos2A−sin2A=2cos2A−1=1−2sin2A! !! !"#$% "#$!!"#$ ! = !"# !"# !"# !"# ABC== !"

Page 3 of 24 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2025 1 A home-based sticker company that prints stickers has calculated its profit, $P, for each order using the equation , where x is the number of stickers produced in hundreds. (a) Express in the form , where a, b and c are constants. [2] (b) Using your answer in (a), explain clearly if the company should accept an order for printing 400 stickers. No. As he will be making a loss of $40 if he accept the order. [2] !"!# $# %#!" "=! ! !"!# $# %#!" "=! ! ! "#!" # $ %=! + !"!# $# %#!" "=! ! !"#$ % & %#!!=! ! ! !! !"#$ % & !' & !' ( %#!!=! ! + ! ! ! ! !"#$ !% &#!=! ! +

Page 4 of 24 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2025 2 Find the exact coordinates of the stationary point on the curve , . Stationary point = [6] ! !" !"# != !!!!! ! !! "# $" $"!$ " $"% $ "# $ !!"# ! $ $ ! "! ! != ! " #$ % & ' !!" ! ! != ! " !#$ % " !"! ! != ! " !# $ %& " !"! ! != !!" # "!"!> !"!!=!!=! " !"= ! "#$ %& !

Page 5 of 24 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2025 3 Solutions to this question by accurate drawing will not be accepted. The diagram shows a quadrilateral OABC where O is the origin. The point A is and the point B is (3, 7). The perpendicular bisector of AB meets the x-axis at C. (a) Find the coordinates of C. Gradient of AB = = Gradient of Perpendicular to AB = -2 Midpoint of AB = = Equation of perpendicular bisector of AB: [M1] At x-axis, y = 0 Coordinates of C = (4, 0) [4] (b) Find the area of the quadrilateral OABC. [2] Area = = = 25 units² !" # $ %! !" #$ ! + !"!" #$%&& !+ +"# $%&' ( )!" # !" # $ %!"!= ! ! !"!"=! + !" # $ ! ! =! + = !"# $!$ !!% & !' ! !"#$ %& !' $( #$ ) $ (*% ++ + ! ! + + y

Page 6 of 24 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2025 4 The diagram shows an inverted right circular cone with a diameter, 30 cm and height, 50 cm. The cone was initially empty. It is being filled with water at a constant rate of 100 cm³/s. The surface of the water remains horizontal as it fills. At time t seconds, the radius of the horizontal water surface area is r cm and the depth of the water is h cm. (a) Express r in terms of h. [2] (b) Show that the volume of water, V cm³, in the cone is . [1] !" "# !"= ! "#!"= !! "## !" != !" #!" #!= !"#$%#" &!" "!= !! "## !" != h cm 30 cm 50 cm r cm

Page 7 of 24 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2025 (c) Find the rate of change of the depth of the water when h = 35 cm. When h = 35 cm, [3] !" #$$ !" #!# != !" !" !# !$ !# !$=! !"#$$ %&'(#$$ !" !#!=" !"#$$%&' !"#$( ) *+,- .!" #$ %!&==

Page 8 of 24 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2025 5 The diagram shows part of the graph of , . O is the origin and points P and R lie on the x and y axes respectively. Point Q is a point on the curve such that OPQR forms a rectangle. (a) Given that the x-coordinate of P is h, write down an expression for the area of the rectangle, A units2, in terms of h and show that . When x = h, A = = [3] !!"=! !!< !" # ! $# !" " " != ! !!"=! !!!! !! ""!#$ # $ # % $ #! $ # % $# ! $" !" ## #!# ! =! ! + ! ! "! ! ! ! !=+ ! ! !"# $ !# !! ! !+ != ! !" # ! $# !" " " != ! y x O P R Q

Page 9 of 24 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2025 (b) Find the stationary value of A and determine the nature of this stationary value. =0 A = =2 units² h Slightly less than 2 2 Slightly more than 2 + 0 - Shape / - \ Or = At h = 2, Therefore Stationary value A = 2 is a maximum value. [4] !" # ! $# !" " " != ! !" #!!=!!=!" !! !" !# !" # # # !#$ %$ & % ' $ & %& % $ & %! &( # () % $ & !! !" !! !!! !! ! ! = ! ! "! ! ! "!" !#<

Page 10 of 24 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2025 6 (a) The expansion of in ascending powers of x is . Show that and calculate the value of n. (shown) [4] ( )! ! "#+ !""#$ $ $%%!!++ + ! "!= ( ) !"" # $ # $ % % %"! ! !!"# "# "#!" !"+= + + + #$ #$%& %& ! " # !"!" =#$%& ! " ! # !"!" =#$%& ! " ! " !" " ! = = !"# $ % !& !! "! = ! "# $ % % !& & !! ! !"# =$%&' ! "# $ % #!& !! ! !"# =$%&' !" #$ #% ! ! != !! "!!!=!!=!" #$!% #!==