SCGS 2025 AM Prelim Paper 1

[Turn over SINGAPORE CHINESE GIRLS’ SCHOOL PRELIMINARY EXAMINATION 2025 SECONDARY FOUR O-LEVEL PROGRAMME CANDIDATE NAME CLASS 4 REGISTER NUMBER CENTRE NUMBER INDEX NUMBER ADDITIONAL MATHEMATICS 4049/01 Paper 1 Thursday 28 August 2025 2 hours 15 minutes Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your name, class, register number, centre number and index number at the top of this page. Write in dark blue or black pen on both sides of the paper. 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 electronic 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. FOR EXAMINERS USE Q1 Q5 Q9 Q2 Q6 Q10 Q3 Q7 Q11 90 Q4 Q8 Q12 The Question Paper consists of 19 printed pages and 1 blank page.

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the quadratic equation ax 2 + bx + c = 0 , Binomial Expansion , where n is a positive integer and 2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B sin A sin B tan ( A ± B ) = sin 2A = 2 sin A cos A cos 2A = cos2 A – sin2 A = 2cos2 A – 1 = 1 – 2 sin2 A tan 2A = Formulae for D ABC a 2 = b 2 + c 2 - 2bc cos A D = ab sin C ! !"##$ ! "!!±!= ( ) !" " !" ! !! ! ! " " !!! !#$ # #$ # $ # $ $ " !! !"# "# "#+= + + + + + +$% $% $%&' &' &' !! ( ) ( )( )!!" "" " ! !! ! "! " "!" " !! +"#==$% !&' ! !!"# !"# $ !"# !"# !" !" ± ! ! ! "#$ % "#$ ! !! !"# !"# !"# !"# ABC == !"

3 [Turn over 1. Oil is dripping into an empty inverted right circular cone at the rate of 16 cm3 per second. The height, h cm, of the cone is three-quarters its radius, r cm. [Volume of a right circular cone = ] (a) Show that the volume, V cm3, of the cone is . [1] (b) Calculate, at the instant when the radius is 8 cm, the rate of change of the radius. [3] !" # !"!"" " !" #!" !=

4 2. The straight line intersects the curve at the points A and B. Show that the length AB is units. [5] !" =+!" !"## =+!! !"!" !!

5 [Turn over 3. (a) Find . [3] (b) Hence evaluate , giving your answers in exact form. [4] !"! ! !"#!$ % % ! !!"!!"#$ %$$ & !

6 R S T Q U P V 4. The diagram shows a circle passing through the points P, Q, R and S. Chord SQ and RP intersect at T. The straight line URV is a tangent to the circle at R. The tangent meets the line SQ extended at U such that RU = RS. (a) Show that triangle RQU is isosceles. [3] (b) Show that angle SPT = 2 ´ angle QPT. [3]

7 [Turn over 5. The equation of a curve is , where m is a constant. (a) In the case where m = 1, show that the line is a tangent to the curve. Find the coordinates of the point of contact. [3] (b) Find the range of values of m for which the curve is above the line y = 5. [5] ( )! "# $!" # #"=+ ! + !"!"=!

8 6. (a) By using long division, show that is a factor of . [2] (b) Express in partial fractions. [5] ! !"!+ !" "# "!! !!+ ! ! "! ## $ ## !% ! !! !! ! !! !+ !

9 [Turn over 7. (a) State the amplitude and period of the graph of . [2] (b) Sketch the graph of for . [3] (c) By adding another suitable curve on your sketch, determine the number of solutions of the equation . [3] !"#$ %& !"=+ !"#$ %& !"=+ !"!!"" !"#$% &'#"( (" !!=! !

10 8. The highest point on a circle is (2, 8). The equation of the tangent, T, to at the point (6, 6) is . (a) Find the equation of . [6] A second circle, , is the reflection of in the line T. (b) Find the equation of . [3] !!!!!"" #!"+= !! !! !!!!