2025 Prelim AM P1 (Presbyterian High School QuestionPaper)

Name: Index No.: Class: PRESBYTERIAN HIGH SCHOOL ADDITIONAL MATHEMATICS 4049/01 Paper 1 25 August 2025 Monday 2 hours 15 min PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL 2025 SECONDARY FOUR EXPRESS / FIVE NORMAL (ACADEMIC) PRELIMINARY EXAMINATIONS DO NOT OPEN THIS QUESTION PAPER UNTIL YOU ARE TOLD TO DO SO. INSTRUCTIONS TO CANDIDATES Write your name, index number and class in the spaces provided above. 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. Write your answers in the spaces provided below 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. 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 Examiner’s Use Qn 1 2 3 4 5 6 7 8 9 10 11 12 13 Marks Deducted Marks Category Accuracy Units Notations Others Question No. Setter: Mr Tan Lip Sing Vetter: Ms Sabrina Tan This question paper consists of 23 printed pages and 1 blank page. TOTAL MARKS 90

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0,ax bx c+ += 2 4 2 b b acx a −± −= Binomial expansion 1 22( ) ... ... ,12 n n n n nr r nnn nab a a b a b a b b r −− −  + = + + ++ ++     where n is a positive integer and 2. TRIGONOMETRY Identities 22sin cos 1AA+= 22sec 1 tanAA= + 22cosec 1 cotAA= + sin( ) sin cos cos sinAB A B A B±= ± cos( ) cos cos sin sinAB A B A B±=  tan tantan( ) 1 tan tan ABAB AB ±±=  sin 2 2sin cosA AA= 22 2 2cos 2 cos sin 2cos 1 1 2sinA AA A A= − = −=− 2 2 tantan 2 1 tan AA A= − Formulae for sin sin sin abc ABC= = 2 22 2 cosa b c bc A=+− 1 sin2 bc A∆=

3 1 (a) Differentiate 2 3ln 1 x x   + with respect to x. [3] (b) Hence find 2 2 d1 x xx +∫ . [2]

4 2 It is given that 1cos 2A= and 1sin 2 B=− where 0 90 A°< < ° and 180 270B°< < ° . Find, without using a calculator, the exact value of ( )cos AB− , leaving your answer in the form 26pq + , where p and q are real numbers. [4]

5 3 Baking powder is poured onto a flat surface at a constant rate of 312 cm sπ − , forming a right circular cone. The radius of the cone is always 1 18 of its height. Find the rate of change of the radius of the cone after 3 seconds of pouring. 21Volume ofcone 3 rhπ = [5]

6 4 A and B are the points of intersection of the line 4 21yx= + and the curve 34y x xy−= . (a) Find the coordinates of A and of B. [4] (b) Henry says that the line 245yx−= is perpendicular to the line AB. Is he correct? Justify your answer with workings. [3]

7 5 (a) Write down and simplify the first three terms in the expansion, in descending powers of x, of 8 32 x  − . [2] (b) Given that there is no x term in the expansion of ( ) 8 2 312 2x kx x −− −  , find the constant term in the expansion. [4]

8 6 (a) Express 244 3y xx= −− in the form ( ) 2 px q r++ where p, q and r are constants. [2] (b) Hence, explain whether 24 4 30xx− −= has any real solutions. [2]

9 TURN OVER FOR QUESTION 7

10 7 Peter constructed an open fish tank with a rectangular base of length 4 l m, breadth l m, and height h m. He wanted the total outer surface area of the fish tank to be 5 2m . (a) Show that the volume of the tank, 3mV , is given by ( ) 32 54.5V ll= − [3]