4E Northbrook AM P1 2026

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. This document consists of 21 printed pages and 1 blank page. [Turn over O NORTHBROOKS SECONDARY SCHOOL Preliminary Examination 2026 Secondary 4 CANDIDATE NAME CLASS REGISTER NUMBER ADDITIONAL MATHEMATICS 4049/01 Paper 1 Select a time zone. 13 August 2026 2 hours 15 minutes Candidates answer on the Question Paper. READ THESE INSTRUCTIONS FIRST Write your name, class and index number 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. 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 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 90

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0ax bx c+ + = , a acbbx 2 42 −−= Binomial expansion ( ) nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− ......21 221 , where n is a positive integer and ( ) ! ( 1)...( 1) ! ! ! n n n n n r r n r r r  − − +== − . 2. TRIGONOMETRY Identities 22sin cos 1AA+= 22sec 1 tanAA=+ 22cosec 1 cotAA=+ sin( ) sin cos cos sinA B A B A B =  cos( ) cos cos sin sinA B A B A B=  tan tantan( ) 1 tan tan ABAB AB =  sin 2 2sin cosA A A= 2 2 2 2cos 2 cos sin 2cos 1 1 2sinA A A A A= − = − = − 2 2 tantan 2 1 tan AA A= − Formulae for ABC C c B b A a sinsinsin == 2 2 2 2 cosa b c bc A= + − 1 sin2 bc A=

3 [Turn over 1 (a) Do not use a calculator in this question. A circular cylinder of volume ( )3 5 4 − cm3 has a height of ( )35+ cm and a radius of r cm. Obtain an expression of r2 in the form 5 ,ab c + where a, b and c are constants. [4]

4 (b) The line 3xy−= intersects the curve 22 3 19 0x xy y− + + = at points P and Q. Find the coordinates of P and of Q. [4]

5 [Turn over 2 (a) Two variables, x and y, are related by the equation 6 3 1.4 12 xy =−  Given that both x and y vary with time, find the value of y when the rate of change of y is 12 times the rate of change of x. [4] (b) The function f is given by 2 f ( ) 1 xex x= − for 1x . Show that f is a decreasing function. [3]

6 3 (a) Express ( )( ) 2 2 7 30 3 53 xx xx −+ −+ in partial fractions. [5]

7 [Turn over (b) Differentiate ( ) 2ln 3x + with respect to x. [1] (c) Use your answers to part (a) and (b) to find ( )( ) 2 2 7 30 3 d . 53 xx x xx −+ −+ [2]

8 4 The starting section of a roller coaster track can be modelled by the equation 23 24 103y t t= − + , where y represents the height, in metres, of the first roller coaster car at time t seconds. (a) Explain the meaning of the constant term 103 in this model. [1] (b) Express 23 24 103tt −+ in the form of 2( ) .a t h k−+ [2] It is stated in safety requirements that roller coaster tracks must be at least 25 m above the ground. (c) Determine, with reasons, whether this section of the roller coaster track has met the safety requirements. [1]

9 [Turn over 5 (a) By considering the general term in the binomial expansion of 8 2 12x x  − , explain why no term is independent of x in this expansion. [3] (b) Find the first 3 terms in the expansion of 8 2 12x x  − in descending powers of x. Give the terms in their simplest form. [2] (c) Find the coefficient of 2x in the expansion of 8 32 11 52 xxx   +−     . [2]

10 6 (a) Given that 2 13y x x=− , show that ( )4 15d d 2 1 3 xxy x x −= − . [3] (b) Hence find the value of ( )0 1 2 4 15 d . 2 1 3 xx x x− +− − [3]