4E Northbrook AM P2 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 4049/02 Paper 2 Select a time zone. 24 August 2026 2 hours 15 minutes Candidates answer on the Question Paper. READ THESE INSTRUCTIONS FIRST Write your name, class and register number in the spaces 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. The use of an approved scientific calculator is expected, where appropriate. 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. 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 marks for this paper is 90. FOR EXAMINER’S USE Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 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= − = − = − A AA 2tan1 tan22tan −= 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 Do not use a calculator in this question. Given 3 2cos −=A for 180 360A   , express cot A in the form of 5h where h is a rational number. [3]

4 2 (a) Given that BA tan3tan = , show that ( ) B BBA 2sin21 2sintan +=− . [4]

5 [Turn over (b) (i) Solve, in terms of  , the equation cos 2 3 cos 1xx=− for 02 x  . [4] (ii) Hence solve, in terms of  , cos 3 cos 1 2 xx=− for 02 x  . [1]

6 3 A computer animation shows a cartoon person walking in a straight line such that t seconds after it passes through a fixed point O, its velocity, v cm/s, is given by 212v pt t=− , where p is a constant. (a) Given that the cartoon person attains a maximum walking speed at 2 seconds, show that 48p= and hence find the maximum speed. [3] (b) Explain clearly why the total distance travelled by the person in the first 7 seconds is not obtained by finding the value of the displacement, s, when 7t = . [2]

7 [Turn over (c) Find the total distance travelled by the person in the first 7 seconds. [4]

8 4 A circle 1C , with centre C and radius r units, has equation 0168422 =−+−+ yxyx . (a) Find the coordinates of C and the value of r . [3] (b) A circle 2C is the reflection of the circle 1C in the line 3y= . Find the equation of the reflected circle 2C and justify with reasons whether circles 1C and 2C intersect each other at 2 points, touch each other at 1 point or do not touch each other. [4]

9 [Turn over 5 (a) A formula for working out the braking distance d, for a vehicle travelling at speed v, is 32d av bv=+ , where a and b are constants. Values of d for different values of v have been collected. Explain how a straight line can be drawn to represent the formula, and state how the values of a and b could be obtained from the line. [4]

10 (b) The value, $V, of a limited edition watch has been increasing each year from 2016 to 2025. A watch trader claims that the increase is exponential and so can be modelled by an equation in the form kt oV V e= , where oV and k are constants and t is the time in years since 1st January 2016. The table below gives values of V and t for some of the years from 2016 to 2025. Year 2016 2019 2022 2025 t years 0 3 6 9 $V 12000 12950 13900 15000 (i) Plot lnV against t and draw a straight line graph to show that the model is valid for the years 2016 to 2025. [3]