2026 Zhonghua Prelim P1

ZHONGHUA SECONDARY SCHOOL PRELIMINARY EXAMINATION 2026 SECONDARY 4 EXPRESS/ SECONDARY 5 NORMAL (ACADEMIC) Candidate Name Class Register Number ADDITIONAL MATHEMATICS 4049/01 Paper 1 26 August 2026 2 hours 15 minutes Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your name, class and index number on all the work you hand in. 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 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 of the marks for this paper is 90. Setter: Mdm Seoh Bee Hua Vetter: Mr Francis Tan This document consists of 19 printed pages and 1 blank pages. Mathematical Formulae O 90

2 1. ALGEBRA Quadratic Equation For the equation 02 =++ cbxax a acbbx 2 42 −−= Binomial expansion nrrnnnnn bbarnbanbanaba ++++++=+ −−− 221 21)( , where n is a positive integer and ! )1()1( )!(! ! r rnnn rnr n r n +−−=−=  2. TRIGONOMETRY Identities 1cossin 22 =+ AA AA 22 tan1sec += 22cosec 1 cotAA=+ BABABA sincoscossin)sin( = BABABA sinsincoscos)cos( = BA BABA tantan1 tantan)tan(  = AAA cossin22sin = AAAAA 2222 sin211cos2sincos2cos −=−=−= A AA 2tan1 tan22tan −= Formulae for ABC C c B b A a sinsinsin == Abccba cos2222 −+= Abcsin21= 1 The Richter scale is used to measure the magnitude of an earthquake.

3 [Turn over The magnitude, M, is related to the amplitude, A mm, of the seismic wave recorded by a seismograph according to the model. lg 0.001AM =  It was observed that two earthquakes happened within a month. (a) Find the magnitude of the first earthquake where the amplitude of the seismic wave was recorded at 1200 mm. (b) The second earthquake has a magnitude of 4.6, find the amplitude, A mm, of the seismic wave recorded. (c) A researcher proposed a new model to measure the magnitude of an earthquake. 3log 9AM =  Using this new model, find the magnitude of t he first earthquake with an amplitude of 1200 mm. [1] [2] [2]

4 2 Show that the curve 23 5 4y x x=− + − lies completely below the line 2yx=− . [4]

5 [Turn over 3 (a) Differentiate 2e (3 1)x x− with respect to x. (b) Hence find 2 4d x xe x . [2] [3]

6 4 (a) Prove that 1 cos 2 sin 2 2 tansin 2 1 cos 2AA AAA− +=+ . (b) Hence solve the equation 1 cos 2 sin 2 2 cotsin 2 1 cos 2 7AA AAA− +=+ for 90 90 A−    . [2] [4]

7 [Turn over 5 A straight line graph is drawn for lg y against x to represent the curve xhyk += . (a) Given that the line passes through the points (0, 3) and (10, 23) find the value of k and of h. (b) Hence, find the value of x when lg 4yx= . [4] [2]

8 6 The equation of a curve is ( ) 2 2 22x yk+ − = , where k is a constant. (a) Find the set of values of k for which the line 2yx= meets this curve. (b) In the case where the line 2yx= is tangent to the curve, deduced the value(s) of k , giving your answer in the form of ab where a and b are integers. [5] [1]

9 [Turn over 7 (a) Find the first 4 terms in the expansion of 5 2 4ax − in ascending powers of x, simplifying each term. (b) Given that there is no term in x in the expansion of ( ) 5 5 2 2 4axx −−  , find the value of the constant a. [4] [3]

10 8 A delivery company uses three drones, P, Q and R, for a coordinated flight test. Drone Q is hovering at a fixed point 7 m due south of a control point O. Drone P flies due west from O. At time t seconds, drone P is 2x metres west of O, x is increasing at a constant rate of 5 m/s. Drone R is programmed to continuously maintain an equilateral triangle formation with drones P and Q throughout the flight. (a) Show that the distance between the two drones, PQ, is 24 49x + metres. (b) Find the rate of change of the distance PQ, in m/s, when x = 12. [1] [4] Q 7 m O P 2x m R