2026 Zhonghua AM Prelim P2

ZHONGHUA SECONDARY SCHOOL PRELIMINARY EXAMINATION 2026 SECONDARY 4 EXPRESS/ SECONDARY 5 NORMAL (ACADEMIC) Candidate Name Class Register Number ADDITIONAL MATHEMATICS 4049/02 Paper 2 27 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: Mr Pang Hong Ming Vetter: Mr Francis Tan This document consists of 22 printed pages and 2 blank pages. O 90

2 Mathematical Formulae 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=

3 [Turn over 1 Without using a calculator, find the values of p and q , for which the solution of the equation 108 75 5 xx −= is 15 103 pq+ . [5]

4 2 (a) It is given that 32f ( ) 2 17 41 30x x x x= − + − . Show that (2 3)x− is a factor of f ( )x . [1] (b) Express f ( )x as a product of three linear factors. [3] (c) Hence, solve f ( ) 0x = . [1]

5 [Turn over 2 (d) Information about another function g( )x is given below. 1. g( )x gives a remainder of 2 when divided by ( 3)x− . 2. g( )x gives a remainder of 2 when divided by (3 1)x− . 3. g( )x is a quadratic function. Suggest a possible function for g( )x in the form 2ax bx c++ . [2] (e) Factorise 63 27xy+ . [1]

6 3 For a particular curve, 2 2 d 6sin 2 4cosd y xxx =− . The curve passes through the point ( )0, 5Q and the gradient of the curve at Q is 2. Find the equation of the curve. [5]

7 [Turn over 4 (a) State the range of principal values for (i) 1sin x− , [1] (ii) 1tan x− . [1] (b) Without using a calculator, evaluate 2sin 60 tan 45 cos 45cos30  + . [3]

8 5 A farmer wants to enclose a rectangular pen against a long, straight barn wall. The barn wall forms the entire top side of the pen, so no fencing is needed there. Fencing is required for: • the bottom longer side (length L metres), using standard fencing at $1 per metre. • the two shorter sides and one internal dividing fence (each of length x metres), using reinforced fencing at $4 per metre. The internal dividing fence runs parallel to the shorter sides and splits the pen into two equal sections. The farmer has a total budget of $120. The diagram below shows the rectangular pen. (a) Show that the total area of the pen, 2 mA , is given by 2120 12A x x=− . [3] barn wall x L

9 [Turn over (b) Hence, find the maximum area of the pen. Justify that the area found is a maximum. [4]

10 6 A particle travels in a straight line such that the velocity of the particle, v m/s, t seconds after leaving a fixed point O , is given by 2 9 20v t t= − + . Find the total distance travelled by the particle in the interval 0t = to 7t = . [7]