3E NV AM EOY 2024

NORTH VISTA SECONDARY SCHOOL End-of-Year Examination 2024 Secondary 3 Express CANDIDATE NAME CLASS ADDITIONAL MATHEMATICS 4049 2 OCTOBER 2024 2 hour 15 minutes Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your centre number, index number and name 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. This document consists of 18 printed pages. [Turn over INDEX NUMBER For Examiner’s Use Category Question No. Accuracy Brackets Fractions Units Others Marks Deducted 90

2 1. ALGEBRA Quadratic Equation For the equation 02 =++ cbxax , a acbbx 2 42 −−= . Binomial Theorem ( ) nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− ......21 221 , where n is a positive integer and !)!( ! rrn n r n −=      . 2. TRIGONOMETRY Identities 1cossin 22 =+ AA AA 22 tan1sec += cosec 2 A = 1 + cot 2 A 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 −+= . Area of Abcsin2 1= .

3 [Turn Over 1 Solve the simultaneous equations ( )( ) ( ) 2 322 9 27 1, 1 . xy x y a aa + = = [4]

4 2 (a) Given that the graph of 2 52y mx x= − + lies entirely above the x-axis, find the range of values of m. [2] (b) Find the range of values of x for which 20 4 2 16.x x x −  + [4]

5 [Turn Over 3 In a simplified prey-predator model, fifty wolves were deliberately introduced to an island to curb the population of wild rabbits. The remaining population of rabbits, R, was given by ( )0.02300 6000 10 tR −=+ where t represented the number of days since the introduction of wolves. (a) State the initial number of rabbits. [1] (b) Find how long it would take before the population of rabbits dropped below half the original population. Round your answers to the nearest day. [3] (c) Determine, with reason, whether the number of rabbits would become zero on the island in the long term. [2]

6 4 (a) (i) Sketch the graph of ),1ln(3 −= xy clearly showing the asymptote(s) and the intercept(s) of the graph. [3] (ii) Hence, determine the equation of the straight line needed to be drawn to obtain a graphical solution of the equation .1 3 1 x ex − =− [2] (b) Find the value of the constant p such that ln100 0.x x pe−= [2]

7 [Turn Over 5 (a) Simplify 5 3 3 4 , 2 3 2 2 2 3 27  −+ −+ giving your answer in the form 3 9 hk+ where h and k are integers. [4] (b) Given that 2 3 2 33 5 27 125 2x x x x x+− =   , find the value of .30x [3]

8 6 By considering the general term in the binomial expansion of 11 2 3 2 ,x x  + determine whether the ratio of the coefficient of 7x term to that of 3x term exists. [4]

9 [Turn Over 7 (a) Factorise 333 24xy− completely. [2] (b) Express ( )( ) 3 2 3 10 12 xx xx −− +− in partial fractions. [6]

10 8 (a) A circle has equation 22 8 6 4 0x y x y+ − + − = . Showing your working clearly, determine whether a point T (5, 1) lies inside, outside or on the circle. [3] (b) The diagram shows a simplified map of the Indian Ocean. Geological stations ( )0,3P , ( )0,7Q and ( )11,0R detected an earthquake and a geologist is attempting to locate the epicentre C of the earthquake. Instruments at P and Q detected the earthquake at the same time, indicating that the epicentre C is equidistant from P and Q. Instruments at R detected it a few seconds later in the direction as indicated by the line L in the diagram above. The line L makes an angle of 45 with the x-axis. P (0, 3) Q (0, 7) R (11, 0) L C 45 x y