AMKSS Prelim 2019 4E5N AMath P2 Questions

[Turn Over Class Index Number Name ANG MO KIO SECONDARY SCHOOL PRELIMINARY EXAMINATION 2019 SECONDARY FOUR EXPRESS / FIVE NORMAL ACADEMIC ADDITIONAL MATHEMATICS 4047/02 Paper 2 Wednesday 18 September 2019 2 hours 30 minutes Answer on the question paper. No additional materials are required. READ THESE INSTRUCTIONS FIRST Write your name, index number and class 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 a pencil for any diagrams or graphs. Do not use 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. At the end of the examination, fasten all your work securely together. 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 100. For Examiner’s Use 100 This document consists of 22 printed pages.

2 AMKSS 4E5N Prelim 4047/02/2019 [Turn Over Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 =++ cbxax , a acbbx 2 42 −±−= Binomial expansion ,......21)( 221 nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− where n is a positive integer and ! )1(...)1( )!(! ! r rnnn rnr n r n +−−=−=      2. TRIGONOMETRY Identities AAec AA AA 22 22 22 cot1cos tan1sec 1cossin += += =+ BA BABA BABABA BABABA tantan1 tantan)tan( sinsincoscos)cos( sincoscossin)sin(   ±=± =± ±=± A AA AAAAA AAA 2 2222 tan1 tan22tan sin211cos2sincos2cos cossin22sin −= −=−=−= = Formulae for ABC∆ a2 = b2 + c2 – 2bc cos A Cabsin2 1=∆ C c B b A a sinsinsin ==

3 AMKSS 4E5N Prelim 4047/02/2019 [Turn Over 1 On 1st January 2010, a certain type of bacteria was found at the bottom of a seabed. It is known to grow with time, such that its population P, after t years is given by P = 50 000ekt , where k is a constant. (i) Given that the population doubles in two years, show that k = . [2] (ii) Find the size of the population of bacteria on 1st January 2015, giving your answer correct to the nearest 10 000. [2] (iii) Find the year in which the population reaches 450 000. [3] 2ln2 1

4 AMKSS 4E5N Prelim 4047/02/2019 [Turn Over 2 (a) Given that xey x sin= , prove that 0222 2 =+− ydx dy dx yd . [4]

5 AMKSS 4E5N Prelim 4047/02/2019 [Turn Over (b) A particle moves along the curve 1 162 − += x xy in such a way that both its x-coordinate and y-coordinate are positive and vary with time. Find the x-coordinate of the particle at the instant that the rate of increase of its y-coordinate is twice the rate of decrease of its x-coordinate. [5]

6 AMKSS 4E5N Prelim 4047/02/2019 [Turn Over 3 (a) Solve 310log2lg 2 =− xx . [5]

7 AMKSS 4E5N Prelim 4047/02/2019 [Turn Over (b) Given that ( ) axym =3log and ( ) bxm =2log , express       y x mlog in terms of a and b. [5]

8 AMKSS 4E5N Prelim 4047/02/2019 [Turn Over (c) Without using a calculator, solve for a and b, the simultaneous equations 2184 2ab −×= , 3 3 81.ab = [4]

9 AMKSS 4E5N Prelim 4047/02/2019 [Turn Over 4 (a) The equation of a curve is 3 2 5 10yx kx= + , where k is a constant. Given that its normal at x = 2 will never meet the y-axis, find the value of k. [3]

10 AMKSS 4E5N Prelim 4047/02/2019 [Turn Over (b) The equation of a curve is ln 3 2yx= + . (i) Determine the range of values of x for which the equation of the curve is defined. [1] (ii) Find an expression for dy dx , giving your answer in the simplest form. [2]