ACSI 2016 Y3EXP FYE AMath P1

Anglo - Chinese School (Independent) FINAL EXAMINATION 2016 YEAR THREE EXPRESS ADDITIONAL MATHEMATICS Paper I Friday 7 October 2016 1 hour 30 minutes Additional Materials: Answer Paper (6 Sheets) ________________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Write your index number on all the work you hand in. Write in dark blue or black pen. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions. Write your answers on the separate Answer Paper provided. 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 a 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 number of marks for this paper is 60. ___________________________________________________________________________ This question paper consists of 4 printed pages. [Turn Over

2 ACS(Independent)MathDept/Y3Exp/AM/2016/Final Exam 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) !! ! n n nn n r r rnr r  − −+= = − 2. TRIGONOMETRY Identities 1cossin 22 =+ AA AA 22 tan1sec += AAec 22 cot1cos += sin( ) sin cos cos sinAB A B A B±= ± cos( ) cos cos sin sinAB A B A B±=  tan tantan( ) 1 tan tan ABAB AB ±±=  sin 2 2sin cosA AA= 2 2tantan2 1 tan AA A= − Formulae for ∆ABC C c B b A a sinsinsin == Abccba cos2222 −+= 1 sin2 ab C∆= 22 2 2cos2 cos sin 2cos 1 1 2sinA AA A A= − = −=−

3 ACS(Independent)MathDept/Y3Exp/AM/2016/Final Exam Answer all questions 1 (i) Find the value(s) of x for which 2 1 2 1 3 13 xx = − . [2] (ii) Sketch the graph of 2 1 3 − = xy and of 2 1 3 1 xy = on the same axes, indicating the coordinates of the point of intersection. [2] 2 In triangle ABC shown below, sides AB and AC are )23( +b cm and )23( + cm respectively and ο45=∠BAC . Given that the area of triangle ABC is )232 11( + cm2, (i) show that 2=b , [2] (ii) find the perpendicular distance from B to AC, leaving your answer in the form qp +2 . [3] 3 A biologist conducted a research on a particular type of bacteria. At the start of the experiment, there were 50 bacteria in the growth medium. After 8 hours, there were 960 bacteria. The growth of bacteria is assumed to follow the equation ktePP ο= , where P is the number of bacteria after t hours, οP is the initial number of bacteria and k is the growth constant. (a) Find the value of k . [3] (b) Determine the number of bacteria present after 1 day of growth. [1] (c) Determine the number of hours for the number of bacteria to be 15 times the original number. [2] 4 Express ( )( ) 2 2 13 )3(2 −+ − xx x in partial fractions. [6] )23( +b )23( + A B C ο45

4 ACS(Independent)MathDept/Y3Exp/AM/2016/Final Exam 5 The expression khxxx +++ 23 3 is exactly divisible by )2( −x and has a remainder of 30 when divided by )1( +x . (a) Find the value of h and of k. [5] (b) Hence, or otherwise, (i) determine the remainder when the above expression is divided by )3( −x , and [1] (ii) factorise the expression completely. [3] 6 (a) Solve, for οο 3600 ≤≤ x , the equation 5cot3cos4 2 += xxec . [5] (b) Given that 50 << z , find the maximum value of z such that 6.0)12tan( =−z . [4] 7 (a) (i) Sketch the graph of xy 2cos2= for the domain π20 ≤≤ x , showing clearly the coordinates of the turning point(s). [2] (ii) To solve the equation xx =−ππ 2cos2 , a line must be added to the graph of (i). Find the equation of this line. [2] (iii) Insert the line on the graph in (i). [1] (iv) Hence determine the number of points of intersection in the interval π20 ≤≤ x , for the equation xx =−ππ 2cos2 . [1] (b) Given that 4 3sin =A and Acos is negative, find without the use of calculators, (i) Asec , [2] (ii) )cos( A− , [1] (iii) ).2sin( A−π [2] 8 (a) Solve the equation 2)3(log)1(log 42 =−−+ xx . [4] (b) Solve the equation 18)3( =+xx ee . [3] (c) If ))(lg(lg4)(lg)(lg4 22 yxyx =+ , express y in terms of x. [3] End of Paper

5 ACS(Independent)MathDept/Y3Exp/AM/2016/Final Exam Answers to EOY Y3 AM Paper I 1(i) (ii) x=9 6(a) 104.00, 284.00, 450,2250. 2(ii) 23+ 6(b) 3.91 3(a) 0.369 7(a)(i) (b) 351000 (ii) Y=(x/π)+1 (c) 7.33 hours3 (iii) 2 solutions 4 2)1( 1 )1(4 5 )3(4 3 −−−++ xxx (b)(i) 7 4− 5(a) h = -16 , k = 12 (ii) 4 7− (b)(i) 18 (iii) 4 7− (ii) (x-2)(x+6)(x-1) 8(a) X=7 (b) 1.10 © Y=x2