ACSI 2018 Y3EXP FYE AMath P1

Anglo - Chinese School (Independent) FINAL EXAMINATION 2018 YEAR THREE EXPRESS ADDITIONAL MATHEMATICS PAPER 1 Friday 5 October 2018 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 5 printed pages. [Turn over

ACS(Independent)MathDept/Y3EXP/AMP1/2018/FINAL EXAM 2 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 += Formulae for ∆ABC C c B b A a sinsinsin == Abccba cos2222 −+= 1 sin2 ab C∆=

ACS(Independent)MathDept/Y3EXP/AMP1/2018/FINAL EXAM 3 Answer all the questions. 1 Solutions to this question by accurate drawing will not be accepted. In the diagram below, A(0, 2), B(6, 0) and C(h, k) are the midpoints of the sides of the triangle PQR, and Q is the point )1 ,2( −− . Find the coordinates of C. [4] 2 (a) Simplify 2 180 245 3 125+− , leaving your answer in surd form [2] (b) Solve the equation 1 213 x x = − . [3] 3 When the graph of xy x+ against y is drawn, a straight line is obtained. Given that the line passes through the points ( )5, 2− and ( )1, 4 , find (i) y in terms of x, and [4] (ii) the value of x when y = 2. [2] 4 A rectangle has a length of 2α cm and a width of 2βcm, where α and β are the roots of a quadratic equation. (i) Given that the equation 2 53 196 0xx−+= has roots 2α and 2β , find the quadratic equation in x whose roots are α and β . [5] (ii) Hence, find the dimensions of the rectangle. [2] P C R Q B A y x

ACS(Independent)MathDept/Y3EXP/AMP1/2018/FINAL EXAM 4 5 The circle 22 1 : 2 8 23Cx y x y+−+= passes through the point ( 5, 2)P −− . (i) Find the equation of the line PQ which is a tangent to the circle at P. [3] (ii) Determine if the point ( 3,1)R − lies within 1C . [3] (iii) Find the equation of circle 2C which is a reflection of 1C about the line 3x=− . [2] 6 Answer the whole of this question on a single sheet of graph paper. An experiment was done to estimate the population of bacteria, P, present in a culture at time T days after the start of the experiment. The results are shown in the table below. Time, T (days) 3 5 10 20 40 Population of Bacteria, P 900 1884 10000 16000 47315 It is known that P and T are related by the equation bP aT= , where a and b are constants. (i) Using 4 cm to represent 1 unit on the vertical axis and 2 cm to represent 0.2 units on the horizontal axis, plot the graph of lg P against lg T. [2] (ii) Use your graph to (a) estimate the value of a and of b, [3] (b) identify the abnormal reading and estimate its correct value, [2] (c) estimate the population of bacteria after 4 days. [2] 7 (a) (i) Solve the equation 23x−= . [2] (ii) Sketch the graph of 23yx= −−+ , labelling clearly the intercepts and the vertex. [2] (iii) Find the equation of the straight line which is to be inserted in the diagram in (ii) in order to solve the equation 54 2 xx+= − . [2] (b) Sketch the graph of cos 2yx= for π≤≤ x0 . By adding a suitable straight line, state the number of solutions to the equation 2 2 cos 2xxππ−= . [4]

ACS(Independent)MathDept/Y3EXP/AMP1/2018/FINAL EXAM 5 8 (a) Simplify 2 2log log log nm m mn m × . [2] (b) It is given that 2log xr= and 4log ys= . If 2 64xy = and 2 128xy = , determine the value of r and of s. [6] (c) Solve (5 ) 1 1 lnlog xe xe− = + , leaving your answer in terms of e. [3] END OF PAPER ONE Answer Key 1 (8, 3) 2 (a) 45 (b) 1 4x= 3 (i) 11 2 23 xy x −= + (ii) 5 6x= 4 (i) 2 9 14 0xx−+= (ii) 4 cm by 14 cm 5 (i) 3 13yx= + (ii) R lies outside of 1C (iii) 22 14 8 25 0xy xy++ ++= 6 (ii) (a) a = 158.5, b = 1.52 (b) 5623 (c) 1413 7 (a)(i) 5x= or 1x=− (ii) (iii) 24yx= −− 7 (b) 2 solutions 8 (a) 4 (b) 4 3s= ; 5 3r = (c) 5 ex e= −