ACSI 2019 Y3EXP FYE AMath P1

INDEX NUMBER Anglo - Chinese School (Independent) FINAL EXAMINATION 2019 YEAR THREE EXPRESS ADDITIONAL MATHEMATICS PAPER 1 Friday 4 October 2019 1 hour 30 minutes Candidates answer on the Question Paper. No additional materials are required. READ THESE INSTRUCTIONS FIRST Write your index number on boxes above. 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 in the spaces provided under each question. 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. 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 15 printed pages and one blank page. [Turn over For Examiner’s Use 60

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∆=

3 Answer all the questions. 1 Find the range of values of x for which 23 4 67xx x− +≤ . [2] 2 Given that the line kxy += 2 does not meet the curve xy −= 1 2 , find the least integer value of k. [4] [Turn Over

4 3 On the same axes, sketch the graphs of 3cos 2y θ= and siny θ= for 02 θπ≤≤ , showing clearly the point(s) of intersection. [3]

5 4 Simplify (i) 2 12 log 1 log log 3 km m m m −+ + [2] (ii) 2 2 5 25 log 4 log log log m m m m × × . [3]

6 [Turn Over 5 (a) When the graph of xy is plotted against y, the resulting straight line has a gradient of 2− and an intercept of 0.5 on the xy-axis. Find the gradient and vertical intercept of the graph if 1 xy is plotted against 1 x . [4] [Question 5 continues next page]

7 5 (b) The variables x and y are related in such a way that when ln y is plotted against 2x , a straight line is obtained which passes through ( )1, 2− and ( )4, 7 . (i) Express y in terms of x. [3] (ii) State the value y would take as x approaches zero, leaving your answer in terms of e. [1] [Turn Over

8 6 (i) Solve the equation 13x−= . [2] (ii) On the same diagram, sketch the graphs of xy = and 13 −−= xy . [2] [Question 6 continues next page]

9 6 (iii) Solve the equation 13xx+−= . [3] (iv) Hence, find the range of values of x such that 31 ≤−+ xx . [1] [Turn Over

10 7 The points P and Q have coordinates (3, 5) and (9, 11). (i) Calculate the length of PQ. Express your answer in the form ba . [2] (ii) Find the equation of the circle with PQ as the diameter. [3] [Question 7 continues next page]