ACSI 2015 Y3EXP FYE AMath P2

Anglo-Chinese School (Independent) FINAL EXAMINATION 2015 YEAR THREE EXPRESS ADDITIONAL MATHEMATICS PAPER 2 5 October 2015 1 hour 30 minutes Additional Materials: Answer Paper (8 sheets) Graph Paper (1 sheet) 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 quest This question paper consists of 4 printed pages. [Turn over…

ACS(Independent)MathDept/Y3AddMathP2/2015/FinalExam 2 Mathematical Formulae ALGEBRA Quadratic Equation For the equation 02 =++ cbxax , a acbbx 2 42 −±−= Binomial expansion ( ) 1 22 ... ... ,12 n n n n nr r nnn nab a ab a b a b b r −− −  + = + + ++ ++     where n is a positive integer and ( ) ! ( 1)...( 1) !! ! n r n nn n rC rnr r − −+= = − . 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/Y3AddMathP2/2015/FinalExam 3 Answer all questions. 1 Solve the equation xxx 2142 532 −−+ = . [4] 2 A herd of 20 deer is introduced to an island as part of a conservation project. The population of deer is predicted to increase so that after a period of t years, the population, D, is given by te kD 14.041 −+ = . (i) Find the value of k. [1] (ii) Find the predicted number of deer after 10 years. Give your answer correct to the nearest integer. [2] (iii) Find the number of years it will take for the population to reach 70. [3] 3 (i) Solve the equation xx 42 =− . [3] (ii) On the same diagram, sketch the graphs of xyxy 4 and 2 =−= , labelling the axes intercepts and their intersection point(s) clearly. [3] (iii) Hence, or otherwise, solve xx 42 >− . [1] 4 (i) The line 135 =++− n y m x cuts the x-axis and y-axis at P and Q. Find the coordinates of the midpoint of PQ in terms of m and n. [3] (ii) If 32 1 += xy is the perpendicular bisector of PQ, find the value of m and of n. [5] 5 (i) Prove the identity xx x x x sec2sin1 cos cos sin1 ≡−+− . [3] (ii) Prove that 1)costan)(sin1)(sin1(sin 222 −≡++−+ xxxxx . [3] (iii) Hence, solve )costan)(sin1)(sin1(sin4sin1 cos cos sin1 222 xxxxxx x x x ++−+=−+− for x between 0o and 360o . [2] 6 (a) (i) Given that CBxxAxxxx ++−+≡+++ )1)((1 2224 , determine the values of A, of B and of C. [3] (ii) Hence, state the remainder when 124 +++ xxx is divided by 12 −x . [1] (b) Express 1 27 2 3 − ++ x xx in partial fractions. [5]

ACS(Independent)MathDept/Y3AddMathP2/2015/FinalExam 4 7 (a) If ax=tan , express x x 2 2 cos sin23+ in terms of a. [3] (b) Solve, for π≤≤ x0 , the equation xx cotcos2 = , leaving your answers in terms of π . [4] 8 Answer the whole of this question on a single sheet of graph paper. The variables x and y are related by the equation xqpxy += 2 , where p and q are constants. The table shows experimental values of the variables x and y. x 1 2 3 4 5 y 5.5 12.2 19.2 38.0 56.7 (a) Plot the graph of xx x y against . [3] (b) Using your graph (i) estimate the values of p and q, [3] (ii) estimate the value of x when xy 352 = , [2] (iii) identify the abnormal reading and estimate its correct value. [2] (c) If instead, a graph of 2x y is plotted against 2x x , state the value of the gradient of this straight line. [1] End of Paper ANSWER KEY 1. 0.922 2. (i) 100 (ii) 50 (iii) 16.0 3. (i) 5 2 (iii) 5 2<x 4. (i)      +− 2 3,2 5 nm (ii) 9=m , 5=n 5. (iii) °° 240120 or 6. (a)(i) A=2, B=1, C=3 (ii) 3+x (b) 1 3 1 5 ++−+ xxx 7. (a) 253 a+ (b) 6 5,2,6 πππ 8. (b) (i) 3,2 ≈≈ qp (ii) 73.3≈x (iii) abnormal reading = (3, 19.2); correct reading (3, 23.6) (c) Gradient = q )3(≈