ACSI 2017 Y3EXP FYE AMath P2

Anglo-Chinese School (Independent) FINAL EXAMINATION 2017 YEAR THREE EXPRESS ADDITIONAL MATHEMATICS PAPER 2 Wednesday 11 October 2017 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 5 printed pages. [Turn over…

ACS(Independent)MathDept/Y3AddMathP2/2017/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/Y3AddMathP2/2017/FinalExam 3 1. Solve the equation ( )x x 224 4 = . [3] 2. The roots of the equation ( ) 232 −−= xkx , are 1+α and 4+α where k > 0. Find the value of k in the form ba+ . [4] 3. Express ( )( ) 2 2 132 7 −+ +−− xx xx in partial fractions. [4] 4. A metallic sphere is heated to a temperature of 287oC and left to cool in an air conditioned room. As the sphere cools, its temperature, ToC, t mins after it is left in the room is given by kteT −+= 26522 , where k is a constant. When t = 5, the temperature of the sphere reaches 200oC. (i) Find the value of k correct to 3 significant figures. [2] (ii) Find the temperature of the sphere after 20 mins. [1] (iii) Explain what happens to T as t becomes very large. [1] 5. It is given that ( )( ) ( ) baxxQxxxxxxx ++−+=++−+− 11232432 2345 , where ( )xQ is a polynomial and a and b are constants. (i) State the degree of ( )xQ . [1] (ii) Find the value of a and b. [3] (iii) Hence, state the remainder when 32432 2345 ++−+− xxxxx is divided by 12 2 −− xx . [1] 6. (i) Solve the simultaneous equations: 13 −−= xy 24 1 += xy [3] (ii) Sketch on the same axes, the graph of 13 −−= xy and 24 1 += xy for 0 ≤ x ≤ 10. [3] (iii) Hence, state the range of values of x for which 1234 +≤− xx . [2]

ACS(Independent)MathDept/Y3AddMathP2/2017/FinalExam 4 7. The diagram shows the quadrilateral PQRS. Points P and Q are on the x-axis. Point U lies on RQ. (i) Find the equation of SR. [2] (ii) Given that SP is perpendicular to SR, find the coordinates of P. [2] (iii) Given that the area of PQRS is 41 units, find the coordinates of Q. [3] (iv) If RU : UQ = 2 : 3, find the coordinates of U. [2] (v) Explain with working, if the point U lies on the perpendicular bisector of SP. [3] 8 (a) Solve, for oo x 3600 ≤≤ , 3cos 3tan2 −= xx . [4] (b) (i) Prove that xx x x x cot21sec sin 1sec sin =++− . [3] (ii) Hence, or otherwise, solve ( ) ( ) 1 32 12sec2sin12sec2sin −=−++ xxxx for −π < x < π, leaving your answer in terms of π. [4] [ Please turn over y R (4, 10) x O P Q S (0, 4) U

ACS(Independent)MathDept/Y3AddMathP2/2017/FinalExam 5 9 Answer the whole of this question on a single sheet of graph paper. The table shows some experimental values of two variables, x and y. x 0.8 1.0 1.2 1.4 1.6 1.8 y 21.0 14.9 10.6 9.2 4.5 2.1 It is believed that x and y are related by the equation x baxy += , where a and b are constants. (i) Using 2 cm to represent 1 unit on the horizontal axis and 2 cm to represent 2 units on the vertical axis, plot the graph of xy against x2 [3] (ii) Use your graph to (a) estimate the value of a and of b, [2] (b) identify the abnormal reading and estimate its correct value, [2] (c) find the value of y when x = 1.5. [2] -----------------------------------------------------------The End --------------------------------------------------------- ANSWERS: 1. 2 12=x 2. =k 3 + 17 3. ( )( ) ( ) 22 2 1 1 1 1 32 1 132 7 − +−−+= −+ +−− xxxxx xx 4. (i) 0796.0=k (ii) 75.9oC (iii) 22=T 5. (i) 3 (ii) a = 3 and b = 2 (iii) 23 +x 6. (i) 0, 8 (iii) 80 ≤≤ x 7. (i) 42 3 += xy (ii) ( )0,6 (iii) (9, 0) (iv) ( )6,6 (v) U does not lie on the ⊥ bisector 8. (a) oooo andx 360300,60,0= (c) 6 π− , 3 2π− , 3 π and 6 5π 9. (iia) a = − 4.95, b = 20.00 (iib) y = 7.14 (iic) y = 5.73