ACSI 2015 Y3EXP FYE AMath P1

Anglo - Chinese School (Independent) FINAL EXAMINATION 2015 YEAR THREE EXPRESS ADDITIONAL MATHEMATICS PAPER 1 Wednesday 30 September 2015 1 hour 30 minutes Additional Materials: Answer Paper (7 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

ACS(Independent)MathDept/Y3Exp/AM1/2015/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/AM1/2015/Final Exam 3 Answer all the questions. 1 A cylinder with volume v cm3 has base radius )323( + cm and height ) 3 135( + cm. Express v π in the form 3ba+ where a and b are integers. [4] 2 (i) Find the coordinates of the point of intersection of 2 1 3xy = and 2 5 xy = for 0≥x . [2] (ii) Sketch the graph of 2 5 xy = and 2 1 3xy = on the same diagram, indicating the point of intersection and any intercepts. [2] 3 The curves xy 3 1= and 06152 =+− yxyx cut at two distinct points A and B. Find the equation of the perpendicular bisector of AB. [6] 4 The equation 632 2 =− xx has roots βα and . Find (i) the value of βα + and αβ , [2] (ii) the quadratic equation whose roots are 3βα and 3αβ . [4] 5 Given the function ( ) 22sin4 += xxf , (i) state the period and the amplitude of the function, [2] (ii) solve the equation 022sin4 =+x for °≤≤° 1800 x , [3] (iii) sketch the graph of ( ) 22sin4 += xxf for °≤≤° 1800 x , showing the intercepts clearly. [2] 6 (a) Find the range of values of k for which kkxx >+ )( for all real values of x . [3] (b) Find the range of values of p for which the line xpy 2+= intersect the curve 522 =+ yx at two distinct points. [4] 7 The equation of a circle, 1C , is 01561022 =−−−+ yxyx . (i) Find the coordinates of the centre and the radius of 1C . [3] (ii) Show that the point (-1, 2) lies inside the circle 1C . [2] (iii) Find the equation of the circle 2C which is a reflection of 1C in the line 1−=x . [2]

ACS(Independent)MathDept/Y3Exp/AM1/2015/Final Exam 4 8 Solve the following equations. (a) )2(log2log)718(log 333 −=−− xx . [4] (b) 1 1 1214 − −     =+ x x ee [4] 9 (a) (i) Solve the equation 02723 23 =−−− xxx . [4] (ii) Hence, find the values of θ between 0o and 360o inclusive such that 2sin7)2sin3(sin2 +=− θθθ . [3] (b) The expressions 6143 ++ xx and 3082 23 −−− xxx leave the same remainder when divided by ax− . Find the possible values of a. [4] ___________________________________END OF PAPER 1 _______________________ Answers 1 1923112 + 2(i) 23.1=x 32.3=y 3 )6 1,2( , )3 2,2 1( Equation: 3 103 −= xy 4(i) 2 3=+βα , 3−=αβ (ii) 0324994 2 =++ xx 5(i) amplitude = 4 Period = °180 (ii) 165,105=x 6(i) 04 <<− k (ii) 55 <<− p 7(i) Centre = (5, 3) Rad = 7 (iii) 0961422 =+−++ yxyx 8(i) 2 5=x (ii) 10.2=x 9(a)(i) 2,3 1,1 −−=x (ii) °°°= 5.340,270,5.199θ (b) 92 −−= ora