ACSI 2016 Y3EXP FYE AMath P2 Solutions

Anglo - Chinese School (Independent) FINAL EXAMINATION 2016 YEAR THREE EXPRESS ADDITIONAL MATHEMATICS PAPER 2 Wednesday 12 October 2016 1 hour 30 minutes Additional Materials: Answer Paper(7 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 question paper consists of 4 printed pages. [Turn over

ACS(Independent)MathDept/Y3Exp/AM2/2016/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/AM2/2016/Final Exam 3 Answer all the questions. 1 Prove that x xecxx cos1 cos1)cos(cot 2 + −≡− [4] 2 Solve )3(253 1 xx −+ −= . [4] 3 Given that the curve 0342 =+− xyx cuts the line 123 += xy at A and B, find the coordinates of A and B. [5] 4 A circle, C1, has equation 02010422 =−+−+ yxyx (i) Find the coordinates of the centre and the radius of 1C . [3] (ii) If circle C1 is reflected on the line 1−=y , find the equation of the reflected circle. [2] 5 (a) Find the range of values of x which satisfy the inequalities 123 ≤− x and 252 2 −<− xx . [4] (b) Find the range of values of k for which the line 1y kx= − does not intersect the curve 2 23yx= − . [4] 6 It is given that 123)( −−= xxf . (i) Solve 0)( =xf . [2] (ii) Sketch the graph of )(xf , indicating the intercept(s) and vertex of the graph. [3] (iii) State the range of values of m for which the line mxy = cuts the graph of )(xf at only one point and that 0>m . [1] (iv) State the value of c if there are infinitely many solutions for 1232 1 −−=+ xcx . [2]

ACS(Independent)MathDept/Y3Exp/AM2/2016/Final Exam 4 7 Answer the whole of this question on a piece of graph paper. The variables x and y are known to be related by the equation y = ln (px2 + q) − 1, where p and q are constants. The following table shows some experimental pairs of values of x and y. x 1 2 3 4 y −0.09 0.72 1.36 1.86 (i) By plotting e y + 1 against x2, draw a straight line graph and use the graph to estimate the value of p and of q, [6] (ii) Find the possible values of x when 5ln=y . [2] 8 (a) The roots of the equation 052 2 =−+ mxx are α and β . If 4 10122 =+βα , find the possible values of m . [4] (b) The roots of an equation xx 8122 =+ are α andβ . Obtain an equation whose roots are 2 1 α and 2 1 β . [5] 9 The solution to this question by accurate drawing will not be accepted. The diagram shows a kite ABCD whose diagonals meet at M. The coordinates of A, B, C and D are (6, 5), (16, a), (2, −3) and (0, 3) respectively where a is a constant. Find the (i) coordinates of M, [1] (ii) equation of BD, [3] (iii) value of a, [1] (iv) area of the kite ABCD, [2] (v) ratio of DBDM : [2] ____________________________END OF PAPER 2_______________________ y x 0 D (0, 3) A(6, 5) C (2, −3) B (16, a) M

ACS(Independent)MathDept/Y3Exp/AM2/2016/Final Exam 5 Solutions 1. LHS 2)cos(cot ecxx−= xececxxx 22 coscoscot2cot +−= xx x x x 222 2 sin 1 sin cos2 sin cos +−= x x 2 2 sin )cos1( −= x x 2 2 cos1 )cos1( − −= )cos1)(cos1( )cos1( 2 xx x +− −= x x cos1 cos1 + −= [Proved] 2. )3(253 1 xx −+ −= 23.53.3 2 −= xx 023.53.3 2 =+− xx 0)13)(23.3( =−− xx 3 23 =x or 13 =x 369.0−=x 0=x 3. 123 += xy 3 12 += xy 033 1242 =+     +− xxx 09483 22 =+−− xxx 0945 2 =−+ xx 0)1)(95( =−+ xx 5 9−=x or 1=x 15 13−=y 1=y 4(a) 02010422 =−+−+ yxyx 49)5()2( 22 =++− yx

ACS(Independent)MathDept/Y3Exp/AM2/2016/Final Exam 6 Centre = )5,2( − Radius = 7 (b) New centre = )3,2( Equation: 49)3()2( 22 =−+− yx 5(a) 123 ≤− x 1≥x 252 2 −<− xx 0)2)(12( <−− xx 22 1 << x Ans: 21 <≤ x 5(b) 32)1( 2 −=− xkx 321222 −=+− xkxxk 04)22(22 =+−−+ xkxk 0<D 0))(4(4)22( 22 <−−− kk 0123 2 >−− kk 0)1)(13( >−+ kk 3 1−<k or 1>k 6(i) 123 =− x 123 =− x 4=x Or 123 −=− x 8=x (ii) Min point (6, -1) B1 y-intercept B1 Shape B1 (ii) 2 1≥m x y O 2 4 8 (

ACS(Independent)MathDept/Y3Exp/AM2/2016/Final Exam 7 (iii) 4−=c 8(a) 052 2 =−+ mxx 2 m−=+βα 2 5−=αβ 54)2 5(24 22 22 +=−−=+ mmβα 4 10154 2 =+m 812 =m 9±=m (b) 8=+βα 12=αβ 22 22 22 11 βα βα βα +=+ 2 2 12 )12(28 −= 18 5= 2222 11.1 βαβα = 144 1= 0144 1 18 52 =+− xx 0140144 2 =+− xx 9(i) ( )1,42 35,2 62 =     −+=M (ii) gradient of AC = 226 35 =− + gradient of BD 2 1−= Equation of BD: 32 1 +−= xy (iii) when 16=x , 53)16(5.0 −=+−=a

ACS(Independent)MathDept/Y3Exp/AM2/2016/Final Exam 8 (iv) Area of kite 35 06 53 160 2 12 −×= )3048(1880 −−+= 80= (v) By similar figure, 4 1 )5(3 13 =−− −