ACSI 2017 Y3EXP FYE AMath P1

Anglo - Chinese School (Independent) FINAL EXAMINATION 2017 YEAR THREE EXPRESS ADDITIONAL MATHEMATICS PAPER 1 Friday 6 October 2017 1 hour 30 minutes Additional Materials: Answer Paper (8 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/AM1/2017/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/AM1/2017/Final Exam 3 Answer all the questions. 1 Given that 32() 2 2f x x ax x=+ ++ for all values of x. (a) Find the value of a if ()fx has a remainder of 1− when divided by ( )1x+ . [2] (b) Find the quotient when the expression 32 22x ax x+ ++ is divided by ( )1x+ . [2] 2 The total resistance of two resistors in an electric circuit is given by the formula 12 111 RR R= + . Given that 31R= − and ( )1 1 313R = + , find the value of R2 in the form 3ab+ where a and b are integers. [4] 3 (i) Find the coordinates of the point of intersection of 3y x= and 1 42yx= for 0x> . [2] (ii) Sketch the graph of 1 42yx= for 0x≥ and 3y x= for 0x> on the same diagram, indicating the point of intersection. [3] 4 The variables x and y are related by the equation pxxq y−= , where p and q are constants. When y is plotted against y x , a straight line graph is obtained. The line has gradient 3 and it passes through ( )1, 5 . (i) Find the value of p and of q. [4] (ii) Find the value of y when 20xy−= . [2] 5 (a) Find the range of values of x for which ( )( )61 8xx− + ≥− . [3] (b) Find the range of values of m for which the curve 2 8y mx= − meets the line 54ym x+= . [4]

ACS(Independent)MathDept/Y3Exp/AM1/2017/Final Exam 4 6 Solve the following equations. (a) 8 16xxee −= − . [3] (b) ( ) ( ) 2 93 9 12log 2 log 4 log 3xx − −= . [4] 7 A circle, C1, has equation 22 2 6 60xy xy+ − − += . (i) Find the coordinates of the centre and the radius of C1. [3] The points ( 4, 11)A and ( 8, 7 )B lie on a second circle C2 whose centre lies on the line 4x= . (ii) Find the equation of C2. [4] (iii) Determine, with working, whether C1 and C2 intersect. [2] 8 Given the function ( ) 3sin 2 2fx x = − , (i) state the period and the amplitude of the function, [2] (ii) solve the equation 3sin 2 2 0x−= for 0 180x°≤ ≤ ° , [3] (iii) sketch the graph of ( ) 3sin 2 2fx x = − for 0 180x°≤ ≤ ° , showing the x – and y – intercepts clearly. [2] 9 (a) Given that ( ) 2() 2 9 2f xxx= −+ , (i) prove that ( )21x− is a factor, [1] (ii) solve the equation ( ) 2 2 9 20xx − += , [3] (iii) hence, or otherwise, find the values of θ between 0 and 360 inclusive such that 222 sin (sin 4) cos 1θθ θ −+ = − . [3] (b) If 13sec 5θ = and 180 360θ<< , find without the use of a calculator, the value of (i) ins θ (ii) ( )tan 180 .θ− [4] ____________________________________END OF PAPER 1

ACS(Independent)MathDept/Y3Exp/AM1/2017/Final Exam 5 Answers: 1a. 0a= 1b. 2 3xx−+ 2. 2 32R = + 3i. (1.38,2.17) 3ii. 4i. 2; 3pq= = 4ii. 8y= 5a. 2 25 m−≤≤ 5b. 27 x−≤≤ 6a. 1.39x= 6b. 2.55 or 7.05x= − 7i. ( )11centre of 1,3 ; radius of 2 unitsCC = 7ii. ( ) ( ) 22 24 74xy− +− = 7iii. Distance between the centres of 1 C and 2C = ( ) ( ) 2 24 1 7 3 5 units−+− = Since 12 245rr+ =+> 12 and intersects. CC∴ 8i. Period = 180 Amplitude = 3 8ii. 20.9 , 69.1x=  8iii. 9aii. 1 or 4.45 or 0.4492x= − 9aiii. 30 ,150 ,206.7 ,333.3θ =   9bi 12sin 13θ =− 9bii ( ) 12tan 180 tan 5θθ−= − = x y 1 42yx= 3y x= (1.38, 2.17) O [Correct shape of graph, period, amplitude] [Correct x, y – intercepts, and turning points] ( ) 3sin 2 2fx x = − y x