ACSI 2020 Y3 IP AdvMath Paper1

ACS(Independent)MathDept/Y3IPAdvancedMathPaper1/2020/FinalExamination 1 FINAL EXAMINATIONS 2020 YEAR 3 INTEGRATED PROGRAMME ADVANCED MATHEMATICS PAPER 1 Monday 5 October 2020 1 hour and 30 minutes Additional Materials: Writing paper (6 sheets) INSTRUCTIONS TO STUDENTS Do not open this examination paper until instructed to do so. A calculator is required for this paper. Answer all questions on the answer sheets provided. Unless otherwise stated in the question, all numerical answers must be given exactly or correct to three significant figures. INFORMATION FOR STUDENTS The maximum mark for this paper is 80. __________________________________________________________________ This question paper consists of 4 printed pages. [Turn over]

ACS(Independent)MathDept/Y3IPAdvancedMathPaper1/2020/FinalExamination 2 Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working. 1. [Maximum marks: 8] It is given that ( ) 3221f x x Ax Bx= + + + is exactly divisible by ( )1x− and leaves a remainder of 9 when divided by ( )2x− . (a) Find the value of A and of B. [4 marks] (b) Hence, prove that the equation ( ) 14f x x =− has only one real root. [4 marks] 2. [Maximum marks: 10] (a) Prove that ( )2x− is a factor of the polynomial, ( ) 326 5 19 10g x x x x= − − + . [2 marks] (b) Factorise ( )gx completely. [4 marks] (c) Hence, express ( ) 53 x gx + in partial fractions. [4 marks] 3. [Maximum marks: 10] (a) When the cubic polynomial, ( )Px , is divided by x , ( )1x+ and ( )2x− , the remainders are 5, 6 and 3− respectively. Given that ( )Px is divisible by ( )1x− , find the polynomial, ( )Px , in the form, 32Ax Bx Cx D+ + + . [7 marks] (b) Hence, find the remainder when P(x) is divided by ( ) 2 2x + . [3 marks]

ACS(Independent)MathDept/Y3IPAdvancedMathPaper1/2020/FinalExamination 3 4. [Maximum marks: 13] The function, h, is defined as :h x ax b + . (a) Given that ( )25h = and ( )1 32h− − =− , find the value of a and of b. [4 marks] Another function, k, is defined as 4:1 xk x e −− . (b) Solve ( ) 0kx = . [2 marks] (c) Sketch the graph of ( )kx , stating the equation of the asymptote and the axes intercepts. [3 marks] (d) Explain why the inverse of ( )kx exists. [1 mark] (e) Solve the equation ( )11 4h k x−− = . [3 marks] 5. [Maximum marks: 8] The functions, f and g, are defined as follow. 1: , 2 2f x x x −+ : 2 3g x x + (a) Express ( )fg x and ( )2fx in similar form. [5 marks] (b) Solve ( ) ( )220f x fg x−= . [3 marks] 6. [Maximum mark: 8] The function, f, is defined as 2:6f x ax x b −+ . (a) Given that ( ) ( )0 2 8ff == , find the value of a and of b. [3 marks] (b) State the largest possible domain for ( )1fx− to exist. [3 marks] (c) Hence, find ( )1fx− . [2 marks]

ACS(Independent)MathDept/Y3IPAdvancedMathPaper1/2020/FinalExamination 4 7. [Maximum mark: 15] (a) Find all the angles x, where  3600 x , such that (i) 112sin 24x = [3 marks] (ii) 23sin 2cos sinx x x= [5 marks] (b) Solve the following equations for 02 x  , leaving your answers in terms of  . (i) 2 4sec 3x= [3 marks] (ii) 22cos 5cos 2xx=− [4 marks] 8. [Maximum mark: 8] The diagram below shows the graph of ( )y f x= which passes through ( )1,2A and ( )1, 2B −− . The graph of ( )y f x= is mapped onto the graph of ( )1 2 2 1y f x=− − + after undergoing a series of transformations. (a) Describe fully the transformations which map the graph of ( )y f x= onto the graph of ( )1 2 2 1y f x=− − + . [4 marks] (b) Write down the co-ordinates of A and B after the transformations. [2 marks] (c) Hence, sketch the graph of ( )1 2 2 1y f x=− − + . [2 marks] End of Paper y x A(1,2) B(-1,-2)

ACS(Independent)MathDept/Y3IPAdvancedMathPaper1/2020/FinalExamination 5 Answers 1. (a) 1A =− , 2B =− 2. (b) ( )( )( )2 2 1 3 5x x x− − + (c) ( ) ( ) 12 3 2 3 2 1xx −−− 3. (a) 1A= , 2B =− , 4C =− , 5D= (b) ( )( ) ( ) 22 2 9 6x x x− + + − 4. (a) 2a= , 1b= (b) 4x= (e) 147.4x =− 5. (a) 25,2 5 2 x xx + −+ (b) 3 2x =− 6. (a) 3a= , 8b= (b) 1x or 1x (c) For 1x , ( )1 51 3 xfx− − = + OR For 1x , ( )1 51 3 xfx− − = − 7. (a) (i) 14.4 ,345.6x =   (ii) 33.7 ,180 ,213.7x =    (b) (i) 5 7 11, , ,6 6 6 6x    = (ii) 5,33x = 8. (a) ( )1 2T f x=− : Horizontal translation of 2 units along the positive x-axis. ( )2 2T f x=− − : Reflection on the x-axis. ( )3 22T f x=− − : Stretch, factor 2, along the y-axis. ( )4 2 2 1T f x=− − + : Vertical translation of 1 unit along the positive y-axis. (b) ( )1 3, 3A − , ( )1 1,5B