ACSI 2021 Y3 IP AdvMath Paper2

FINAL EXAMINATION 2021 YEAR 3 INTEGRATED PROGRAMME ADVANCED MATHEMATICS PAPER 2 Tuesday 12 October 2021 1 hour 30 minutes Additional Materials Writing paper (7 sheets) INSTRUCTIONS TO STUDENTS Do not open this examination paper until instructed to do so. A graphic display calculator may be used in this paper. Answer all the questions on the answer sheets provided. At the end of the examination, attach the answer sheets in a single bundle. 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/Y3IPAdvancedMathPaper2/2021/FinalExamination 2 Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. In particular, solutions found from a graphic display calculator should be supported by suitable working, e.g. if graphs are used to find a solution, you should sketch these as part of your answer. Where an answer is incorrect, some marks may be given for correct method, provided this is shown by written working. You are therefore advised to show all working. Answer all the questions on the answer sheets provided. 1. [Maximum mark: 7] The polynomial, ( ) 32 22P x Ax x Bx= + + − , is exactly divisible by ( )21x− and has a remainder of 42 when divided by ( )2x− . (a) Find the value of A and of B. [4 marks] (b) Hence, show that ( ) 0Px = has only one real solution. [3 marks] 2. [Maximum mark: 7] (a) Express 2 29 2 5 3 x xx − +− as partial fractions. [4 marks] (b) Hence, solve the equation, 2 14 63 15 22 5 3 3 x x x x − =++ − + . [3 marks] 3. [Maximum mark: 9] The function, f , is defined by ( ) 2 :2f x x p q −+ , for the domain 4px . (a) Given that ( )06f =− and ( )18f =− , find the value of p and of q. [5 marks] (b) Explain why ( ) 1fx− exists and hence, find an expression for ( ) 1fx− . [3 marks] (c) State the domain of ( ) 1fx− . [1 mark]

ACS(Independent)MathDept/Y3IPAdvancedMathPaper2/2021/FinalExamination 3 4. [Maximum mark: 9] The functions, f and g, are defined as 2: 2 xfx x− , 2x , and : 2 1g x x − . (a) Obtain an expression, in similar form, for ( ) 1 gf − and s tate the value(s) of x for which ( ) 1 gf − is undefined. [5 marks] (b) Hence, solve ( ) ( ) 121g g x gf −− = . [4 marks] 5. [Maximum marks: 8] The graph of ( )y f x= is mapped onto the graph of ( )1 2 2 2y f x= + − after undergoing a series of transformations. (a) Describe these transformations