ACSI 2023 Y3EXP FYE AMath P2

1 INDEX NUMBER Anglo - Chinese School (Independent) FINAL EXAMINATION 2023 YEAR THREE EXPRESS ADDITIONAL MATHEMATICS PAPER 2 4049/02 Wednesday 11 October 2023 1 hour 30 minutes Candidates answer on the Question Paper. No additional materials are required. _______________________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Write your index number in the space at the top of this page. Write in dark blue or black pen. You may use an HD pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all questions. 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 an approved scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. 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 document consists of 13 printed pages and 1 blank page. [Turn over For Examiner’s Use 60

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 = + +c bx ax, a ac b bx 2 42 − ± −= Binomial expansion ( ) , ......2 1 2 21 nr r nnnnn b b ar nb anb ana b a + +     + +     +     + = + −−− where n is a positive integer and ( ) ! ( 1)...( 1) !! ! n n nn n r r rnr r  − −+= = − 2. TRIGONOMETRY Identities 1cos sin22 = +A A AA 22 tan1sec + = AAec 22 cot1cos + = Formulae for ∆ABC C c B b A a sin sin sin= = A bc c b acos22 2 2− + = 1 sin2 ab C∆=

3 Answer all the questions. 1. It is given that a and b are the roots of the quadratic equation 2 2 10xx− −= and that a > b. Show that 3 22a b = −− . [5] [Turn over

4 2. A circle with centre O passes through the points ( )1, 7P − and ( )0,8Q . (a) State the relationship between the perpendicular bisector of PQ and the point O. [1] (b) Find the coordinates of O , given that the line 22yx= − passes through the centre of the circle. [5] (c) Hence find the equation of the circle. [2]

5 3. The polynomial ( )fx is given by ( ) 329 30 23 4fx x x x=− −− . (a) Factorise ( )fx completely. [4] (b) Hence, prove that the equation 9 23 4 30xx x x−= + has only one real root. Find the solution. [3] [Turn over

6 4. (a) Solve the equation 12 23xx+− += . [4] (b) The equation of a curve is 224y x ax b=−+ where a and b are non-zero constants. Explain why 0y> if 22ba> . [3]

7 5. (a) Find an expression for x, in terms of e, for which ( )ln 3 ln 3xx−= + . [3] (b) Solve the equation 5 25log log 4xx+= . [3] [Turn over

8 6. (a) (i) Prove the trigonometric identity: ( ) 2 2 2 cosec 2cot cosec cos sin AA A AA + = + [4] (ii) Hence solve the equation: ( ) 22cosec 2cot 4 cos sinA A AA+= + for 0 360x≤≤ ° . [4]

9 6. (b) Given that 1 sin 2cos 11 2sin cos xx xx ++ =++ and x is acute, find the exact value of cos x. [4] [Turn over

10 7. (a) (i) Write down the first four terms in the expansion of ( ) 6 14 x− . [2] (ii) Hence find the coefficient of 3x in the expansion of ( ) ( ) 623 14xx+− . [2]