AM4 4049 BSSS 2023 P2 QP

BEDOK SOUTH SECONDARY SCHOOL PRELIMINARY EXAMINATION 2023 4E5N CANDIDATE NAME CLASS REGISTER NUMBER ADDITIONAL MATHEMATICS Paper 2 4049 / 02 23 August 2023 2 hours 15 minutes READ THESE INSTRUCTIONS FIRST Write your name, class and register number on all the work you hand in. Write in dark blue or black. You may use a HB pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. The use of an approved scientific calculator is expected, where appropriate. If the degree of accuracy is not specified in the question and if the answer is not exact, give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees. For π, use your calculator value or 3.142. 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. If working is needed for any question it must be shown in the space below the question. Omission of essential working will result in loss of marks. The total number of marks for this paper is 90. Setter : Mr Loh Jia Perng For Examiner’s Use Student Check & Sign Marks before deduction Parent’s Signature TOTAL 90 This paper consists of 18 printed pages.

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0ax bx c+ += , Binomial Expansion , where n is a positive integer and 2. TRIGONOMETRY Identities Formulae for ABC

3 Answer all questions. 1. The cubic polynomial ()fx is such that the coefficient of 3x is 2 and the roots of the equation () 0fx = are 2, 1 2− and k. Given that ()fx has a remainder of 6− when divided by 1x− . Find the value of k. [3]

4 2. A merry-go-round horse ascends and descends as it travels along a circular path. The distance between the bottom of the horse and the ground, d cm, can be modelled by the equation, 3 2cosd kt= − , where t is the time in minutes after the merry-go-round ride starts and k is a constant. Each merry-go-round ride lasts 5 minutes and the horse ascends 10 times on each ride before coming to rest. (a) Find the value of k in radians per minute. [2] (b) For how long, during one merry-go-round ride, is d > 2? [4]

5 3. The diagram shows part of the curve 2ln ( 1)yx= − . The point P (a, 2) lies on the curve. (a) Find the value of a, in terms of e. [2] (b) Show that 2 0 d2xy e =∫ . Hence explain why 1 2 2ln( 1) d 2 e xx + −=∫ . [6] 2ln ( 1)yx= −

6 4. (a) Prove that 2 2 sec 2 1 sec cos 2 3 x xx=++ . [3] (b) Hence solve the equation 2 2 sec 2 4 1 sec 2 7 x x =+ for 03 x≤≤ . State the number of solutions for 22 xππ− ≤≤ . [4]

7 5. (a) Sketch the graph of the equation 2xye −= . [2] (b) The gradient function of a curve is given by 22d d xxy eex −= + . Explain why the function is an increasing function. [2] (c) Find the equation of the function, given that y-intercept is 3. [3]

8 6. The diagram shows a circle passing through points A, B, C and D. GAH is a tangent to the circle at point A. ∠ECF = ∠ECB. E and F are the midpoints of AC and DC respectively. (a) Prove that ∠DAG = ∠ECB. [2] (b) Prove that ECF is similar to BCE. [4] (c) Show that 1 2EC BE AD BC×= × . [2]

9 7. Using a suitable substitution in the form xya= , where a is a constant, show that the equation ( ) ( ) 13 22 2 74 52x xx+ += + can be expressed as 322 7 5 40yyy− − += . Hence, given that x is an integer, solve the equation ( ) ( ) 13 22 2 74 52x xx+ += + . [8]

10 8. Two particles A and B, each moving in a straight line and in the same direction, passes point O at the same instant. The velocity of particle A, t seconds after passing O, is given by 2 47AVt t=−+ m/s. Particle B passes O with a velocity of 23 m/s and moves with an acceleration, 6 16Bat = − m/s2, where t is the time in seconds after passing O. (a) Calculate the range of values of t for which the velocity of the particle A is greater than that of particle B. [6]