NYJC TJC VJC FM 2024 Prelim P2

1 TEMASEK JUNIOR COLLEGE JC2 Preliminary Examination Higher 2 FURTHER MATHEMATICS 9649/02 Paper 2 12 September 2024 3 hours Additional Materials: Answer Booklet List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet will be provided with this question paper. You should follow the instructions on the front cover of the answer booklet. If you need additional answer paper ask the invigilator for a continuation booklet. Write your Civics Group and name on all the work that you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the 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. You are expected to use a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. 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. This document consists of 7 printed pages and 1 blank page. [Turn over

2 Section A: Pure Mathematics [50 marks] 1 (a) Solve the equation 3 32 32 3iz =− , giving your answers 1z , 2z and 3z in exact form ie,r  where 0r ,   −   and 1 2 3arg arg argz z z . [3] (b) The roots 1z , 2z and 3z in part (a) are represented by points 1P , 2P and 3P . Find the exact perimeter of triangle 1 2 3PP P . [2] 2 The Fibonacci numbers nF are defined by the conditions 0 0F = , 1 1F = and 11n n nF F F+−=+ for all 1n . (a) Compute 23,FF , 4F and 5F . [1] (b) Compute 2 11n n nF F F+− − for 1, 2,3, 4n= . [2] (c) Conjecture and prove by induction, for all 1n , an expression for 2 11n n nF F F+− − . [6] 3 At the start of the month in January 2023, Sandy started her career as a lifestyle vlog content creator with 60 subscribers for her channel. By the end of each month, she lost 5% of her existing subscribers but gained 100 new subscribers each month through her publicity efforts. Let nu be the number of subscribers at the end of the nth month taking January 2023 as the first month. (a) Write down a recurrence relation to model the number of subscribers at the end of the nth month. [2] (b) Find the general formula of nu in terms of n. Hence show that the number of subscribers at the end of December 2023 is 952, correct to the nearest integer. [4] (c) Based on the model in (a), comment on the long-term prospect of Sandy’s career. [2] At the start of January 2024, Sandy decided to collaborate with another well -known content creator to gain more exposure for her own channel. By the end of each month, she gained k% of the number of existing subscribers but lost 100 existing subscribers. Let nw be the number of subscribers at the end of the nth month taking January 2024 as the first month. (d) By considering the general formula of nw , find an inequality in terms of k that will lead to an increase in the number of subscribers in the long run. Hence find the least integer value of k. [4]

3 4 A car travels over a rough surface. The vertical motion of the front suspension is modelled by the differential equation 2 2 d 25 30cos5 d y yt t += where y is the vertical displacement of the top of the suspension and t is time. (a) Find the general solution of the above differential equation. [6] It is given that 1y= and d 0d y t = initially. (b) Find the solution subject to these conditions. [2] A second model of the motion of the suspension is given by 2 2 dd 2 25 30cos5dd yy yttt + + = . (c) Verify that 3sin 5yt= is a particular integral for this differential equation. Hence find the general solution. [3] (d) Compare the behaviour of the suspension predicted by the two models. [1] 5 The ellipse E is given by the equation 22 198 xy+= . P(x0, y0) is a point on E where x0 > 0 and y0 > 0. The tangent to E at P intersects the x-axis at a point Q and O is the origin. (a) Show that ( ) ( ) 8tan tan 9POQ PQO   = . [4] (b) (i) Find the eccentricity of E. [1] F1(−c, 0) and F2(c, 0) are the foci of E, where c > 0. Write down the polar equation of E if the pole is located at (ii) F1, [1] (iii) F2. [1] It is given that 2PF Q = and 2F PQ = . (iv) Using the reflective property of ellipse and both polar equations above, show that ( ) 44 33 cos 3 cos 2  +=+ + + . [5] [Turn over

4 Section B: Probability and Statistics [50 marks] 6 A random sample of 100 university students were taken from the faculty of Medicine, Dentistry and Health Sciences. The sample consists of 32 Medicine students, 16 Dentistry students and 52 Health Sciences students. Out of the 100 students, 65 were females. A test, at the 1% level of significance was carried out on this data and it was found that there is association between the types of courses enrolled and gender. Given that there were 11 female dentistry students and n male medicine students, find the set of possible values of n. [5] 7 The Weibull distribution is a versatile distribution characterised by its ability to model a wide range of data types. It has important applications in meteorological studies such as wind speeds and thunderstorms. The probability density function is given by ( )1f ( ) e k xkkx k x  −−= , 0x where k is the shape parameter and  is a non-negative scale parameter. (a) For the case 2k = , show that f ( )x is a probability density function for any value of  . [2] At a weather station, the wind speed is measured at noon each day. The wind speed, X m/s, is modelled by the Weibull distribution. (b) For the case 2k = and 0.25 = , explain what the shape of the density function indicates about the wind speed at noon at the weather station. Find the most likely wind speed. [2] The average power output of wind turbines near the weather station is related to the wind speed, x. A model is proposed in which the power, Y megawatts, is given by 3Y CX= , where C is a constant. (c) For the case 2k = and 0.25 = , find ( )E Y in terms of C. [2]

5 8 The Health Promotion Board recommends that adults should engage in at least 150 minutes of moderate-intensity aerobic activity per week. The results of a survey from a random sample of 2105 adults in Singapore show that 1577 adults follow the recommendation. (a) Calculate a 95% confidence interval for the true proportion p, of adults in Singapore who engage in at least 150 minutes of moderate-intensity aerobic activity per week. Give the end points of the interval correct to 4 significant figures. [2] (b) Give two reasons why this interval is an approximation. [2] (c) Suppose that a 90% confidence interval is required, and that the width of the interval is no more than 0.04. Determine the smallest sample size that will satisfy the requirement regardless the value of p. [3] 9 Alvin and Bernard take alternate turns at kicking a football at a goal, and their probabilities of scoring a goal on each kick are 1p and 2p respectively, independently of previous outcomes. The first person to score allows the other person one more kick. If the other then scores, the game is drawn. If the other then misses, the first has won the game. Alvin begins a game. (a) Show th