RVHS 2023 JC1 Promos 9649

1 9649 / 01 / 2023 [Turn over RIVER VALLEY HIGH SCHOOL 2023 JC 1 Promotional Examination Higher 2 Name CLASS 2 3 J INDEX NUMBER FURTHER MATHEMATICS Paper 1 Additional Materials: List of Formulae (MF26) Answer Papers Cover Page 9649/01 27 September 2023 3 hours READ THESE INSTRUCTIONS FIRST Do not open this booklet until you are told to do so. Write your name, class and index number in the space at the top of this page. Write your name and class on the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 signi ficant 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 an approved graphing calculator. Unsupported answers from a graphing calculator ar e allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calcul ator are not allowed in a question, you are required to present the mathematical steps us ing mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. Up to 2 marks may be deducted for poor presentation in your answers. At the end of the examination, place the cover page on top of your answer paper and fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 6 printed pages and 2 blank pages.

2 9649 / 01 / 2023 1. (a) Prove by mathematical induction that 52nn is divisible by 3 for all positive integers n. [4] (b) Hence, find the greatest common divisor of the numbers in the set      115 25 2 ,nn n   . (The greatest common divisor of a set of numbers is the largest integer that can divide all the numbers in the set.) [3] 2. (i) Use Simpson’s rule with five ordi nates to find an approximation to 5 24 2 d 87 Ix xx   , giving your answer to 4 decimal places. [2] (ii) Find the exact value of I and deduce from part (i) an approximate value of 1 1sin , 3    giving your answer to 4 decimal places. [3] 3. Solve the differential equation 2 2 2 d 48 e c o s 24 s i n 2d xz zx xx    , given that 0z and d 1d z x  when 0x . [7] 4. A deck of 25 playing cards consists of 3 diffe rent colours – blue, yellow and red. Each card has a picture of a fruit on it. The number of cards of each type are given in the following table: Apple Kiwi Watermelon Grapes Blue 1 3 2 4 Yellow 1 2 1 2 Red 2 2 1 4 A card is drawn at random from the deck. Events B and G are defined as follows: B: The card is Blue. G: The card shows Grapes. (i) Determine if events G and 'B are independent. [2] The card is placed back into the deck a nd now two cards are drawn randomly from the deck. (ii) Find the probability that the two cards contain exactly one Apple card given that at least one card is Red. [5]

3 9649 / 01 / 2023 [Turn over 5. The sequence nx where 2n is defined by the recurrence relation 12 944 2 nnn nxx x  . It is given that 0 1x  and 1 3x  . By considering the sequence ny , where 2 n nnyx  for 0n , show that 1244 0nn nyy y  . Hence, find an expression for nx as a function of n. [7] 6. Loran, short for Long Range Navigation, is a radio-based navigation system that was widely used before the adve nt of Global Positioning System (GPS). It was developed during World War II and continued to be used for navigation purposes until it was largely phased out in the late 20th century. Loran ope rates by using the time difference of arrival (TDOA) of radio signals from multiple ground-based transmitters to determine the position of a receiver, typically a ship or aircraft. Two transmitter stations A and B are 50 km apart. Taking the origin as the mid-point of A and B, a ship is found to be on a hyperbola 1H with A and B as the foci, as shown below. Radio signals are transmitted at a constant speed of 0.3 km per microsecond (µs) from the stations. It is found that the TDOA of signals from stations A and B to the ship is 140 µs. (i) Find the equation of 1H . [5] Another transmitter station C is 40 km away from station B and is located on the line AB produced. Taking the origin as the mid-point of B and C instead, the same ship is found to be on a hyperbola 2H with B and C as the foci, as shown below. The equation of 2H is 22 1324 76 xy  . (ii) Find the coordinates of the location of the ship, with refere nce to the coordinate axes with the origin as the mid-point of A and B. [3] A B x y O B C x y O

4 9649 / 01 / 2023 7. For 0 π , find the range of values θ for which 32 s i n 2 0 . [2] A curve C has polar equation 32 s i n 2 , 0 πr    and 0r . (i) Sketch C, indicating clearly the polar coordina tes of the axial intercepts and any tangents to the curve at the pole. [3] (ii) Find the exact area of the region bounded by C and the line yx . [3] 8. A curve C has parametric equations 4sin , cos2x t y t for π0 2t . Sketch C, indicating clearly the coordinates of the end points. [1] (i) Let A be the area of the surface of revolution formed when this arc is rotated through 2π radians about the y-axis. Show that π 22 0 32π sin cos 1 sin dA tt t t  . Hence find A. [3] (ii) The region bounded by the arc, the line 1y and the y-axis is rotated through2π radians about the y-axis. Using the shell method, find the exact volume of the solid generated. [4] 9. The Kentucky Department of Fish and Wildli fe Reserve models the population growth of white-tailed deer in the state of Kentucky using the following differential equation d1 ()d5 P PN Pt  , where P is the population of white-tailed deer in thousands and t is the time in years. (i) State what N represents and explain what it refe rs to in the context of the model. [2] (ii) Sketch, on the same diagram, the curv es that represent the behaviour of the population P with time when the initial population 0P , in thousands, is such that (a) 00 P N , (b) 0PN . [2] The Kentucky Department of Fi sh and Wildlife Reserve woul d like to set guidelines for hunting in the state, stipulating that white-tailed deer can only be hunted at a constant rate of 100H thousands per year. (iii) Write down a revised differential equati on taking into account the hunting of white- tailed deer. Suppose 0 150P  and N = 900, determine the maximum value of H so that the population does not become extinct. [5]

5 9649 / 01 / 2023 [Turn over 10. The function y = f(x) satisfies the differential equation  4e 5d d3 x x yy x x    , with 2 3y when 0x . (i) Use two steps of Euler method to determine an approximation to y when 0.5x . [2] (ii) Use one step of Improved Euler method to determine an alternative approximation to y when 0.5x . [1] (iii) Discuss the relative merits of th e two methods employed to obtain these approximations. [2] (iv) Show that  2 () e3x pxy x   where p(x) is a polynomial in terms of x. [4] 11. (a) Referred to the origin O, the points P and Q are such that OP  p  and OQ q   and 0pq . The point R lies on PQ such that :: 1PR RQ   . Given that OR  is perpendicular to PQ   , show that 2 22   p pq . [4] (b) Two planes, 12π and π , have the following equations 21 . 2 10 and . 2 7 12               rr respectively. (i) Given that 12π a