NYJC 2023 FM CT2 P1 - modified

JC2 CT2 FM 9649/01 [Turn Over NANYANG JUNIOR COLLEGE JC2 Common Test 2 Higher 2 FURTHER MATHEMATICS 9649/01 Paper 1 26th June 2023 3 Hours Additional Materials: Answer Paper List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name and class on all the work 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, 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 an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing 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. At the end of the examination, 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 5 printed pages.

JC2 CT2 FM 9649/01 2 [Turn over 1 Given that u, v and w are linearly independent vectors, show that the vectors +−u v w , +−v w u , +−w u v are linearly independent. [2] 2 Use the substitution d d yzy x=− to solve the differential equation 2 2 dd 2 3 2edd xyy yxx + − = , given that 1y= and d 2d y x = when 0x= . [7] 3 The mapping ( )22F:  →M  is given by ( ) 1 2 1 F d pq px q r x s xrs −  = + + +  . (a) Show that F is a linear transformation. [3] (b) Find a basis for the null space of F. [3] (c) Hence, state the range space of F. [1] A matrix A is said to be antisymmetric if T=−AA . Let W denote the set of antisymmetric matrices in ( )22M  . (d) Show that W is a subspace of the null space of F. State the dimension of W. [4] 4 Consider the differential equation 3d 2e d xy yx x += with the initial conditions 0y= when 0x= . (a) Use Euler’s method with step size 0.5 to approximate the value for y when 1x= , leaving your answer in 3 decimal places. [2] (b) Use the Improved Euler’s method with step size 1 to approximate the value for y when 1x= , your answer in 3 decimal places. [2] It is given that the exact solution to the differential equation is 3 3 21 1 1e e e5 25 25 x x xyx −= − + . (c) Use an analytical method to explain if the Euler Method gives an under-estimate or over-estimate of the true value of y when 1x= . [3] (d) Determine which of the methods in (a) and (b) gives a better approximation. [2]

JC2 CT2 FM 9649/01 3 [Turn over 5 A garden landscaper is designing a canopy for a rectangular plot of land to make it aerodynamic and efficient at shedding rainwater. The landscaper models the canopy with the following equation f ( , ),z x y= where 2 2/3 f ( , ) 7 0.05 3 ,x y y x= − − with domain   2 ( , ) : 1 1, 2 2D x y x y=  −   −    representing the land; x, y and z are measured in appropriate units. It is assumed that raindrops will follow the path of steepest descent. (a) Write down f ( , ),xy and hence determine the points on the canopy for which raindrops landing on do not necessarily follow the initial path of descent prescribed by the assumption. [2] (b) A surface is considered efficient in shedding rainwater if the rate of descent of a raindrop at any point on the surface is at least 3, that is, the height of a raindrop decreases instantaneously by at least 3 units for 1 unit of displacement from the point. (i) Find an expression for the rate of descent of a raindrop at ( ), ,f ( , )x y x y under the given assumption. [1] (ii) Comment with justification, whether the canopy is considered efficient in shedding rainwater. [1] The landscaper plans to support the canopy with steel beams. Two vertical steel beams are to be erected on the land at ( , ,0)ab− and ( , ,0)ab to the canopy at P and Q respectively. Another steel beam supports the canopy by joining P and Q along the vertical trace of f ( , )z x y= from P to Q. The total length of the three beams is denoted by ( , ).L a b (c) Ignoring thickness of the beams, show that 2/3 1/3 0 4( , ) 2f ( , ) 2 d . a xL a b a b x x +=+  [5] (i) Evaluate ( , )L a b in terms of a and b. Deduce that for a fixed value of a, the largest possible value of ( , )L a b is ( ,0)La . [3] (ii) Hence determine the largest possible value of ( , )L a b . [1]

JC2 CT2 FM 9649/01 4 [Turn over 6 (i) Describe the geometrical relationship between the points on the Argand diagram representing the complex numbers z and iez  . [1] (ii) The distinct complex numbers , and    are represented by the points , and A B C respectively. Given that A, B and C are the vertices of an equilateral triangle taken in an anticlockwise order, state the value of arg    −  − and write down the condition tha t , and    must satisfy. [2] (iii) Deduce from (ii) or show otherwise that the complex number         −− +−− is purely real and give its value. [2] (iv) Hence show that and         −− −− are the roots of the quadratic equation 2 10zz − + = and find their values in the form iab+ where a and b are real numbers. [3] (v) The complex numbers , and      are represented by the points , and A B C   respectively. Given that , and A B C   are respectively the reflections of A, B and C in the imaginary axis, find, in exponential form, the value of   − − . [2] 7 Consider the equation e ln 0 5 x x− +=  . (a) Show that there is exactly one real root. [3] (b) Given that the root of the equation lies in the interval [1,5] , use linear interpolation to find a first approximation of the root, giving your answer to 5 decimal places. [2] (c) Using the Newton-Rhapson method with your answer in (b) as the initial approximation, find the root to 3 decimal places. [2] (d) The Newton-Rhapson method fails when the initial approximation takes a value greater than or equal to k. Find k . [3]

JC2 CT2 FM 9649/01 5 [Turn over 8 Guppies are well known for their high reproduction rate. In a research study, an ichthyologist, a fish scientist, wishes to study the reproduction of guppies when they are given a special diet. He believes that with this special diet, the number of guppies , nG , is dependent on the interactions between the number of male guppies, nM , and female guppies, nF , in n months. He hypothesised that this can be modelled by 1 12 1 2, 2 2 , n nn nn n nM G M F F G M +++ + − = − =− where 0.n Initially, the research starts off with 10 female and 2 male guppies. After a month, there are 4 male guppies. (a) Find a first order r ecurrence relation for nF and a second order recurrence relation for nM . Hence solve the recurrence relations and find a solution for nG . [8] (b) Unfortunately, at the end of the 3rd month, there was a sudden spike in ammonia. This toxicity damaged the gills and internal organs of guppies. This led to the death of 75% of the population of male guppies and 50% of the population of female guppies. The next 3 months saw no growth in population. Subsequently, the ichthyologist assumes that population growth would revert back to normalcy and there would be 5 male guppies in the 7 th month. Find the projected population of guppies at the end of the year. [4] 9 In 2024, the comet C