NYJC 2022 FM CT2 P1 - modified

This document consists of 6 printed pages. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2022 [Turn Over NANYANG JUNIOR COLLEGE JC2 COMMON TEST 2 Higher 2 FURTHER MATHEMATICS 9649/01 Paper 1 5th July 2022 3 hours Additional Materials: 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. 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. The number of marks is given in brackets [ ] at the end of each question or part question.

2 NYJC 2022 JC2 Common Test 2 9649/01 [Turn Over 1 Determine the maximum number of iterations, using the Euler Method , with step size 0.1 on the differential equation 2d40 d ( ),yyy xxx y=− where 1y= when 0x= , such that the error between the actual value of y and the approximated value of y is less than 0.1. [5] 2 Let ( )22M  be a vector space consisting of all 22 matrices with real entries. Let V be a subset of ( )22M  consisting of all symmetric traceless matrices such that ( ) 2 2 : and trace( ) 0TV =  = =A A A AM  , where trace( )A is the sum of the diagonal entries of A . (i) Show that V is a subspace of ( )22M  . [2] Let W be a subspace of ( )22M  consisting of matrices whose first column has both entries equal to zero. (ii) By finding a basis for V, show that ( )22VW += M  . [3] 3 The curve C has polar equation 2cotr = , where r  0 and 0 2  such that  . (i) Show that the lines with cartesian equation 2y= and 2y=− are asymptotes of C. [1] (ii) Sketch C, showing clearly the behaviour of C near the origin. State the cartesian equation of the tangent to C at the pole. [2] (iii) The region R is bounded by C, the lines with cartesian equation 3y= and 0x= . Find the exact area of R. [4]

3 NYJC 2022 JC2 Common Test 2 9649/01 [Turn Over 4 In ecology, a population response model describes the interaction between the availability of food resources and the population of a particular species. The model assumes that the food availability decreases linearly with the population in that same month, while the population growth is proportional to food availability in the previous month. Let Fₙ and Pₙ denote the food availability and population in month n, respectively. The model is defined by: 1,andn n n nF a bP cF P −= − = where a, b, c are positive constants. The ecosystem adjusts population each month so that food availability equals population size, such that Fₙ = Pₙ for all n. (a) In terms of a, b, c, write down the first order linear recurrence relation for Pₙ. Hence solve for Pₙ given that the initial population is P₀. [5] It is known that Pₙ converges to eP , as n → ∞, and that 0 .ePP (b) State the condition for convergence of Pₙ, and write down the value of eP . [2] (c) Describe, with reasons, the behaviour of Pₙ as it approaches eP . [2] 5 It is required to solve the equation ln( 1) 3 0xx− − + = . It is given that there are two roots,  and  , where 1.1 21 .  and 4.1 24 .  . (i) The root  can be found using the iterative formula 1 ln( 1) 3nnxx+ = − + . (a) Using this iterative formula with 1 4.15x = , find  correct to 3 decimal places. You are to show all your working. [2] (b) Explain with the aid of a sketch why this iterative formula will not converge to  for any initial value taken. [2] (ii) Show that the Newton-Raphson iterative formula for this equation can be written in the form ( ) 1 3 2 1 ln( 1) 2 n n n n n x x xx x + − − − −= − . [2] (a) Explain with the aid of a sketch why this iterative formula will not converge to  if the initial value 1 2.2x = is taken. [1] (b) Use this formula with 1 1.2x = to find  correct to 3 decimal places. [1]

4 NYJC 2022 JC2 Common Test 2 9649/01 [Turn Over 6 (a) Define sin dsin n nxIx x= . Prove that, for n > 2, 2 2sin( 1) 1 nn nxI I C n − −− = + − , where C is an arbitrary constant. [2] Hence find the general solution of the differential equation d cot sin 5d y y x xx−= . [5] (b) Given that uy x= , show that 22 2 2 2 3 d 1 d 2 d 2 d d d y u u u x x x x x x=−+ . [2] Hence find the general solution of the differential equation 2 2 d 2 d 25 0dd yy yx x x+ + = , x > 0. [3] 7 Most bacteria can grow to some extent in the presence of oxygen, known as aerobic culture. A scientist in a laboratory tried to adjust the conditions to suit the target bacterium. Rich nutrient or complete media can be helpful when trying to bulk up a pure culture and get the bacterial cells in good condition. As such, the scientist tried to manage the nutrient for the bacteria he intended to culture and postulated that the amount of nutrients, nu , measured in suitable units is such that 0 1u = and 1 2u = , satisfies the following recurrence relation 12 2 1 2 0 2 1 2 22 nn n uu u −− =+ for 2n . (i) Solve the recurrence relation. [5] Subsequently, the scientist found that this recurrence relation is connected to the amount of bacteria, nx (in billions), cultured in a bottle of fermented milk drink in a laboratory in n days. He postulated that ( ) ( )1 1 20 2 . nn n n r r r x u u u − = = + −  (ii) Show that, in long run, the amount of bacteria in the bottle of fermented milk cannot exceed a certain value which is to be determined. [4] A second scientist disagreed with the first scientist’s hypothesis. He felt that that a certain proportion of the bacteria is lost due to temperature fluctuation in the process of culturing. Instead, he postulated that the number of bacteria, nx is more than 80% of 1nx − by 10. (iii) Write down the new recurrence relation relating nx and 1nx − . Hence, find the amount of bacteria, in billion, in the long run according to this new conjecture if 1 1x = . [4]

5 NYJC 2022 JC2 Common Test 2 9649/01 [Turn Over 8 Sketch on an Argand diagram, the locus of the point P representing the complex number z where i1 e cosz  =− and 0  . You are to provide full justification for your conclusion. Find, in terms of  , the modulus and argument of z. By considering nz where n is a non-negative integer, prove that, for 0 π  ,   0 11 cos cos sin cos π2 n r rn r n rnr                             . [11] 9 The roof of the James S. McDonnell Planetarium in Saint Louis (Figure 1) is designed to resemble a hyperboloid, which is created when a hyperbola is rotated about one of its axes of symmetry. Figure 1 Figure 2 Figure 2 shows the outline of the roof. Coordinate axes have been superimposed onto this outline for ease of reference. The defining curve for the outline of the roof is the hyperbola H which contains the points (6,0),P (10,4)Q and 205(2 , 14)R − , as shown on Figure 2. P lies on the x-axis, which is one of the axis of symmetry of H. All distances are in metres. (i) Determine a cartesian equation for H. [2] (ii) Show that the volume enclosed by the roof is 4392  cubic metres. You may assume that the thickness of the roof is negligible. [3] (ii