HCI 2023 JC1 MYE 9649

HWA CHONG INSTITUTION 2023 JC1 BLOCK TEST FURTHER MATHEMATICS Higher 2 Paper 1 Wednesday 28 June 2023 3 hours Additional materials: 12-page Answer Booklet List of Formula (MF26) 4-page Additional Answ er Booklet (upon request) READ THESE INSTRUCTIONS FIRST Write your name and class on the 12-page Answer Booklet and any other additional 4- page Answer Booklets 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 paper clips, highlighters, glue or correction fluid. Do not write anything on the List of Formula (MF26). 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 differ ent level of accuracy is specified in the question. You are expected to use a graphing calculator. Unsupported answers from a graphing ca lculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calc ulator are not allo wed 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. At the end of the examination, slot any additional 4-page Answ er Booklets used in your 12-page Answer Booklet and indicate on the 12-page Answer Book let the number of additional 4-page Answer Booklets used (if any) This question paper consists of 9 printed pages. 9649/01

2 1. Prove by mathematical induction that 22 n nnab a b  for n  , where a and b are positive. [5] 2. A point P lies on the curve C with polar equation 2c o s, 22ra     , where a is a positive constant. The point N is the foot of perpendicular from the pole to the tangent of C at P . (a) Sketch ,CP and N on the same diagram. [2] (b) By considering the polar coordinates of N , show that, as P varies, the locus of N is given by the polar equation (1 cos ), .ra      [4] 3. The sequence  nu is given by the recurrence relation 21 56nn nuu u , n ,      together with terms 12 and ua ub . (a) Find the expression of nu in terms of a and b. [4] (b) Find algebraically th e possible limits of 1 n n u u     . [3]

3 4. (a) A straight line l passes through the focus F of a conic with polar equation 1c o s kr e   and meets the conic at 2 points A and B . Show that 11 2 AF BF k . [4] (b) The diagram above shows an ellipse with foci 1F and 2F . The points ,PQ and R lies on the ellipse with PQ and PR passing through 1F and 2F respectively. By using the result in part (a), or otherwise, show that 12 12 PF PF FQ F R is a constant. [4] 5. The points A and B have Cartesian coordinates (, 0 )a and (, 0 )a respectively, where a is a positive constant. The point P is such that 2AP BP a . The curve C describes the locus of P . (a) Show that C has polar equation 22 2c o s 2ra  , 02  . [4] (b) Sketch the graph of C , indicating all key features and symmetries of the curve. [2] (c) Find the exact area of the region enclosed by the curve C . [3]

4 6. In the diagram, the curve with equation 21 l nyx for 0x divides the rectangle bounded by the axes, the lines 1y  and ex into two regions, S and T. (a) Show that the volume of the solid generated when S is rotated completely about the x- axis is given by  1 0 2F d yy , where F y is a function to be determined. [1] (b) Find the exact value of  1 0 F dyy . [2] (c) By using the result in part (b), find the exact value of 1 e 2 e ln 1 dxx  . [4] (d) The arc of the curve between the x-intercept and ex is rotated through 2 radians about the x-axis. Find the area of the surface generated. Without further calculation, deduce the area of the surface generated when the arc of a curve with equation 21e xy  between the y-intercept and ey is rotated through 2 radians about the y-axis, justifying your answer. [4] y x T O S 1 e 21 l nyx

5 7. It is given that the volume of a water reservoir between two cross-sectional areas can be calculated using  d b a Ax x , where Ax is the cross sectional area at depth x metres from the water surface. In real life, various numerical methods are used to estmiate the value of the integral  d b a Ax x . One of the methods that can be used to estimate the integral  d b a Ax x is Simpson’s rule, which is given by   d4 3 b aM ba LAx x A A A  , where L is the perpendicular distance between the two cross-sections. aA and bA are the cross-sectional areas of the end faces and MA is the cross-sectional area of the face in the middle as shown in Figure 1 below. (a) State a condition on the number of given cross-sectional areas for Simpson’s rule to be used to estimate the volume of a water reservoir. [1] The table below shows the cross-sectional areas of the water reservoir in City D at different depths. When the condition to use Simpson’s rule is not met, it is a usual practice to calculate the volume between two cross-sections using Trapezium Rule. Depth (in metres) Cross-Sectional Area (in ten thousand square metres) 0 190 4 127 8 100 12 76 16 20 19 (max depth) 14 Figure 1

6 (b) Using Simpson’s rule and Trapezium rule together, find an estimated volume of the reservoir. [3] A frustum is a truncated cone where a cone is cut off from the top parallel to its base, as shown in Figure 2. Alternatively, the volume of an inverted frustum can be used to estimate the volume of the water reservoir. [The volume of a cone of base radius r and height H is given by 21 π3 rH ] (c) Show that the volume V of an inverted frustum is given by  12 1 2 1 3Vh A AA A  where h is the distance between the two cross sectional areas 1A and 2A . [3] Hence find an estimated volume of the water reservoir in City D using volume of the frustum of a cone. [2] (d) Find the difference between the estimated volume found in part (b) and (c). [1] (e) Due to climate change, City D is experiencing dry spell when there is very little rainfall. The water usage of the city is 700 thousand m3 per day. Find, to the nearest number of days, the water in the reservoi r can last before it reaches 20% of its estimated volume found in part (b). Without further calculation, ju stify whether there would be a difference in the number of days if the estimated volume found in part (c) is used instead. [2] Removed Figure 2

7 8. The function g is given by  3 2g2 1x xx  for 0x . Show that the equation g0 x  has only one real root, x  . State an integer n such that 1nn  . [3] (a) To find an approximate value for , the following rearrangements of g0 x  are suggested as a basis for the iteration method of the form 1 fnnx x  . (1) 23 21 14xx    (2) 12x x  (3) 53 22x xx (i) By considering f x , identify the iteration method which converges to  . Use a graph to explain why the chosen iteration method converges to . [4] (ii) Using the appropriate iteration method found in part (a)(i), and 0x n , find the value of  , correct to 4 decimal places, an d demonstrate how to verify its correctness. [3] (b) (i) Show that a Newton Raphson iteration method for the root  is given by 31 22 1 11 22 22 32 nn n nn xxx xx   