2019 J2 CJC H2 Math Prelim (with ans)

CJC 2019 H2 Mathematics Prelim Paper 1 1 Differentiate 2 e x with respect to x. Hence find 23edxxx ³ . [3] 2 Without using a calculator, solve the inequality 2 343 21 xxx x . [4] 3 The curve C has the equation 22(5 )(3 ) 116 9 xy . (i) Sketch C, showing clearly the coordinates of the stationary points and the vertices. [2] The region R is bounded by curve C, the line x = 9 and the x axis. (ii) Find the volume of the solid generated when region R is rotated completely about the x axis. [3] (iii) Describe fully a sequence of two transformations whic h would transform the curve C to the curve 22(2 5) 116 9 xy . [2] 4 (i) Show that 33 3 3 3( 1 ) 1 (1 ) (1 ) rr A B rr r r , where A and B are to be determined. [2] (ii) Hence find 33 33 3 3 71 9 6 (2 1) 1...( 1 )( 2 ) ( 2 )( 3 ) ( 2 )( 2 1 ) nn nn . [3] (iii) Use your answer in part (ii) to find 33 1 3( 1 ) 1 1 (1 ) 2 r r rr rr f ªº §·«» ¨¸ ©¹«» ¬¼ ¦ . [3] www.KiasuExamPaper.com 102

5 A developer has won the tender to build a stadium. Sunrise Si ngapore, who would finance the construction, wants to have a capacity of at least 50 000 s eats. There is a limited land area to the stadium, and in order for the stadium to have a full sized track and football pitch, the first row of seats can seat 300 people, and every subsequen t row has an additional capacity of 20 seats. ( i ) W h a t i s t h e l e a s t n u m b e r o f r o w s t h e s t a d i u m m u s t h a v e t o meet Sunrise Singapore’s requirement? [3] Assume now that the stadium has been built with 60 rows, with row 1 nearest to the football pitch. (ii) For the upcoming international football match between Riv erloop FC and Gunners FC, tickets are priced starting with $60 for Category 1, with each s ubsequent category cheaper by 10%. How much would a seat in the 45 th row cost? [1] Category 1 Rows 1 – 20 Category 2 Rows 21 – 40 Category 3 Rows 41 – 60 There is a total crowd of 51 000 for the football match. (iii) Assuming that all the tickets from the cheapest catego ry are sold out first before people purchase tickets from the next category, calculate the total revenue collected for the football match. [5] 6 (i) Given that cosxye x , show that 2 2 2 2 dd d 2dd d yy y yy yxx x §· ¨¸ ©¹ . [4] (ii) Find the Maclaurin series for cosxye x , up to and including the term in 3x . [4] (iii) Verify the correctness of the expansion found in part (ii) using standard series found in the List of Formulae (MF26). [3] www.KiasuExamPaper.com 103

7 (i) A water trough, shown in the diagram below, in the sh ape of a triangular prism is used to collect rainwater. The trough consists of two rectangul ar zinc sheets of negligible thickness, each with fixed dimensions 10 m by 2 m, a nd two triangular zinc sheets with height h m, as shown in the diagram below. Use differentiation to find the maximum volume of the trough, proving that it is a maximum. [5] (ii) Water collected in the trough is drained at a rate of 0.0 01m3 /s into a container consisting of two cylinders, as shown below. The larger cylinder has radius 1 m and height 0.5 m. The smaller cylinder has radius 0.6 m and height 0.5 m. Find the rate at which the depth of water is increasing after 29 minutes. [4] (iii) Let H be the height of the water level in the container at time t. Sketch the graph of d d H t against t. [3] 8 Do not use a calculator in answering this question. (a) Showing your working clearly, find the complex number z and w which satisfy the simultaneous equations 3i 1 2 9 i , 2*3 1 6 2 3 i . zw zw [5] www.KiasuExamPaper.com 104

( b ) I t i s g i v e n t h a t z1 is a root of the equation 432 348 0zzz , where 1 13 iz . (i) Express 432 348 0zzz as a product of two quadratic factors with real coefficients. [4] ( ii) G iven that i5 1e pq z , determine the exact values of p a n d q, where qSS . [4] 9 Line l 1 has equation 12 81 35 r O §· § · ¨¸ ¨ ¸ ¨¸ ¨ ¸ ¨¸ ¨ ¸ ©¹ © ¹ where O is a real parameter, and plane p has equation 17 0 a b §· ¨¸˜ ¨¸ ¨¸ ©¹ r . It is given that l1 lies completely on p a n d t h e p o i n t Q has coordinates (1 ,2 , 3 ) . ( i) S how that a = – 1 and b = 2. [3] (ii) Find the foot of perpendicular from point Q to p oi n t P. Hence find the shortest distance between point Q and p in exact form. [4] Given that the shortest distance between l 1 and the foot of perpendicular of Q on p is 5 63 . (iii) Using the result obtained in part (ii) or otherwise, fin d the shortest distance between Q and l1. [2] The line l 2 is parallel to p, perpendicular to l1 and passes through P and Q. (iv) Show that the Cartesian equation of the line l2 is 1 232 x yz . [3] (v) Find the vector equation of line l3, which is the reflection of l2 about p. [3] www.KiasuExamPaper.com 105

10 Tumours develop when cells in the body divide and grow at an excessive rate. If the balance of cell growth and death is disturbed, a tumour may form. A medical scientist investigates the change of the tumour size, L mm at time t days of a particular patient using models A and B. For both of the models, it is given that the initial rate of the tumour size is 1 mm per day when the tumour size is 1 mm. ( i) U nder Model A, the scientist observes that the patient’s turmour is growing at a rate proportional to the square root of its size. At the same time, the tumour is reduced by radiation at a rate proportional to its size. It is further observed that the patient’s tumour is decreasing at 2 mm per day when the tumour is 4 mm. S how that L and t are related by the differential equation d 32d L LLt . [2] (ii) Using the substitution 2Ly where y > 0, show that the differential equation in part (i) can be written as d3 2 d2 yy t . Find y in terms of t and hence find L in terms of t only. [7] (iii) Under Model B, the scientist suggests that L and t are related by the differential equation 2 22 2 d2 d( 1 ) Lt tt . Find the particular solution of this differential equation. [4] (iv) Find tumour sizes predicted by Models A and B in the long run. [2] www.KiasuExamPaper.com 106

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CJC 2019 H2 Mathematics Prelim P2 Section A: Pure Mathematics (40 Marks) 1 The function h is given by 32h( ) , xxa xb x x c  , where a, b a n d c a r e r e a l constants. The graph of y = h(x) passes through the point 131, 4 §· ¨¸ ©¹ . The point ( 8,642) lies on the graph of y = h(|x|) and the point 14, 97 §· ¨¸ ©¹ lies on the graph of 1 h( )y x . Find the values of a, b and c. [4] 2 Given that kp = (p.q)q where k is a positive constant, p and q are non – zero vectors. (i) What is the geometrical relationship between p and q? [1] (ii) Find | q| in terms of k. [3] 3 The diagram below shows the graph of y = f(x). The curve has a minimum point (5,10) and a maximum point (1 ,2 ) . The lines x = 2 and y = x + 2 are asymptotes of the graph. www.KiasuExamPaper.com 108

(i) Sketch the curve f' ( )yx , indicating clearly the coordinates of the points where the graph crosses the x axis and the equations of any asymptotes. [3] (ii) State the range of values of x for which the graph of y = f (x) is (a) strictly decreasing, [1] (b) concave upwards. [1] 4 The function f is defined by f: | 3 | , , 2 3xx x x xo d ,2,2,2 . (i) Explain, with the aid of a sketch, why the inverse functio n 1f exists. (ii) Find 1f( )x and state the domain of 1f . [4] It is given that 22 , 0 2 g( ) 10 2 ,2 4 .1 xx x x