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 differentiati