RI_H2_MATHS_P1_Qns

RI H2 Mathematics 2017 Prelim Exam Paper 1 Question 1 A local wholesaler sells Pikachi plushies in two sizes, small and large. The number of Pikachi plushies bought by three particular retailers and the total amount they paid are shown in the following table. Retailer Small Large Total Amount paid A 30 50 $1375 B k 2k $2704 C 2 k k $2522 Find the price of each small and each large Pikachi plushy and determine the value of k. [4] 2 A right circular cone has base radius r cm and height h cm. As r and h vary, its curved surface area,   22rrh  2cm , remains constant. It is given that when 2r cm, the magnitude of th e rate of change of h is 10 times the magnitude of the rate of change of .r Given also that ,hr find the height of the cone at this instant. [4] 3 (a) Find  2 2 d. 18 4 x x xx      [4] (b) Use the substitution 2secx  to find the exact value of  4 2 2 1 4dx xx   . [4] 4 A curve C has equation  f,yx where   2f, ax cx xb    and , and ab c are constants. It is given that C has a vertical asymptote 1x and a minimum point at  0, 1 . (i) Find the values of , a n d .ab c [4] (ii) Sketch the graph of f,y x stating the coordinates of any point(s) of intersection with the axes and the equation(s) of any asymptote(s). [3]

(iii) Hence, solve the inequality  f4 0 .x   [2] 5 The diagram shows the curve f( ) .yx The curve has maximum points at  5, 4 and the origin, and crosses the x-axis at  4, 0 . The lines y = 0, x = 3 and y = x + 2 are the horizontal, vertical and oblique asymptotes to the curve respectively. On separate diagrams, draw sketches of the graphs of (a)  1 ,fy x [3] (b) f,yx  [3] (c) 1f, 2 xy   [3] labelling clearly the equation(s) of any asymptote(s), coordinates of any axial intercept(s) and turning point(s) where applicable. 6 (i) Given that  ln 1 sin 2yx , show that 22 2 ddee 4 s i n 2 dd yy yy xxx   . Find the first three non-zero terms in the Maclaurin’s series for y. [5] (ii) It is given that the three terms found in part (i) are equal to the first three terms in the series expansion of (1 ) nax bx for small x. Find the exact values of the constants a, b and n and use these values to find the coefficient of 4x in the expansion of (1 ) , nax bx giving your answer as a simplified rational number. [5] y = f(x) x = 3 y = x + 2 (4, 0) (5, 4) y = 0 y

7 Mr Tan is planning to set up a home theatre in his spacious rectangular living room. A projector screen with height a metres is to be positioned against one of the walls b metres above the eye level (see Fig. 1). He is trying to decide on the horizontal distance between the sofa and the screen so that the viewing angle  of the projection screen is as large as possible. (i) Show that 11tan tanab b x x   , where