H2_Math_Paper_1_Question

Name:____________________________________________ Class:____________ JURONG JUNIOR COLLEGE Preliminary Examination MATHEMATICS 9740/01 Higher 2 28 August 2012 3 hours Additional materials: Answer Paper List of Formulae (MF15) Cover Page READ THESE INSTRUCTIONS FIRST Write your name and civics class on all the work 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 staples, paper clips, highlighters, glue or correction fluid. 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 a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic 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. At the end of the examination, fasten all your work securely together, with the cover page in front. This document consists of 6 printed pages. [Turn over

2 1 Without using a calculator, solve the inequality 4 13 x x ≥− . Hence find the range of values of x that satisfies 4 13 x x ≥− . [5] 2 A sequence 012, , , uuu … is such that 0 0u = and ( ) 2 1 24 2n nnuu n n − −=− − + , for all n +∈ ] . (i) Use the method of mathematical induction to prove that 2 2 n nun −= , for all n +∈ ] . [4] (ii) Find () 2 1 24 2 N n n nn− = ⎡⎤−− +⎣⎦∑ . [2] (iii) State the sum to infinity of the series in part (ii). [1] 3 (a) (i) Differentiate 2 1x − with respect to x . [1] (ii) Hence find 1 1cos dx xx − ⎛⎞ ⎜⎟⎝⎠∫ . [3] (b) Use the substitution xu 1= to find the exact value of 6 2 3 1 d 9 x xx −∫ . [4]   4 Let R be the region bounded by the curves y = 4 x and y = x2 + 1. (i) Find the area of region R. [3] (ii) If the line x = c divides region R into two equal areas, find the value of c. [3] (iii) Find the volume of the solid generated when region R is rotated through 2π radians about the y-axis. [3]   5 (i) Given that ()1 ln 1 tan2y x=+ , show that 22 d2e secd y y xx = . Hence write down the first two non-zero terms in the Maclaurin series for y. [4] (ii) Given that the first two non-zero terms in the Maclaurin series for y are equal to the first two non-zero terms in the series expansion of x ab x+ , where a and b are constants, find a a