SAJC_H2_Math_P2_Question

[Turn Over ST ANDREW’S JUNIOR COLLEGE PRELIMINARY EXAMINATION MATHEMATICS Higher 2 9740/2 Monday 15 SEP 2015 3 hours READ THESE INSTRUCTIONS FIRST Write your name, civics group and index number 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. Answer all the questions. Total marks is 100. Give non-exact numerical answers correct to 3 signi ficant 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 a llowed unless a question specifically state otherwise. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematic 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. This document consists of 6 printed pages includi ng this page.

2 [Turn Over Pure Mathematics (40 marks) 1 It is given that ( ) ( ) ( ) ( ) 2 2 6 3 f 3 9 x x x x − = − + . (i) Using partial fractions, find ( ) f d x x ∫ . [5] The diagram above shows the curve with equation ( )fy x = . (ii) Find the exact area bounded by the curve ( )fy x = − and the x-axis, between the values of 0x = and 3x = . [3] 2 (a) Adrian has signed up at a driving centre to learn how to drive. His first lesson is 40 minutes long. Each subsequent lesson is 5 minutes longer than the previous lesson, so that the second lesson is 45 mi nutes long, the third lesson is 50 minutes long, and so on. The centre requires a student to have attended at least 60 hours of lessons before he is qualified to take the driving test. Find the minimum number of lessons that Adrian has to attend before he can take the test. [4] (b) A sequence of real numbers 1 2 3 , , ,... u u u , where 1 0u ≠ , is defined such that the ( )1 th n + term of the sequence is equal to the sum of the fi rst n terms, where .n +∈ ℤ Prove that the sequence 2 3 4 , , ,... u u u follows a geometric progression. [3] Hence find 1 1 N r r u + = ∑ in terms of 1 and . u N [2] y x 0

3 [Turn Over 3 The equations of lines 1l and 2l are given as follows: 1 31 : 1 4 , 20 l λ λ     = − + −    ∈          r ℝ , and 2 3 6 : 3, 5 1 y z l x − − == − . (i) Find the coordinates of N, the foot of