EJC_H2_MATH_P2

EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2018 General Certificate of Education Advanced Level Higher 2 MATHEMATICS Paper 2 [100 marks] 9758/02 21 September 2018 3 hours Additional Materials: Answer Paper List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name, civics group and question number on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use paper clips, highlighters, glue or correction fluid. Answer all 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 an approved graphing calculator. Unsupported answers from a graphi ng calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing 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. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 7 printed pages.

Section A: Pure Mathematics [40 marks] 1 (i) Given that f is a continuous function, explain, with the aid of a sketch, why the value of 11 2 2lim f f ... f n nn n nn n n→∞  ++     ++ +         is 2 1 f( ) dxx [2] (ii) Hence evaluate 1 1lim →∞ = + n n r n nn r exactly. [3] Suggested Solution (i) As shown in the diagram, the area under the curve y = f( x) from x =1 to x =2 can be approximated by the total areas of the n rectangles with width 1 n and heights given by 123f1 , f1 , f1 , , f1 . n nnn n    +++ +       L As n increases, the approximations will get better and approach the exact area as a limit, i.e. 2 1 11 2 2lim f f ... f f ( ) d n nn n xxnn n n→∞  ++     ++ + =          [Shown] (ii) 2 11 11lim f ( ) d where f ( ) n n r n xx xnn r x→∞ = ==+  [] 2 2 1 1 1 d l n| | l n2 l n1 l n2xxx== = − = … x y O 1+ y = f(x) 1+ 1+ 1+ 2 1

2 The diagrams below show the graphs of f( )yx= and f' ( )yx= . On separate diagrams, sketch the graphs of: (i) f( 2 ) 1yx=+ [2] (ii) 1 f( )y x= ′ [3] (iii) f( )yx= [3] showing clearly, in each case, the intersection(s) with the axes, the coordinates of the turning point(s) and the equation(s) of the asymptotes. Suggested Solution y x O 1 5 (8, 3− ) 3− y x O 1 2 y=2 (5, 3) 1

(i) (ii) 1 f( )y x= ′ (iii) y = f(x) 3 (a) A retireme