JJC H2 MATH P1

JURONG JUNIOR COLLEGE J2 Preliminary Examination MATHEMATICS 9740/01 Higher 2 18 August 2009 Paper 1 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 signif icant 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 5 printed pages and 1 blank page. [Turn over

2 1 The diagram below shows the graph of f( )y x . Sketch, on separate diagrams, the following graphs, indicating clearly any asympt otes, axial intercepts and turning points, where possible. (i) , [2] f( ) 2yx  (ii) fy x , [1] (iii) f' ( )y x . [3] y f( )y x x 0 −2 A (2, −2) 2 The functions f and g are defined to be f: and 4 2 , , 0xxe x x    1g: 1 , , 0 , 1 . 1xx x x x    (i) Sketch the graph of f and state its range. [2] (ii) Prove that the composite function gf exists and define it giving its rule and domain. [3] (iii) Sketch the graph of g and hence find the range of gf. [2] 3 On the same Argand diagram, sketch the loci (i) 23 2 2zi  , [2] (ii) , [2] * 8zz i (iii) arg 3 3zi   . [2] The complex number w is represented by the point of in tersection of the loci in parts (i), (ii) and (iii). Find , in the form w x iy , giving the exact values of x and . [1] y

3 4 A student has $1000 in his savings account init ially. He decides to deposit $5 from his pocket money each week into the account, wh ich pays a compound interest rate of 1% per year. Taking a year to consist of 52 weeks and P to be the amount of money in the account after t years, show that the differential equation d 0.01 260d P Pt  may be used to model the balance in the student’s bank account. [1] (i) Express P in terms of t . [4] (ii) How long will it take for the balance to exceed $2000 ? [2] 5 (a) Find 2 14 d 41 x x x   . [3] (b) By means of the substitution tanx  , find the exact value of  2 221 d 1 x x x   . [5] 6 (a) On a farm there are 3 different types of animals : chickens, horses, and sheep. The farmer confirms that the number of sheep is twice the number of chickens. He also counted a total of 1250 animal legs. Due to his handwriting, he was not sure if there were 250 animals or 350 animals in total. Find the correct number of chickens, horses and sheep. [4] (b) By using an algebraic method, solve the inequality 21 8 63 x xx   . [3] Hence, solve the inequality 2ln 18 ln 6ln 3 x xx   . [3] [Turn over

4 7 (a) A sequence of integers is defined by 01 2,, ,uuu  0 1u  and . 1 37nnuu  Prove by induction that 1 75 32 n nu     for all non-negati ve integral values of n. [4] (b) Prove that for all positive integers n ,  11 1 !1 ! ! (nn n n 1 ) !  . [2] Hence evaluate 1 1 !1 N n nn !  in terms of . [2] Deduce that N 1 1 2!n n    . [2] 8 A geometric series has first term a, common ratio r and nth term denoted by Gn. An arithmetic series has first term a, common difference d and nth term denoted by An. It is given that a, r and d are non-zero and the two series are related by the following equations 42AAG 3 and 5556 9Aa G  . Show that . [4] 4291 01rr  0 It is also g iven that r > 0, . 1r  (i) Deduce that the geometric series is converg ent and show that its sum to infinity is 3 2 a . [3] (ii) Find the least value of N for which 11 10 N n nn A    nG  where a > 0. [4] 9 Relative to a fixed point O, the position vectors of the points A, B and C are given as follows : ,ikOA  2 i ,jOB   42 i j kOC    . It is known that A , B and C exist in the plane 1 . (i) Show that the equation of plane 1 is 2 3 1x yz  . [2] (ii) The point D has position vector 5i j k  1 . Find the position vector of the foot of perpendicular from D to plane  and hence find the exact distance from D to plane . [4] 1 (iii) It is further known that the point D is the reflection of point C about another plane . Show that the equation of plane 2 2 is 2 3xy z  . Find the equation of the line of intersection between planes 1 and . [5] 2

5 10 (a) Given that i( 2) and 4iwz w z   , where Im(w) < 0. Find exact values of z and in the form w iab , where a and b are real. [3] (b) Find the fourth roots of 16i  , giving your answers exactly, in the form ire , where   . [3] (c) Express 13z i in modulus-argument form. [2] Hence, find the set of positive integral values of n for which is real and negative. [3] nz 11 (a) A curve is defined by the parametric equations 3x t , 3y t The point P on the curve has parameter 2t  . Show that the normal at P has equation . The normal meets the curve again at the point Q. Find the value of t at Q. [6] 2 8 45yx (b) The volume V and surface area A of a closed cylinder of radius r and height h are increasing at a rate of 15 and 30 respectively. Given that its height rem ains constant, obtain an expression for r in terms of h. 3m / s 2m / s Deduce that h > 1. [6]