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RAFFLES JUNIOR COLLEGE JC2 Preliminary Examination 2007 MATHEMATICS  9740/01  Higher 2 Paper 1 12 September 2007 3 hours  Additional materials : Answer Paper List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your name and CT group 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.  Answerall 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. This document consists of6 printed pages. R AFFLESJUNIORCOLLEGE RJC IES 2007 Math Department [Turn over www.teachmejcmath-sg.webs.com teachmejcmath.sg@gmail.com

RJC 2007  9740/01/S/07  [Turn over  2   1  Find the set of values of x for which 2 e 8 01  x  x  x   . Hence solve 2 e 8 01  x  x  x   . [4] 2 Prove by induction that, for n  Z,  2 2 2 2 1 2(1)(2)(3)...(1)(1) (1) .2 n nn n n n n            [5] 3  A sequence of negative numbers is defined by 1 2 3 4 n n n  x x  x  , where 1 1 7  x   . (i) Write down the values of2 x and3 x, giving your answers correct to 3 significant figures. [2] (ii)Given that asn   ,n x  , find, without the use of a graphic calculator, the value of.  [3] 4  The graph of y= f( x) undergoes in succession, the following three transformations:  A: A translation of 1 unit in the negative x-direction.  B: A reflection about the y-axis. C : A stretch parallel to the x-axis (with y-axis invariant) with a scale factor of 2. The equation of the resulting curve is y = g( x) where g( x) = 2 2 3 2  x x  x    . (i)Express f( x) in the form g(ax+b), wherea andb are constants to be found. [2] (ii) Sketch the graph of y = g( x), showing clearly all the intersections with the axes, the asymptotes and the coordinates of turning points (if any). [4] 5 Given that 1f()r  r   , wherer  is a positive integer, find a single expression for f()f ( 1).r r    Hence, find the sum to 2n terms of the series 1 1 1 1 ...2 6 12 20    . [4] Deduce that the sum to 2n terms of the series2 2 2 2 1 1 1 1 ... 2 3 4 5      is less than 1. [2] www.teachmejcmath-sg.webs.com teachmejcmath.sg@gmail.com

RJC 2007  9740/01/S/07  [Turn over  3 6 Given that 3 2i2 z+- £ and 52i 3i z z++ ³ - , illustrate the locus of the point representing the complex number z in an Argand diagram. [4] Hence find the least possible value of arg z. [2] 7  The graphs of y = | h( x) | and y2 = h( x) for x >3 are as shown below:  (i) Explain why h( x) < 0 for1 < x < 2. [1] Hence sketch the graph of y = h( x) for x >3, showing clearly the asymptote and the coordinates of the stationary point. [1] (ii)Sketch the graphs of   (a) y = h’( x) where h’ is the derivative function of h, [2] (b) y = 1 h() x  for x >3, [3] showing clearly all the asymptote(s) and the coordinates of the stationary point(s). 8 Find a)  1 1tan d  x  x   , [3] b)  1  d  2 1  x  x x , by using the substitution 2 1u x . [4] 9 (i) The region R is bounded by the curves 4 4 1 y  x  , ln y x , the x-axis and the line x = 2. Calculate the area of  R. [3]  (ii) The regionQ is bounded by the curves 4 4 1 y  x  , ln y x  and the line 1 x . Find the volume of the solid of revolution formed whenQ is rotated completely about the  y-axis. [4]  y  x (1 6 ,  2 3 )   y = | h( x) |  203 1  y  y2 = h( x)   x203 1 www.teachmejcmath-sg.webs.com teachmejcmath.sg@gmail.com

RJC 2007  9740/01/S/07  [Turn over  4 10 A glass window, with fixed perimeterP, is in the shape of a semicircle with radius r , and a rectangle as shown in the diagram below. The semicircle is made of tinted glass and the rectangle is made of clear glass. The clear glass lets through a constant amount of light per unit area, L, while the tinted glass lets through 1 3 as much light per unit area as the clear glass.Let X  be the total amount of light that the entire window lets through. (i) Show that 5 26 X LrPr             . [3] (ii) Prove that, to let maximum amount of light through, the radiusr  of the window is   3 5 12 P    . [4]  11 (i)  Find 20 1 d  1 4  x t  t  . [2]  (ii) Expand 1 2214 x    as a series in ascending powers of x up to and including the term in4 x, simplifying the coefficients. Write down the set of values of x for which the expansion is valid. [3] (iii) By using the results from parts(i) and(ii) above, show that the Maclaurin’s series for  1sin2 x  is given by 3 54 122 ...3 5 x x x    . [2]  (iv)Hence, show that 2009 6 3840 π   . [2] tinted glass clear glass r www.teachmejcmath-sg.webs.com teachmejcmath.sg@gmail.com

RJC 2007  9740/01/S/07  [Turn over  5 12 The line1l has equation 12 x y z    . The line2l passes through the point( 2,1,7) A   and is parallel to the vector3 5  ijk . (i)Write down the vector equations of lines1l and2l . [2]  (ii) Show that the lines1l and2l intersect and find the coordinates of E , the point of intersection of1l and2l. [3] (iii)The acute angle between the lines1l and2l is denoted by . By finding cos ,  show that 22sin 35    . Hence find the shortest distance from Ato the line 1l, leaving your answer in exact form. [4] 13On 1st Jan 2000, Selina deposits $1000 into an account which pays a fixed  interest of $200  per year, credited into the account at the end of every year. On 1st Jan 2005, Hebe deposits $1000 into an investment account and receives an interest of $100 on 31st Dec 2005. Thereafter, the interest returned at the end of the year is 1.5 times the interest returned in the previous year. Taking year 2000 as the first year, (i) Write down the amount of savings that Selina has in her account at the end of thenth  year.  [1]  (ii) Show that the amount of savings that Hebe has in her account at the end of nth year is $(1000 + 200(1.5n5 1)). [2]  (iii) At the end of thek thyear, Hebe saw that her savings finally exceeds Selina’s savings for the first time. Find the value ofk  and the interest that Hebe receives in thek th year.   [4]  Afterk  complete years, the interest that Hebe receives for subsequent years is 1/3 of the interest received in the previous year. Determine, showing your reasons clearly, the maximum value of Hebe’s investment.  [3] www.teachmejcmath-sg.webs.com teachmejcmath.sg@gmail.com

RJC 2007  9740/01/S/07  [Turn over  6 14 A curve has parametric equations cc , ,  x ty t  where c is a positive constant. Three  points, ,cPcp  p      , ,cQcq q       and , c Rcr  r       on the curve are shown in the diagram. (a) Show that the equation of the tangent at the point , cPcp  p       to the curve is given by 2 2 0 xpy cp   . [2] (i) This tangent meets the x-axis and the y-axis at A and B respectively. Write down the coordinates of A and B. Prove that the area of triangle AOB is 22c. [3] (ii) Given that the linePO meets the curve again atQ and the straight line BQ meets the  x-axis at L. Expressq in terms of p and find the area of the triangleQOL in terms ofc. [4] (b) If the normal at , c Rcr  r       meets the curve again at ,cScs s     , show that3 1 0rs . [3] E N D O  x  y P  R Q www.teachmejcmath-sg.webs.com teachmejcmath.sg@gmail.com

2007 RJC JC2 Preliminary Examination H2 Mathematics Paper 1 (9740/01) – Page 1 of 10 2007 J