TJC H2 Maths P1

TEMASEK JUNIOR COLLEGE, SINGAPORE Preliminary Examination 2014 Higher 2 MATHEMATICS 9740/01 Paper 1 1 September 2014 Additional Materials: Answer Paper 3 hours List of Formulae (MF 15) READ THESE INSTRUCTIONS FIRST Write your Civics Group and Name on all the work that 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. 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 6 printed pages. © TJC/Prelim Exam 2014/MA H2 9740 [ Turn over

TJC/Prelim Exam/2014/MA H2 9740/01 2 1 (i) Find the derivative of 24 x with respect to x. [1] (ii) Given the differential equation 2 2 2 d41 d yx x  , find y in terms of x. [4] 2 (a) The point A has coordinates (3, a, b) where ,ab  . Given that A lies on the xy-plane and the magnitude of the position vector of A is 5, find the values of a and b. [ 3 ] (b) The real numbers c and d are such that the vectors dc mi j k and cdni j k are perpendicular to each other. Show that  2 1cmn . [3] 3 Given  2f zp z q z r  where p, q and r are complex numbers such that f1 2 i . The equation  f0 z  has roots 1i and 12 i . Find p, q and r. [6] 4 Without the use of a graphic calculator, solve the inequality 1025 2x x  . [3] Hence find the solution to the inequality 102cos 5 2c o s   , where 02   . [3] 5 A souvenir company received an order to produce a souvenir that must satisfy all of the following conditions: (1) The souvenir is a solid cuboid with a square base. (2) The souvenir is made using 1m3 of superior clay. (3) The external surface of the souvenir must be coated with a special-mixed glow paint. Find the dimensions of the souvenir, in m, such that the amount of special paint needed i s t h e m i n i m u m . [ 7 ] [Turn over

TJC/Prelim Exam/2014/MA H2 9740/01 3 6 A contagious disease was found to infect a village with a population of 10000 people. Let P, in thousands, be the number of infected people t days after the start of the outbreak. The disease spread at a rate that was proportional to the product of the number of infected people and the number of non-infected people. It was found that when P reached half the initial population of the village, the disease was spreading at a rate of 1000 people per day. Show that the spread of the disease can be modelled by the differential equation  2 25 5d d2 5 PP t  . [2] Given that 100 people were infected by the disease initially, find P in terms of t. [3] Explain what would happen to the village population in the long term. [2] 7 A convergent geometric sequence of positive terms, G has first term a and common ratio r. Write down, in terms of a and r, an expression for the nth odd-numbered term of G. [ 1 ] If the sum of first n odd-numbered terms of G is equal to the sum of all terms of G after the nth odd-numbered term, show that 22 121 0nnrr   . (i) Hence find the value of r when 5n . [3] (ii) In another sequence H, each term is the reciprocal of the corresponding term of G. If the nth term of G and H is denoted by nu and nv respectively, show that a new sequence whose nth term is ln n n u v    , is an arithmetic progression. [4] [Turn over

TJC/Prelim Exam/2014/MA H2 9740/01 4 8 The functions f and g are defined by f : x  2 81 3xx  , x  , x 4 , g : x  a e x , x  . (i) Show that 1f exists and express 1f in a similar form, stating the domain clearly. [3] (ii) Determine the largest integer value of a such that fg exists. [2] (iii) For the largest value of a obtained in (ii), find fg(x) and state the domain and the range of fg. [4] 9 Given that ln e xy , show that  2 2 dd e1dd xyy xx   . [2] (i) Find the Maclaurin’s series for ee x y , up to and including the term in 3x . [ 4 ] (ii) Find the first three non-zero terms of the Maclaurin series for ee xxy  . [2] Hence find in terms of e, the approximate area bounded by the curve ee xxy  , the x-axis, the y-axis and the line 0.5x . [2] [Turn over

TJC/Prelim Exam/2014/MA H2 9740/01 5 10 The region R is bounded by the x-axis, the y-axis, the line y = 1 and the curve lnyx where ,0xx . The area of R may be approximated by the total area, A, of n rectangles each of height 1 n , as shown in the above diagram. Show that 1 11e 1e n A n    . [ 4 ] Another finite region S is bounded by the x-axis, x = e and the curve lnyx where ,0xx . Explain how A can be used to approximate the area of region S and state, with a reason, whether it is an underestimation or overestimation. [3] Find the exact volume of the solid formed when region S is rotated completely about the y- a x i s . [ 3 ] [Turn over y x lnyx 0 1 n 2 n 3n n  2n n  1n n  1 1y

TJC/Prelim Exam/2014/MA H2 9740/01 6 11(i) Prove by the method of induction that    2 1 1 12 16 n r rn n n    . [4] (ii) It is given that   4f rr . Show that    32ff 1 1r r ar br ar    , for constants a and b to be determined. [2] Hence find a formula for 3 1  n r r , leaving your answer in a fully factorised form. [6] 12 Two planes 1p and 2p have equations 3ax y z b  and 42x yb z a  respectively. They intersect at the line l which contains the point 1, 0, 1A  . (i) Find the values of a and b. [2] (ii) Without the use of a graphic calculator, find a vector equation of the line l. [ 2 ] Given that the point ( 4, 6,12)N  is the foot of perpendicular from point  1, ,B cd to the line l, show that 61 3 2 1 7cd  . [3] Another plane 3p is parallel to the plane 2p and contains B. Given that the distance between planes 3p and 2p is 5 21 . Find the values of c and d. [5] Hence write down two possible equations of plane 3p . [2]