TJC H2 Maths P2

TEMASEK JUNIOR COLLEGE, SINGAPORE Preliminary Examination 2014 Higher 2 MATHEMATICS 9740/02 Paper 2 15 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 7 printed pages. © TJC/Prelim Exam 2014/MA H2 9740 [ Turn over

TJC/Prelim Exam/2014/MA H2 9740/02 2 Section A: Pure Mathematics [40 marks] 1 It is given that 1f( ) 3 15 xx  . Find the roots  and  of the equation f( )x x , where   . [2] A sequence of real numbers 123,,, . . .x xx satisfies the recurrence relation given by 1 fnnx x  for 1n , and 1x is a value that lies between  and. The diagram shows how the graphs of f( )y x and y x are used to obtain 2x from 1x . (i) Copy the above diagram, and illustrate how 3x can be obtained. [1] (ii) Describe the behavior of the sequence in this case. [1] 2 A circle C1 has equation given by 22 43 0xx y   . Another curve C2 with equation   22 ax b y ca   is obtained from circle C1 after going through the following successive transformations:  Stretching with scale factor 3 parallel to the y-axis followed by  Translation of 1 unit in the negative y-axis direction. (i) Find the values of a, b and c. [3] (ii) Sketch, on the same diagram, the two curves, C1 and C2, indicating clearly the points of intersection of C1 and C2. [3] (iii)  1,x k and  2 ,x k are points on C1 and C2 respectively such that 2123 xx , where k is a real constant. State the range of values of k. [1]  fy x y x yx 1x  1f x 2x

TJC/Prelim Exam/2014/MA H2 9740/02 3 3 A curve has equation f( )y x , where  2 for 2,f 2 h( ) for 2. ax bx c xx x xx       The curve has a turning point at  3,5 . The only asymptotes are 1yx and 2x . (a) Find the values of a, b and c. [3] (b) It is given that the tangent to the curve f( )y x  at the point 70, 5   is parallel to the line 230yx . Sketch the graph of  f'yx , indicating clearly all asymptotes and axial intercepts. [3] (c) The function g is such that h( ) d g( )  x xx k for 2x  , where k is an arbitrary constant. By sketching the graph of  hy x for 22 x  , express  2 2 hd x x   in the form g( ) g( )p uq v , where ,,,p quv are constants to be determined. [3] y x y = 1 – x x = 2 7 5  3, 5 f( )y x

TJC/Prelim Exam/2014/MA H2 9740/02 4 4 The parametric equations of a curve C are given by sin cos x and sec 2y where 5 12 2    . (i) Sketch the curve C, indicating the coordinates of the endpoints clearly. [2] (ii) Show that the area of the region bounded by the curve C, x-axis and the lines 2 2x and 1x , can be written as 2 5 12 1 dsin cos    . [3] (iii) Prove that sin cos 2 sin 4      . [1] Hence show that the exact area of the region is   2 ln 2 1 3 22   . [3] 5 (i) Show that 2ii1e 2 c o s e 2    . [2] (ii) Solve the equation  6 610 zz  , giving the complex number z in the form of eir  where 0r  and   . [4] (iii) Two of the roots are such that  Im 0z  . One of these roots is 6i 1 3 e3z   . State the other root 2z . [1] (iv) Given that  is the complex number such that  1 5arg 6  z , find the minimum value of 2 z . [4] Section B: Statistics [60 marks] 6 Explain what is meant by the term ‘random sample’. [1] The Ministry of Education wishes to conduct a focus group discussion with teachers from primary schools, secondary schools and junior colleges to find out how they use ICT in the classroom. Give a reason why simple random sampling may not give a representative sample of teachers’ feedback across different schools and levels. [1] Explain why a stratified sample is more appropriate in this context. [2]

TJC/Prelim Exam/2014/MA H2 9740/02 5 7(a) (i) 6 men and 3 women signed up for a dance class. Find the number of ways of distributing them into 3 equally-sized groups labelled A, B and C. [2] (ii) After the groups are finalised, the participants are asked to sit in a row of 11 chairs so that they can view a dance demonstration video. Find the number of possible arrangements so that members of each group are seated together. [3] (b) Find the number of ways that 6 men and 3 women can sit in a circle of 9 chairs so that the women are seated together and two particular men are not seated next to each other. [3] 8 Anand, Beng and Charlie patronised a restaurant that offered lucky draws to its customers depending on what they ordered. Anand and Beng ordered set meals, and each got to participate in a lucky draw from the “Blue Box”, which had a 25% chance of awarding a prize. Charlie ordered a la carte, and got to participate in a lucky draw from the “Gold Box”, which had a 40% chance of awarding a prize. Find the probability that (i) Anand, Beng and Charlie all won prizes from their lucky draws. [2] (ii) at least one of Anand, Beng and Charlie won a prize from their lucky draws. [2] (iii) Charlie won a prize from his lucky draw, given that at least one of Anand, Beng and Charlie won a prize from their lucky draws. [4] 9 Amongst various exhibits at an art gallery, only paintings and sculptures are placed for sale. On average, the number of paintings sold from the art gallery in a week follows a Poisson distribution with mean 1. The number of sculptures sold from the same art gallery in a week follows a Poisson distribution with mean 0.25. You can assume that the sale of the sculptures and paintings are independent. Taking a month to be 4 weeks, (i) find the probability that one painting is sold in a month. [1] (ii) find the probability that over a period of 4 w eeks, the first sculpture is sold in the fourth week. [2] (iii) If the total sales exceed 4 pieces in a month, the gallery owner will consider that month a “Good” month. Taking a year to be 12 months, use a suitable approximation to find the probability that there are no more than 30 “Good” months in 5 years. [6]

TJC/Prelim Exam/2014/MA H2 9740/02 6 10 Flour is packed by a manufacturer into bags labelled 5.00 kg each. Recently, the manufacturer received complaints from some customers who claimed that the flour they bought weighed less. To test the claim, th e manufacturer takes a random sample of 60 bags and records down the weight, t kg, of the flour in each bag. The results are summarised by ∑ t = 291.6 and ∑(t – 4.86)2 = 18.344 Test, at 5% significance level, if the complaints are justified. [5] The manufacturer decides to re-label the weight of flour in each bag as 0 kg. Find the range of values of 0 so that the data previously collected provides insufficient evidence at