YJC H2 MATH P2

This question paper consists of 5 printed pages. YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIOR COLLEGE YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIOR COLLEGE YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIOR COLLEGE YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIOR COLLEGE YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIOR COLLEGE YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIOR COLLEGE YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIOR COLLEGE YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIOR COLLEGE YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIOR COLLEGE YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIOR COLLEGE YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIOR COLLEGE YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIOR COLLEGE YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIOR COLLEGE YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYSIHUNJUNIOR COLLEGE YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYSIHUNJUNIOR COLLEGE YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYSIHUNJUNIOR COLLEGE YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYSIHUNJUNIOR COLLEGE YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYSIHUNJUNIOR COLLEGE YISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYISHUNJUNIORCOLLEGEYSIHUNJUNIOR COLLEGE YISHUN JUNIOR COLLEGE 2015 JC2 PRELIMINARY EXAMINATION MATHEMATICS 9740/02 Higher 2 Paper 2 25 August 2015 TUESDAY 0800h – 1100h Additional materials : Answer paper Graph paper List of Formulae (MF15) TIME 3 hours READ THESE INSTRUCTIONS FIRST Write your name and CTG in the spaces provided 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 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, w rite down the question number of th e questions attempted, model of calculator used on the spaces provided on the cover page. Tie your cover page on top of the answer scripts before submission. The number of marks is given in brackets [ ] at the end of each question or part question.

Yishun Junior College  2015 JC2 Preliminary Exam  H2 Mathematics 9740/02 Page 2 of 5 Section A: Pure Mathematics [40 marks] 1 (a) Using an Argand diagram, show that any complex number iab , where a and b are real numbers, may be expressed in the form (cos isin )r  , where 0r and      . State the geometrical meaning of r and  . [3] Write down the cartesian equation of the locus of the point representing 3(cos isin ) as  varies. [1] (b) Given that cos isinz  , show that 1 2cosz z  and 1 2isinz z  . Hence, find 2 2 1 1 z z   in terms of  . [5] 2 In the given diagram, the curve 1C has equation 2 4xy and the curve 2C has equation   2 8x y k where 0k  . The region bounded by 1C and 2C is shaded. A bowl is generated by rotating the shaded region through  radians about the y-axis. (i) Show that the diameter of the rim of the bowl is 4√(2k). [2] (ii) Show that the volume of the material used in making the bowl is equal to the capacity of the bowl. [5] (iii) If k = 4, find the area of the shaded region, giving your answer correct to 3 decimal places. [2] 3 The curve C has equation 2 7 2 x qxy x   , where q is a negative constant. (i) Obtain the equations of the asymptotes of C. [2] (ii) Show that 2q if yx is an asymptote to C, and sketch C in this case, stating the coordinates of any points of intersection of C with the axes. [4] (iii) Hence, by sketching another suitable curve on the same diagram, solve the inequality   2 2 27 1 2 16 xx xx    . [5]

Yishun Junior College  2015 JC2 Preliminary Exam  H2 Mathematics 9740/02 Page 3 of 5 4 The functions g and h are defined by g : ln , , 0x x x x  , 1h : , , 0x x x x  . (i) Explain why the function 1g exists. [1] Find 1g ( )x and write down the domain and range of 1g . [2] (ii) Explain why the composite function hg does not exist. [1] (iii) Find the composite function gh in a similar form and write down its range. [3] (iv) Find 2h ( )x and hence 9h ( )x in simplified form. [2] (v) State a relationship between the graphs of (a) g and gh, (b) g and 1g . [2] Section B: Statistics [60 marks] 5 A school has 600 male and 400 female students. 5% of the students are to be sampled to find out their opinions about the PE programme. (i) Describe how the sample could be obtained using stratified sampling. [2] (ii) State one reason why a stratified sample is preferable in this context. [1] 6 The voltage of a battery is a random variable with the distribution   2N, . The mean voltage of 4 randomly selected batteries is denoted by X . It is given that    P 22.2 P 26.2 0.0128XX    . State the value of  and find the value of  . [4] 7 A box contains 5 black balls and 3 red balls. 3 balls are drawn randomly from the box, one at a time, without replacement. The colour of the ball that is drawn is noted. (i) Find the probability that at least 2 red balls are drawn. [2] (ii) Given that the third ball drawn is red, find the probability that the first ball drawn is also red. [3] (iii) Justifying your conclusion, determine whether the events “The first ball drawn is red.” and “At least 2 red balls are drawn.” are independent. [2] 8 Under ordinary conditions, the mean time required by a person to complete an arithmetic sum is 50 s. Sixty people were ra ndomly selected to do the sum in a room with a temperature set at 15 C . The time taken for this sample had an average of 55 s and a standard deviation of 19.3 s. (i) Test, at the 5% significance level, whether there has been an incre ase in the mean time required by a person to complete the sum. [5] (ii) Explain whether any assumptions about the population are needed in order for the test to be valid. [1] (iii) The same sample is now used to carry out a test at the 5% significance level, of whether the mean time required by a person to complete the sum has changed. State with a reason using (i), the conclusion of this test. [2]

Yishun Junior College  2015 JC2 Preliminary Exam  H2 Mathematics 9740/02 Page 4 of 5 9 Five-letter codewords are formed using the letters of the word MATHEMATICS. (a) Find the number of codewords that can be formed if there are no restrictions. [4] (b) Find the probability that a codeword (i) contains both M’s and both A’s, [1] (ii) has both A’s separated,