2012 H2 Math Prelims Paper 2 Qn Paper (final)

© DHS 2012 This question paper consists of 8 printed pages (including this cover page). Name: Index Number: Class: DUNMAN HIGH SCHOOL Preliminary Examination Year 6 MATHEMATICS (Higher 2) 9740/02 Paper 2 17 September 2012 3 hours Additional Materials: Answer Paper List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your Name, Index Number and 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 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, attach the question paper to the front of your answer script. The total number of marks for this paper is 100. For teachers’ use: Qn Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Total Score Max Score 6 6 6 10 12 4 4 6 8 8 10 10 10 100

1 A w t 2 A A hemisphe initially fill of 20 cm 3 p where h is t the water su Given that find, at this (i) the r a (ii) the r a A curve C1 Sketch a cle Describe a Sketch C3 asymptotes DH Sec erical goldf led with wa per min. T h the depth of urface in cm the minimu s instant, ate of chang ate of decrea is defined p early labelle sequence o which is s and any po S 2012 Year 6 ction A: Pu fish tank wi ter. The tan he volume f water at th m, is given b um depth o ge of the dep ase of the ra parametrica 2 1x t  an ed diagram f geometric 1 1x t  a the recipr o oints of inte 2 6 H2 Math Pre ure Mathem ith radius 1 nk has a def of water i n he centre of by 30r h of water ne e pth of water adius of the ally by nd 4 1y t  of C1. cal transform and 4y t  ocal functi o ersection wit eliminary Exam matics [40 m 5 cm (as s h fect and wat n the tank i s f the tank in 2h h . eded for th e r, and water surfa 1 , 1t   mations whi ,1 1t   on of C1, s th the axes. mination marks] hown in the ter is leakin s given by cm. Show t e goldfish t ace. . ich maps C1 . stating the e figure abo ng at a const 453V h that r, the r to survive i 1 to C2 defin equations [Tu ove) was tant rate  23h h , radius of [1] is 5 cm, [2] [3] [2] ned by [2] of any [2] urn over

3 DHS 2012 Year 6 H2 Math Preliminary Examination 3 In a single Argand diagram, sketch the following loci, labelling each locus clearly. (i) 5,z  (ii) 88 i .zz The two complex numbers that satisfy the above equations are represented by p and q, where arg 0.p q   Find p and q exactly. [5] State the exact value of (5 )arg .(5 ) p q   [1] 4 A finite sequence {}na has 50 terms and is such that 1 0.15nnaa  for 1, 2,3, , 49.n  (i) Given that 50 1 99 ,aa  show that 1 0.075.a  [2] (ii) Find, without using a calculator, the value of 50 1 .n n a   [2] Another infinite sequence {}mb is such that 15 0ba and 1 0.98m m b b   for 2.m (iii) Determine the smallest value of k such that 25.kba [2] (iv) Find the least value of h such that the sum of the first h terms of {}mb is more than 99% of its sum to infinity. [2] (v) If 1 0.98m m b b   instead, find 13 0 .m m b     [2] [Turn over

4 DHS 2012 Year 6 H2 Math Preliminary Examination 5 The equations of planes 1p and 2p are given by 1 2 :7 2 0 , , 2 3 :5 7 , . p p            r r . .   (i) Verify that the point (1, 2, 3)A lies in 1p and find the value of  if A lies in 2p as well. [2] (ii) Find the equation of the line of intersection l between 1p and 2p in terms of . If l is coincident with anothe r line with equation given by 14 22 , , 31 kk               r  show that 3.  [3] (iii) Find the acute angle between 1p and 2 .p [2] (iv) Another point B has coordinates (2, 4,8). Find the position vector of the foot of the perpendicular from B to l. Hence find the reflection of the line through A and B about l. [5] Section B: Statistics [60 marks] 6 An office received a total of 810 loan a pplications on a particular day. The applications were categorised into business, house and study loans. Due to a shortage of staff, the office was only able to process 30 applications on that day. (i) Describe how you would choose a systematic sample of size 30 from the loan applications. [2] (ii) Explain why a systematic sample may not ensure that all categories of loan applications were processed within that day. [1] (iii) Explain why a stratified sample may be preferred over a systematic sample in the context of this question. [1] [Turn over

5 DHS 2012 Year 6 H2 Math Preliminary Examination 7 The number of Green Top taxis and EZCab taxis arriving in a randomly chosen 10- minute period at the airport taxi bay may be assumed to be independent Poisson variables with mean 3 and 5 respectively. (i) Show that the probability th at at least 7 taxis arrive at the taxi bay in a randomly chosen 10-minute period is 0.687. [2] (ii) In a randomly chosen 10-minute period, at least 7 taxis arrived. Find the probability that all these taxis were EZCab taxis. [2] 8 How many six-digit numbers can be formed from the digits 1, 2, 3, 4, 5, 6 and 7 if (i) repetition of digits is not allowed? [1] (ii) repetition of digits is allowed? [1] (iii) every digit in the number appears at least twice? [4] 9 Data is gathered on how exam scores, y, vary with the number of hours, x, students spent studying for Mathematics each week. The data for 7 students is shown below: Study hours, x 1 2 3 6 12 24 36 Exam score, y 41 61 71 86 92 93 95 (i) Sketch a scatter diagram for the data and determine the linear product-moment correlation coefficient. Comment on whether a linear model would be appropriate here. [3] (ii) The following are two possible models for the data above: Model A: 2 bya x Model B: dyc x  , where a, b, c, and d are real constants. State, with a reason, which of the above is a better model. [2] (iii) For your choice of the model in (ii), find the corresponding equation of the least squares regression line. Use your equation to estimate the value of y when x = 65. Comment on the reliability of your answer. [3] [Turn over

6 DHS 2012 Year 6 H2 Math Preliminary Examination 10 The waiting time for a patient to see a doctor in a clinic is T minutes. The waiting times of a randomly chosen sample of 120 patients are summarised by  15 123,t and   2 15 2504.t (i) Find an unbiased estimate of the populat ion mean and show that the unbiased estimate of the population variance is 19.983. [2] The management of the clinic claims that the mean waiting time is 15 minutes. (ii) Carry out a test at the 5% significan ce level to determine whether the mean waiting time differs from 15 minutes. [3] (iii) In another sample, the waiting times of n patients (where 50n ) were recorded and a one-tail test was conducted. Given that the sample mean was 15.5t  minutes and the null hypothesis was not rejected at