DHS H2 MATH P2 QP

© DHS 2015 This question paper consists of 6 printed pages (including this cover page). Name: Index Number: Class: DUNMAN HIGH SCHOOL Preliminary Examination Year 6 MATHEMATICS (Higher 2) 9740/02 Paper 2 23 September 2015 3 hours Additional Materials: Answer Paper Graph 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 an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing 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. For teachers’ use: Qn Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Total Score Max Score 10 10 10 10 4 5 7 7 8 8 10 11 100

2 DHS 2015 Year 6 H2 Mathematics Preliminary Examination [Turn over Section A: Pure Mathematics [40 marks] 1 A sequence 12 3, , , ...uu u is such that 1 1 4u  and 1 22 4 , for all (1 ) (2 ) 1.nnuu n nn n     (i) Use the method of mathematical induction to prove that 22 1 .(1 ) nu nn  [4] (ii) Hence find 1 22 1 .(1 ) (2 ) N n nn n  [2] (iii) Give a reason why the series in part (ii) is convergent and state the sum to infinity. [2] (iv) Use your answer to part (ii) to find 2 22 1 .(1 ) N n nn  [2] 2 [It is given that a right circ ular cone with base radius r and height h has volume 21 3 π ,rh and the curved surface area is π ,rl where l is the slanted height of the cone.] The model of a silo is made up of three parts.  The roof is modelled by the cu rved surface of a right circular cone with base radius 12 x cm and height 5x cm.  The walls are modelled by the curved surface of a cylinder of radius 12x cm and height y cm.  The floor is modelled by a circular disc of radius 12x cm. The three parts are joined together as shown in the diagram. The model is made of material of negligible thickness. The curved surface of the c one is made of material A at the cost of $0.05 per cm 2. The curved surface of the cylinder and the base of the cylinder are made of material B at the cost of $0.02 per cm2. Given that it costs $100 to make the model, show that the volume V cm3 of the model is given by 330000 2964 π .Vx x Hence find, using differentiation, the values of x and y which give a model of maximum volume. [10] 12x 5x y

3 DHS 2015 Year 6 H2 Mathematics Preliminary Examination [Turn over 3 (i) Given that the complex number z satisfies the equation 4 ,*z z show that 2.z  [1] (ii) Express 1i  in the form ie.r  [1] (iii) On a single Argand diagram, sketch the loci (a) 4 ,*z z (b) 21 i .zz   [3] (iv) The complex numbers 1z and 2z satisfy the equations 4 *z z and 21 i ,zz    where 12arg( ) arg( ).zz  Find the exact values of 1z and 2 ,z giving your answers in the form i.x y [4] (v) Another complex number w satisfies 4 *w w and 1 2 arππ g.w  Explain why 12 1 2arg( ) arg( ) π.zw zw  [1] 4 A farmer owns a parcel of land th at is partially covered with weed. The area that is covered with weed increases by 80 m 2 each week. The farmer decides to star t weeding. At the start of the first week of weeding, 500 m2 of the land is covered with weed. (i) In option 1, the farmer removes weed from 10% of the area covered with weed at the end of each week. (a) Find the area covered with weed at the end of the second week. [2] (b) Show that the area covered w ith weed at the end of the nth week is given by 2(0.9 (500) (1 0.9 )) m ,nn k where k is a constant to be determined. [3] (c) Find the area covered with weed at the end of a week in the long run. [1] (ii) In option 2, the farmer removes weed from an area of 50 m 2 at the end of the first week. He removes weed from an additional area of 10 m2 at the end of each subsequent week. Thus he removes weed from an area of 60 m 2 at the end of the second week, and 70 m2 at the end of the third week, and so on. (a) Show that the change in the area covered with weed in the nth week is given by 2(40 10 ) m .n [1] (b) Hence, or otherwise, find the area covered with weed at the end of the nth week in terms of n. [3]

4 DHS 2015 Year 6 H2 Mathematics Preliminary Examination [Turn over Section B: Statistics [60 marks] 5 A student decides to conduct a su rvey in his secondary school. His school consists of four levels, with 400 students in each level. (i) The student randomly surveys 10 students from level 1, 20 students from level 2, 30 students from level 3 and 40 students from level 4. Explain whethe r this method is stratified sampling. [1] (ii) Describe how a quota sample of size 40 might be obtained, and state one disadvantage of quota sampling. [3] 6 The Amazing Hair Salon offers hairstyling serv ices to both male and female customers. The times spent (in minutes) at the hair salon by male and female customers, denoted by M and F, respectively, are modelled as ha ving independent normal distributi ons with means and standard deviations as shown in the table. Mean Standard Deviation M µ 18 F 71 35 (i) Given that ( 32) ( 67),PM PM  state the value of µ. [1] (ii) Find the probability that the average time spent by five randomly chosen female customers is more than 3 times the time spent by a randomly chosen male customer. State clearly the mean and variance of any normal distribution you use in your calculation. [3] (iii) A statistician commented that the normal di stribution given above for the times spent by female customers is not an appropriate mode l. What could be his reason for making this comment? [1] 7 Data is transmitted in bytes, where each byte c onsists of 8 bits. The probability of a bit being corrupted during its transmission is 0.03. A byte is considered ‘corrupted’ if it contains at least 2 corrupted bits. Assume that all bits are not corrupted prior to their transmission. (i) Show that the probability that a randomly chosen byte is corrupted during its transmission is 0.0223. [1] (ii) Given that a randomly chosen byte is no t corrupted during its transmission, find the probability that it contains no corrupted bits. [3] (iii) Using a suitable approximation, find the proba bility that between 5 and 10 bytes are corrupted during the transmission of 100 bytes. State the parameter(s) of the distribution you use. [3]

5 DHS 2015 Year 6 H2 Mathematics Preliminary Examination [Turn over 8 A family of seven, consisting of two sisters, th ree brothers and their parents, is queuing up in a line outside a restaurant. Find the number of ways in which the family can queue up if (i) one parent is standing at the fr ont and one parent is standing at the back of the queue, [1] (ii) the person standing in between the pa rents is one of the brothers. [3] Upon entering the restaurant, the fam ily is seated at a round table. (iii) Find the number of ways they can