H2 Math Paper 2 Question

Name:____________________________________________ Class:____________ JURONG JUNIOR COLLEGE Preliminary Examination MATHEMATICS 9740/02 Higher 2 11 September 2012 3 hours Additional materials: Answer Paper List of Formulae (MF15) Cover Page READ THESE INSTRUCTIONS FIRST Write your name and civics 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, fasten all your work securely together, with the cover page in front. This document consists of 6 printed pages. [Turn over

2 Section A: Pure Mathematics [40 marks] 1 An offshore oil well is located in the ocean at point W, which is 8 km from point A on land as shown in the figure above. Oil is to be piped to point B, which is 12 km from point A, using a straight pipe laid underwater from W to some point P between A and B, and then on to point B via another straight pipe laid on land. If the cost of laying a pipe is $60 000 per km underwater and $45 000 per km on land, how far should point P lie from point A to minimise the cost of laying the pipe? [5] 2 In the theory of learning, the amount of material memorised by a person, y units, in time t minutes satisfies the differential equation t y d d = k(10 – y), where k is a positive constant. Initially, zero units of material were memorised. (i) Find y in terms of t, given that five units of material were memorised in two minutes. [6] (ii) Sketch the graph of y against t and explain what will eventually happen to the amount of material memorised. [3] (iii) The rate at which material is forgotten is known to be proportional to the amount memorised in time t. Write down a differential equation involving y when forgetfulness is taken into account. [2] 8 km Land W P B A Ocean 12 km

3 [Turn over 3 The complex number z satisfies the relations 2i 4z− < and 46 i 2 izz+ −≤ − . (i) Illustrate both of these relations on a single Argand diagram. [3] (ii) Find the range of values of arg z. [3] Another complex number w is such that ( ) ( )arg 2 4i arg 1 iw− −= − − . (iii) Sketch the locus of the points representing w on the same Argand diagram in part (i) and hence find the exact least value of zw− . [5] 4 The planes 1Π and 2Π have equations ( )24− +=ir i j k and 3( ) ( 4 ) λ μ=− ++− + ++ +ri j ki j ki j respectively, and meet in a line l1. (i) Find the acute angle between 1Π and 2Π . [3] (ii) Find a vector equation of l1. [2] (iii) The points A and B have coordinates ( )6, 3, 5− and ( )2, 3, 1 respectively. Find the length of projection of AB JJJG onto the line l1. [2] The line 2l passes through the point C with position vector (2 1) 3pp+ +−i j k and is parallel to 33qq−+i j k , where p and q are positive constants. Given that the perpendicular distance from C to 1Π is 15 6 and that the acute angle between 2l and 1Π is 1 2sin 6 − ⎛⎞ ⎜⎟⎝⎠ , find the values of p and q. [6] Section B: Statistics [60 marks] 5 A company has 1500 employees. 35% of employees are in the 21-40 age group, 50% are in the 41-60 age group, and the rest are in the 60 and above age group. A sample of 50 employees is to be chosen to take part in a survey on staff morale. A list of the employees in alphabetical order by name is generated. A random number k from 1 to 30 is chosen. The kth employee on the list is chosen and thereafter, every 30 th employee on the list will be chosen until a total of 50 employees are chosen. Give a disadvantage of this sampling method. [1] State the name of a method of sampling that would not have this disadvantage, and describe how such a sample could be carried out. [3]

4 6 Find the number of ways in which 4-letter code-words can be obtained from the word ENDANGERED if (i) there are no repeated letters, [1] (ii) there are three “E”s, [2] (iii) there is at least one repeated letter. [3] 7 A tennis match is played between two players, A and B. The match consists of at most three sets. Each set is won by either A or B, and the match is won by the first player to win two sets. The probability that A wins the first set is p. For each set after the first, the conditional probability that A wins the set, given that A won the preceding set, is 0.7. For each set after the first, the conditional probability that B wins the set, given that B won the preceding set, is 0.6. Construct a probability tree showing this information. [2] (i) When a match is decided by playin g all three sets, the probability of A winning is the same as that of B winning. Show that the value of p is 8 11. [2] (ii) Find the probability that B wins the first set, given that A wins the match. [3] 8 Fruits are sold by weight. Apples are sold at $2 per kg and oranges at $3 per kg. The masses, in kg, of apples and oranges are modelled as having independent normal distributions with means and standard deviations as shown in the table. Mean mass Standard deviation Apples 0.2 0.05 Oranges 0.3 0.08 (i) Find the probability that the difference between the weight of 3 apples and 1 orange is less than 0.2 kg. [3] (ii) Find the probability that the total selling price of 5 apples and 5 oranges exceeds $5. [3] (iii) Find the least value of n such that there is a probability of less than 0.3 that the mean weight of n randomly chosen apples is greater than 0.21 kg. [3]

5 [Turn over 9 To maintain the good condition of library books, any book with more than 7 defects will be donated away. The mean number of defects per book in the reference section is 3. The number of defects in a reference book is the random variable R. (i) State, in the context of this question, two assumptions needed to model R by a Poisson distribution. [2] (ii) Find the probability that a reference book is donated away. [1] (iii) There are 1000 reference books in the library. Use an appropriate approximation to find the probability that at least 10 but no mo re than 15 reference bo oks will be donated away. [3] The mean number of defects per book in the children’s section is λ. (iv) Given that λ = 5, find the probability th