LUNG CHEUNG GOVERNMENT SECONDARY SCHOOL Mock Examination 2006 / 2007 Mathematics and Statistics

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1 S.7 LUNG CHEUNG GOVERNMENT SECONDARY SCHOOL Mock Examination 006 / 007 Mathematics and Statistics Maximum Mark: 100 Date: Time: This paper consists of Section A and Section B.. Answer ALL questions in Section A and any FOUR questions in Section B. 3. Unless otherwise specified, all working must be clearly shown. 4. Unless otherwise specified, numerical answers should be either exact or given to 4 decimal places. SECTION A (40 marks) Answer ALL questions in this section. 3 + x Ax + B C 1. Given that +, where A, B and C are constants. (1 + x )(1 + x) 1 + x 1 + x (a) Find the values of A, B and C. (b) (i) Using the result in (a), or otherwise, expand of x as far as the term in x 3. (1 + x 3 + x (ii) State the range of values of x for which the expansion of )(1 + x) (1 + x in ascending powers 3 + x )(1 + x) is valid.. The number of book titles x(t) that a small publisher produces per month, after t months of production, can be modelled by 0.1t 0.t x ( t) = 54e 56e +. (a) After how many months, correct to the nearest integer, will the production first exceed 11 titles per month? (b) On average, each title will bring revenue of $60,000 to the publisher. The running cost is $100,000 and the production cost is $10,000 per title. The monthly profit of the publisher is $P(t) after t months of production. (i) Express P(t) in terms of t. (ii) The publisher will cease producing any book titles when the monthly profit falls below $150,000. After how many months, correct to the nearest integer, will the publisher cease producing any book titles? Explain your answer briefly. S.7 Mathematics and Statistics p.1 of 7

2 3. The rate of change of a certain bacteria (in millions) in a sample can be modelled by t f ( t) = e, where t ( 0) is the time measured in seconds. (a) Use the trapezoidal rule with 4 sub-intervals, estimate the total amount of bacteria in the sample from t = 0 to t = 0.1. d f ( t) (b) Find. dt (c) Determine whether the estimate in (a) is an over-estimate or under-estimate. 4. The following back-to-back stem and leaf diagram show the performance of two classes in an examination. Class A (Leaf units) Stem (tens) Class B (Leaf units) (a) Find the median and interquartile range of class A. (b) Draw two box-and-whisker diagrams to compare the result of two classes. (c) Determine which class has better performance in the examination. Explain your answer briefly. 5. Given that a is a real number such that 0 < a < 1. A population consists of seven number, 1,, 3, 4, 5, 6 and 7, whose probability distribution is as follows: Number Probability a A number is drawn from the population according to the above probability distribution. Define the following events: E: The outcome is an odd number. F: The outcome is a multiple of 3. (a) Find the value of a. (b) (i) Find P(E), P(F) and P(E F). (ii) Are the two events E and F independent? Explain your answer. (c) If two numbers are drawn randomly with replacement from the population, find the probability that at least one of the numbers is odd or a multiple of 3. S.7 Mathematics and Statistics p. of 7

3 6. Records show that among all car theft cases in Hong Kong, 30% occurred on Hong Kong Island, 0% in Kowloon and 50% in the New Territories. The probabilities that a car stolen in these three districts can be recovered are respectively 0.45, 0.35 and 0.5. (a) Find the probability that a stolen car can be recovered. (b) Suppose that two cars were stolen and that the two cases were independent of each other. (i) Find the probability that the two cases occurred in the same district. (ii) Given that the two cases occurred in the same district, find the probability that both cars can be recovered. SECTION B (60 marks) Answer any FOUR questions in this section. Each question carries 15 marks. 7. Define ax + b x 1 f ( x) = for all x c and g ( x) = for all x. x + c x + Let C 1 and C be the curves y = f (x) and y = g( x) respectively. It is given that C 1 passes through the origin and has a common vertical asymptote with C. The horizontal asymptote of C 1 is y = 4. (a) Find the values of a, b and c. (4 marks) (b) (i) Show that g(x) is an increasing function for x?. (ii) Sketch the graphs of C 1 and C on the same diagram. Indicate the points of intersection, intercepts and asymptotes of the two curves. (c) Find the area bounded by curves C 1, C and the line x = 1. (4 marks) 8. A toy factory discharges wastes into a river. The concentration y (in mg/l) of chemical Q in the river is given by y x = kxe, for x 0, where x is the number of hours after discharging wastes and k is a positive constant. dy d y (a) (i) Find and in terms of k. dx dx (ii) When will the concentration of chemical Q in the river be the maximum? (iii) Find the value of k if the maximum concentration of chemical Q is 80 mg/l. (9 marks) (b) A policy is approved to control the amount of chemical Q discharged in the river. A factory will be penalized with a $M fine for discharging y mg/l of chemical Q in the river, where M = 100(y 0), for y > 0. (i) Show that M is an increasing function for y > 0. (ii) Find the maximum penalty for a factory if k = 10. Give your answer correct to the nearest dollar. (iii) Find the range of values of k so that a factory will not be penalized. S.7 Mathematics and Statistics p.3 of 7

4 9. The director of a newspaper finds that the sale is not good. Therefore he decides to reduce the price to $1. From the past experience the sale N(t) (in ten thousands) each day on the t-week 10 after the reduction can be modeled by N ( t) =. 1 bt + ae In the last reduction of price the following data is obtained t N(t) (a) Use the table and the graph paper on page 7 to estimate the values of a and b graphically. Correct your answers to 1 decimal place. (5 marks) (b) As time passes, the sale will approach a number M. Find the value of M. (1 mark) (c) The price of the advertisement will be raised at the time when the rate of increasing of the sale is greatest. Using the values obtained in (a) and (b) to determine at what time should the price of advertisement be raised. (d) If the sale increases to 95% of M, the price of the newspaper will be raised to $5 again. Determine when the price of the newspaper will be recovered. Correct your answer to 1 decimal place. ( marks) 10. The number of customers of a shop in the morning follows a Poisson distribution with mean 18, while the number of customers of the shop in the afternoon follows a Poisson distribution with mean 16. (a) Find the probability that (i) the number of customers in the morning is between 16 and 18 inclusive, (ii) there are 16 customers in the morning and 18 customers in the afternoon, (iii) there are 34 customers in one day and at least 16 customers in the morning and afternoon respectively. (8 marks) (b) The amount that each customer spends in the shop follows a normal distribution with mean $5 and standard deviation $4. Find the probability that a customer spends between $0 and $30 in the shop. (3 marks) (c) Find the probability that there are 34 customers in one day, at least 16 customers in the morning and afternoon respectively, and at least 3 customers spent between $0 and $30. (4 marks) S.7 Mathematics and Statistics p.4 of 7

5 11. An organization has a car park of 10 parking spaces for its employees and there are 1 employees own a car and on a weekday (Monday to Saturday) each of them has a probability of 0.6 driving to the office. (a) On a weekday, find the mean number of cars in the car park. ( marks) (b) On a weekday, find the probability that the car park is full. (3 marks) (c) Find the probability that the car park is full in at least days in a week. (3 marks) (d) Today the car park is full. How many days after today would you expect before the car park will be full again? ( marks) (e) A manager has just obtained his driving license and buy a new car and he has a probability of 0.8 driving to the office. (i) Find the probability that on a weekday there are seven cars in the car park. (ii) On a weekday, it is found that there are seven cars in the car park. Find the probability that there does not contain the manager s new car. (5 marks) 1. A bus from FIRST BUS runs at regular interval between Central and Mongkok daily. The delay time, which is the difference between the actual and the scheduled arrival times at either terminus, following a normal distribution with a mean µ minutes and a standard deviation σ minutes. A negative delay time denotes an early arrival. (a) Records show that the bus arrives early with a probability of 0.1 and that it arrives at least 15 minutes late with a probability of Find the values of µ and σ. (b) A captain at the terminus finds that a bus arrives early at a certain moment. (i) How many buses would he expect to observe before the next bus makes an early arrival. (ii) Find the probability that the 11th bus arrives is the second one giving an early arrival after that moment. (iii) It is known that a bus was late. Find the probability that it gave more than 15 minutes late. (8 marks) (c) The CITYBUS company claims that the survey of the same statistic giving the values of µ and σ are both less than FIRST BUS company. Base on these data, can we conclude that CITYBUS offer a faster travelling. (1 mark) END OF PAPER S.7 Mathematics and Statistics p.5 of 7

6 Table: Area under the Standard Normal Curve z Note: An entry in the table is the proportion of the area under the entire curve which is between z = 0 and a positive value of z. Areas for negative values of z are obtained by symmetry. A(z) A( z) = z 0 1 e π x dx 0 z S.7 Mathematics and Statistics p.6 of 7

7 Name: Class: Class No.: 9. Fill in the details above and tie this sheet INSIDE your answer book. 10 ln - 1 Ł N( t ) ł t 1 S.7 Mathematics and Statistics p.7 of 7