RVHS 8865 2022 Prelim

1 ©RIVER VALLEY HIGH SCHOOL 8865/01/2022 Section A: Pure Mathematics [40 marks] 1. Find the exact range of values of k for which ( ) 21 24k x x k− − − is negative for all values of x. [4] 2. (i) Differentiate ( ) 3 212 x− with respect to x. [1] (ii) Integrate 1 12 x− with respect to x. [2] (iii) Express 65x− in the form ( )12a x b−+ , where a and b are constants. Hence, using the results in parts (i) and (ii), find 65 d 12 x x x − − . [4] 3. The curve C has equation 2e84 x xy= − + . (i) Use differentiation to find the exact x-coordinates of the turning point(s) of C. [3] (ii) Sketch C, stating the coordinates of any points of intersection with the axes and the turning point(s). [2] (iii) By adding an appropriate graph on the same diagram in part (ii), solve 2e ln( ) 04 x x x− + − = . [3] 4. The curve C has equation 123 2e xy −=+ . (i) Sketch C, stating clearly the equation of any asymptotes and the exact coordinates of any points of intersection with the axes. [2] (ii) Find the exact equation of the tangent to C at 2x= , giving your answer in the form y mx c=+ . [4] (iii) Find the area of the region bounded by C, the tangent to C at 2x= and the y- axis. [2]

2 ©RIVER VALLEY HIGH SCHOOL 8865/01/2022 5. Mr Dodo is the manager of Todo Fishball Company which relies solely on man ual production of fishballs. Mr Dodo wants to keep track of the monthly production of fishballs in the fiscal year 2021. The rate, x kg per month, at which fishballs are produced and the time, m months, can be modelled by the equation 3230 585 1980 12000x m m m= − + + , for 0 12m . (i) Using differentiation, determine the stationary points of the graph of the above equation. State the nature of the stationary points. [5] (ii) Sketch the graph of x against m, stating clearly the coordinates of the turning points and axial intercepts (if any). [2] (iii) Find the area of the region bounded by the above graph , the x-axis, and the lines 0m= and 12m= . Give an interpretation of the value of this area. [3] To improve the efficiency of the company, Mr Dodo implemented a compulsory upgrading programme for all the employees. Each employee will be able to produce y kg of fishball on the dth day after the completion of the programme . The model of the production by each employee is 0.240 5e dy −=− . (iv) Determine the amount of fishball produced by an employee immediately after the training programme. [1] (v) Sketch the graph of y against d and comment on its suitability. [2] Section B: Statistics [60 marks] 6. In a darts shooting competition, each competitor throws 2 darts onto a target board. The probability of a randomly chosen competitor hitting the bull’s-eye with his first throw is 0.8. The probability of hitting the bull’s-eye on his second throw is • 0.9 if the first throw hits the bull’s-eye, • the same as the probability of the first throw if the first throw does not hit the bull’s-eye. (i) Draw a tree diagram to represent the above information. [2] (ii) Find the probability that a randomly chosen competitor hits the bull’s -eye in his second throw. [1] (iii) If a randomly chosen competitor hits the bull’s -eye on his second throw, find the probability that he hits the bull’s eye on his first throw. [3]

3 ©RIVER VALLEY HIGH SCHOOL 8865/01/2022 7. A 7-letter codeword is formed using the letters A, U, X, Y, Z. Repetition of letters are allowed. Find the number of possible codewords if (i) there are no restrictions, [1] (ii) letter Z occurs only once in the centre. [1] The codeword is of the form AUXY***, such that * is any of the letters A, U, X, Y, Z. (iii) Show that if *** is in alphabetical order and consists of exactly two identical letters, the number of possible codewords is 20. [1] (iv) Hence find the number of possible codewords if *** is in alphabetical order. [3] 8. In a particular region, a person is either left -handed or right-handed. On average, 8% of the people are left-handed. A school in the region has 50 classes, each with 30 students. (i) State, in the context of the question, one assumption needed for the number of left- handed students in the school to be well modelled by a binomial distribution. [1] Assume now that the number of left-handed students in the school follows a binomial distribution. (ii) State the expected number of students who are right-handed in the school. [2] (iii) Find the probability that there are at least 4 left-handed students in a randomly chosen class. [3] The school has 4 lecture theatres each with a maximum capacity of 250. Following feedback from students, the school now wants to include chairs with swivel board s that are custom-made for left-handed students in the 4 lecture theatres. (iv) Determine the minimum number of such chairs needed for a randomly chosen lecture theatre in order to be 90% certain of meeting the needs of left -handed students. [3]

4 ©RIVER VALLEY HIGH SCHOOL 8865/01/2022 9. An institution offers classes in three different subjects: Sciences, Arts and Business. The number of students studying different combinations of these subjects are shown in the Venn diagram below. All students study at least one subject. The number of students who study all three subjects is x, and the number of students who study Arts and Business but not Science is y. One of the students is chosen at random. S, A and B are events defined as follows. • S is the event that the student studies Science. • A is the event that the student studies Arts. • B is the event that the student studies Business. A and B are not mutually exclusive. Write down a relationship satisfied by x and y. [1] A and S are independent, find an expression for y in terms of x. [3] It is further given that ( ) 3P| 8BS = . (i) Find the value of x and y. [2] (ii) Find ( )( )P' S A B . [1] 10. The ages x (in months) and the resale prices $ y of a random sample of 9 bicycles of a certain model are given in the following table. Age (x) 11 17 24 33 37 43 48 57 63 Price (y) 108 102 88 73 74 63 65 50 43 (i) Sketch a scatter diagram of the data. [2] (ii) Determine the product moment correlation coefficient and comment on its value in the context of the question. [2] (iii) Find the equation of the regression line of y on x, in the form y mx c=+ , leaving the values of m and c correct to 2 decimal places. Interpret the value of c in the context of the question. Sketch the regression line on the scatter diagram drawn in part (i). [3] (iv) Estimate the price of a 72-month old bicycle and comment on the reliability of this estimate. [3] Science Arts Business 15 5 5 20 10 x y

5 ©RIVER VALLEY HIGH SCHOOL 8865/01/2022 11. A mobile application states that the mean time taken to travel from Town A to Park B is 14.5 minutes. A data analyst would like to test if the mean travel time is more than 14.5 minutes. The data analyst gathers a random sample of 35 travel times of various people. The travel times, x minutes, are summarized below. 2539 8647.4xx== (i) Calculate the unbiased estimates of the population mean and variance of the travel time from Town A to Park B. [2] (ii) Test, at the 5% level of significance, whether the mean travel time from Town A to Park B is more than 14.5 minutes. You should state your hypotheses clearly and define any symbols used. [4] (iii) Explain, in the context of the question, the meaning of ‘at the 5% level of significance’. [1] It is now known that the variance of travel times from Town A to Park B is 9.6 minutes2. A rival mobile application claims that the data collected is incorrect and hence proceeds to collect a new set of travel times from Town A to Park B from a random sample of 100 peopl