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