NJC 8865 2022 Prelim

* © NJC 2022 [Turn over NATIONAL JUNIOR COLLEGE SENIOR HIGH 2 Preliminary Examination NAME SUBJECT CLASS 2ma1 REGISTRATION NUMBER H1 MATHEMATICS 8865/01 12 September 2022 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name, class and registration number on the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use paper clips, glue or correction fluid. Answer all the questions. Write your answers in the spaces provided in the Question Paper. 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. Up to 2 marks may be deducted for improper presentation. The number of marks is given in the brackets [ ] at the end of each question or part question. Question Number Marks Possible Marks Obtained 1 4 2 4 3 5 4 5 5 10 6 12 7 5 8 6 9 8 10 9 11 10 12 10 13(a) 6 13(b) 6 Presentation Deduction –1 / –2 TOTAL 100 This document consists of 21 printed pages and 3 blank pages.

2 © NJC 2022 Section A: Pure Mathematics [40 marks] 1 Find the exact set of values of m such that the line 3y mx=− intersects the curve ( ) 2 12y mx m x= − − − . [4] 2 By using a suitable substitution , solve the inequality 23e e 70xx+ , giving your answer in logarithmic form. [4] 3 Tammy deposits $100 000 in three different savings plans for 2 years as shown in the table below: Savings Plan Returns I Simple interest of 1.4% per annum. II One time interest of 3.5% for a 2-year period. III Quarterly payout of $50 for every $10 000 invested. Investments are made in multiples of $10 000. At the end of 2 years, she received an interest of $3 560. The amount of money she deposits in savings plan II is twice the amount she deposits in savings plan I. (i) Find the amount of money she deposits in each savings plan. [3] For the amount of money deposited in savings plan II and III, Tammy was given an option to deposit the amount in savings plan IV, which involves a variable return, as shown below. Savings Plan Returns IV Daily interest rate that fluctuates between 3.7% to 3.9% for a period of 2 years. (ii) Explain if Tammy should take up savings plan IV, justifying your answer mathematically. [2] 4 Find the exact value of 24 1 13d xx  −  , expressing your answer in the form lna b c+ , where a, b and c are constants to be determined. [3] Hence find the value of p such that 24 31 1 13 1e d 3 d p x xx x + − =−   , leaving your answer correct to four decimal places. [2]

3 © NJC 2022 [Turn over 5 The curves C1 and C2 have equations 25 2 xy x −= − and 1e2 xy −=+ respectively. (i) Express 25 2 x x − − in the form p + 2 q x− , where p and q are integers to be determined. [1] (ii) On the same axes, sketch the graphs of C1 and C2, showing clearly the coordinates of any points of intersection between C1 and C2, axial intercepts and the equations of any asymptotes. [3] (iii) Find the exact area of the region bounded by the curves C1, C2 and the y-axis. [3] (iv) Without the use of a calculator, find the integers a and b, where y = xb a + is the equation of the tangent to C1 when x = 4. [2] (v) Hence, using the values of a and b found in part (iv), find the set of values of x such that 1e2 x xb a − ++ . [1] 6 Mark owns a factory that manufactures bottled drinks. He wants to model the production cost, C thousand dollars per month over a period of t months. He believes that 32 7 8 20, for 0 6C t t t t= − + +   . (i) Use differentiation to find the exact values of t which give stationary points on the graph of C against t. For each point, justify whether it is minimum or maximum. [5] (ii) Use your calculator to find the value of ( ) 6 32 0 7 8 20 dt t t t− + + . In the context of the question, what does this value represent? [2] Mark is also interested in modelling his monthly profit, P thousand dollars, which is related to C by the equation ( ) 0.25100 e 3.2CP −=+ . (iii) When t = 6, show that d d P C = kem, where k and m are integer to be found. [3] (iv) Hence, find the rate of decrease in profit per month when t = 6. [2] Section B: Probability and Statistics [60 marks]

4 © NJC 2022 7 In a forest, 30% of the plants have heights less than 0.83 m and 30% have heights greater than 2.41 m. The standard deviation of the height of the plants is 1.5 m. (i) State the mean height of the plants in the forest. Give a reason why a normal distribution, with this standard deviation, would not give a good approximation to the distribution of the height of the plants. [2] (ii) A random sample of 100 plants in the forest were taken and their heights measured. Find the probability that the mean height of the sample lies between 1.2 m and 1.6 m. [3] 8 A vehicle insurance company classifies the drivers it insures as class L, M and H according to whether they are of low risk, medium risk or high risk with regard to having an accident respectively. The company estimates that 30% of the drivers who are insured are class L and 50% are class M. The probability that a class L driver will have at least one accident in a year is 0.01, the corresponding probabilities for class M and class H are 0.03 and 0.06 respectively. (i) The company insures a driver and within a year, the driver had at least one accident. Show that the probability that the driver is of class H is 0.4. [3] (ii) Three drivers insured by the company are chosen randomly. Find the probability that all three drivers are of class H and exactly one of them has at least one accident in a year. [3] 9 A group of 12 students comprises of 5 girls and 7 boys. Among the 5 girls are Ann and Alice. (a) The 12 students are seated in a row to take a group photo. (i) Find the number of different s eating arrangements in which Ann and Alice are separated from each other. [2] (ii) Find the number of different seating arrangements in which there are exactly 3 boys between Ann and Alice. [3] (b) A teacher wants to form two teams of 5 students for an activity from the group of 5 girls and 7 boys. Ann is selected as the leader of one team and Alice is selected as the leader of the other team. Find the number of possible teams that can be formed with exactly one team comprising of 2 girls. [3] 10 A financial magazine publishes an annual ranking of the financial services companies in the world. The ranking is based on sales, profit, assets and market value. A random sample of 7 pairs of assets, US$ s billion and profit, US$ p billion is shown in the table below.

5 © NJC 2022 [Turn over Assets, s 4914.7 4301.7 2832.2 4159.9 394.5 491.9 326.7 Profit, p 65.8 39.3 17.9 31.3 2.6 3.4 2.0 (i) Give a sketch of the scatter diagram for the data, labelling the axes. [2] (ii) Find the equations of the least-squares regression line of p on s and s on p. Draw the least- squares regression line of p on s and plot ( s , p ) in the scatter diagram in part (i). [3] (iii) Find the product moment correlation coefficient between s and p and comment on its value in the context of the data. [2] (iv) Using an appropriate regression line from part (ii), calcu