2017 HCI H1 Maths Prelims Questions

3   H1 Mathematics 2017 Preliminary Exam Paper Question Section A : Pure Mathematics [40 Marks] 1. Whole Food Grocer was having sales and some food items were on of fer. Organic feed eggs were having a 15% discount. There was also a $1 discount for every 2 packets of chia seeds purchased. There was no promotion for organic quinoa. The table below shows the total bills and the number of packets of orga nic quinoa, organic feed eggs and chia seeds Stephanie, Weiwei and Leo bought from Whole Food Grocer . Calculate the original selling price for one packet of each of the 3 food items, giving your answers correct to the nearest cent. Quinoa Eggs Chia seeds Total Bill ($) Stephanie 3 1 2 71.28 Weiwei 2 2 5 91.85 Leo 6 3 3 144.43 [4] 2. Differentiate the following with respect to x. (a)  2 lnx x , [ 2 ] (b) 1 2e. x    [ 2 ] 3. On the same diagram, sketch the graphs of 1 23 xy x   and 2 3yx x  , indicating clearly the equation(s) of any asymptote(s) and any point(s) of inte rsection. Hence solve the inequality 21 3.23 x xxx    [5] 4. (i) Express 221 4 x x   in the form 4 CAx B x  , where A, B and C are constants to be determined. [2] (ii) Hence, without the use of a calculator, find 26 5 21 d4 x xx   in exact form. [3]

4   5. (i) Sketch the curve C with equation   31yx x ,  stating clearly the coordinates of the turning point and the axial intercepts. [2] (ii) The line yx k intersects C twice. Find the set of values of k. [3] (iii) Without the use of a calculator, find the area of the region bounded by C and the line 5yx .  [5] 6. (i) Kim wants to fence up a vegetable plot in his backyard. The ve getable plot to be fenced up will occupy a rectangle of 2x m by y m together with half of a regular hexagon with sides of x m each, as shown in the diagra m below. It is given that the area of the vegetable plot is 15 m2. Show that the perimeter 15 3 35. 4Px x x Find, using differentiation, the values of x and y such that P is minimum. [7] Two companies provide the cost for the fencing. Company A $90 per metre or part thereof * Company B $95 per metre for the first 10 metre $84 for the subsequent metre or part thereof * For example, it costs $180 to bu ild a fence of 1.2m using Company A (ii) Find the range of the length of fencing to be built such that it is cheaper to engage Company B. [3] y xx x 2x y

5   (iii) Hence conclude which company Kim shoul d engage to fence his backyard when P is minimum. [2] Section B: Statistics [60 marks] 7. In an IT department, a staff is tasked to form 7-letter codes (need not be valid words) using the given word ‘SPECIAL’. Find the number of codes that can be formed if (a) there are no restrictions ex cept the code ‘SPECIAL’ cannot be formed, [2] (b) all the 3 vowels cannot be together, [2] (c) the first and the last letters are consonants. [2] 8. A company uses 2 production lines, A and B, to produce lunch boxes. If the lunch box cannot be closed tightly, it will be considered as fa ulty. Of all the lunch boxes produced, 5% are faulty and 3% of the lunch boxes produced by B are faulty. Among the lunch boxes that are faulty, 60% of them are produced by line A. One lunch box is selected at random. (i) Show that the probability that it is produced by line B is 2 3 . [2] (ii) Find the probability that it is fa ulty given that it is produced by A. [2] Two lunch boxes are chosen at random. (iii) Find the probability that both lunch boxes are produced by B given that exactly one is faulty. [3] 9. The probability of a diner choosing a burger during his visit to Cheeky Chick Café is 0.05. Among the diners who visited Cheeky Chick Café, 20 diners are randomly chosen. (i) Find the probability that more than 3 diners choose a burger. [2] (ii) Find the smallest value of n such that there is more than 90% chance of less than n diners choosing a burger. [3] (iii) The probability of a diner buyi ng a drink in the cafe is p , where 0.5p . Given that the variance of a diner buying a drink is 4.55, find the value of p. [2]

6   10. The accountant of a company monitors the number of items produced per month by the company, x (in thousands), together with the total cost of production, $ y (in thousands). The following table shows 12 sets of data collected for a random sample of 12 months. Number of items produced (x) 21 39 48 24 72 75 15 35 62 81 12 56 Production cost ($y) 40 58 67 45 89 96 37 53 83 102 35 75 (i) Draw a scatter diagram to illustrate the data. [2] (ii) Find the product moment correlation coefficient and comment on its value in the context of the data. [2] (iii) Find x and y , and mark the point  x ,y on your scatter diagram drawn in part (i). [2] (iv) Find an equation for the regression line of y on x in the form ym x c , giving the values of m and c, correct to 2 decimal places. Sketch this line on your scatter diagram. Interpret the meanings of m and c in this context. [3] (v) Use the equation of your regression line to calculate an estimate for the production cost of 70 thousand items. Comment on the reliability of your estimate. [2] (vi) The selling price of each item produced is $2.20. Find the minimum number of items to be produced per month at which the company does not suffer a loss. [2]

7   11. The weight of a packet of Calhwa potato ch ips is known to have a m ean of 84 grams and standard deviation 5 grams. The manufacturer claims that the average weight of a packet of potato chips is at least 84 grams. To test th is claim, a random samp le of 100 such packets of potato chips are selected and tested. The average weight of the 100 packets of potato chips in the sample is 82.9 grams. (i) State appropriate hypotheses for the test, defining any symbols you use. [2] (ii) Test, at the 1% significance le vel, whether the manufacturer’s claim is valid. [3] (iii) State what you understand by the expression ‘at the 1% significance level’ in part (ii). [1] (iv) State, giving a reason, whether it is necessary for the weight of the packets of potato chips produced by a manufacturer to follow a normal distribution for the test in part (ii) to be valid. [1] (v) Another random sample of 100 packets of potato chips from another batch gives an average weight of t grams. Find the range of values of t such that there is enough evidence to conclude that the aver age weight of the packets of potato chips has changed at the 5% level of significance. [5] 12. (a) The continuous random variable X has the distribution   2N ,. It is given that P 15 0.841X  and  P 9 15 0.682X  , find the values of  and  .