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.