ACJC 2023 JC1 H1 CA1

1 Name: ______________________________ Tutorial Class: _________________ ANGLO-CHINESE JUNIOR COLLEGE MATHEMATICS DEPARTMENT MATHEMATICS Higher 1 2 May 2023 JC1 CA1(25 Marks) Time allowed: 45 mins 1 Find the set of values of k for which 2 1x x k+ − + is more than 2 for all real values of x. [3] 2 Given that 1, 9log 2=x y and 52log 5, =xy find the value of log . x [4] 3 A piece of metal is heated and then allowed to cool in a room with constant temperature. Its temperature C,oT when it has cooled for time t minutes, is modelled by the formula 24 72(0.9) .=+ tT (i) Find the initial temperature at which the cooling starts. [1] (ii) Explain what happens to the temperature for large values of t. [2] (iii) Sketch the graph of T against t, stating the coordinates of any points of intersections with the axes and the equations of any asymptotes. [2] (iv) Find the time taken for the temperature of the metal to reach 28 C.o [1] 8865

2 4 (i) Sketch the graph of 1 ln ,=+yx stating the coordinates of any points of intersections with the axes and the equations of any asymptotes. [2] (ii) By adding a suitable curve to your sketch in part (i), solve 11ln 10 −= − xx x [4]

3 5 A company produces three types of sports shoes: Runners, Joggers and Walkers. It is given that the manufacturing cost of a pair of Runners, Joggers and Walkers is $ R, $J and $W respectively. The total manufacturing cost for 13 pairs of Runners, 18 pairs of Joggers and 40 pairs of walkers is given by 13 18 40R J W++ = $2830. Given that the manufacturing cost of a pair of Runners, Joggers and Walkers is reduced by 20%, 25% and 10% respectively. With this reduction, the total manufacturing cost for 18 pairs of Runners, 30 pairs of Joggers and 40 pairs of Walkers is $2934. It is also given that 10 4 165WR=+ . (i) Find the manufacturing cost of a pair of Walkers before the manufacturing cost was reduced. [2] Explain in the context of the question what does the equation 10 4 165WR=+ represents. [1] The company wants to increase their profit by in vesting in a new type of sports shoes, Sprints. An economist predicts that the profit $P when x pairs of Sprint shoes are produced and sold can be modelled by 2 70 210 xPx=− + − . (ii) Find the maximum profit the company can earn from this i nvestment and the corresponding number of pairs of shoes the company needs to manufacture. [2] (iii) Find the maximum number of pair of shoes the company