ACJC 2024 Prelim H1 Final

Section A: Pure Mathematics [40 marks] 1 A company sells three types of cars: gas-powered, electric and hybrid. The total selling price of 4 electric cars is $172 800 more than the total selling price of 3 gas -powered cars. The selling price of a hybrid car is 1.2 times that of a gas -powered car. The company sold 3 hybrid cars, 2 gas-powered cars and 2 electric cars for $716 880. Find the selling price of an electric car. [3] 2 Find the set of values of k for which 2 2( 36) ( 6) 0.5 0    k x k x for all real values of x. [4] Hence find the set of values of k for which the expression 2 2ln[( 36) ( 6) 0.5)]k x k x    is defined for all real values of x. [1] 3 (a) Find   425 4 d x x x   . [2] (b) Express 4 3 in the form , where and are integers.4 4 x B A A Bx x    Hence find 4 3 d ,4 x xx   where 4x . [3] 4 The curve C has equation 5ln( 1) 2 2y x x     . (i) Show that curve C has no stationary points. [3] (ii) Sketch the graph of C, stating the x-coordinates of any points of intersection with the x-axes and the equation(s) of any asymptotes. [3] (iii) Find the numerical value of the area bounded by C, the x-axis and the lines x = 3 and x = 5. [1]

2 ANGLO-CHINESE JUNIOR COLLEGE 2024 H1 MATHEMATICS 8865/01 5 The diagram shows the curve C with equation 2 2 ,y k x  where k is a positive constant. The x- coordinate of A is positive and the y-coordinate is 23 4 k . (i) Find the equation of the tangent to C at the point A. Give your answer in the form y mx c  , where m and c are in terms of k. [4] (ii) Find the area bounded by the curve C, the tangent to the curve at point A and the x-axis, giving your answer in the form 3,p kq where p and q are integers. [4] 6 Mr Lim has just set up a company selling handphones. Over a period of 9 months , his total profit P thousand dollars per month can be modelled by 0.2 2(e 4) 9,tP   where t is the time in months after he set up his company. (i) Use differentiation to find the value of t which gives a stationary point on the graph of P against t. Justify whether this point is a maximum or minimum point. [4] (ii) Sketch the graph of P against t for 0 9 t  . [1] (iii) Using integration , find 9 0.2 2 0 (e 4) 9 d ,t t   correct to the nearest integer , showing your workings clearly. Explain, in the context of the question, what this value represent. [4] O C x y A

3 ANGLO-CHINESE JUNIOR COLLEGE 2024 H1 MATHEMATICS 8865/01 (iv) Mr Lim has a budget for advertising his handphones for sale. At any time, t months, measured from the start of his adve