HCI_8865_2022_Prelim

HCI_H1Mathematics_Prelim Paper 1 Determine the set of values of k such that the equation 2 30x kx k− + + = has real roots. [4] 2 (a) Find ( ) 33 d xx x x −  . [2] (b) (i) Differentiate 3 26ln 3 4x + with respect to x, simplifying your answer. [3] (ii) Hence find 2 d .34 x xx + [2] 3 A window at a grand church has the shape of a semicircle AEB joined to a regular trapezium ABCD of height x m, and a rectangle DCGF as shown in the diagram. It is given that 2 mAB x= , 4DC x= m and the angle ADC and the angle BCD are both equal to o45 . The total perimeter AEBCGFD of the window is 10 m. (i) Show that 5 2 22DF x = − + + m. [2] (ii) Hence use a non-calculator method to find the exact value of x such that the area of the window is maximum. [5] G F C D B A E 4x m 2x m

2 © HCI 2022 [Turn over 4 The curve C has equation 33e xy . (i) Sketch the graph of C, stating the exact coordinates of the points of intersections with the axes and the equation of the asymptote. [3] (ii) Find the equation of the tangent to C at the point where the curve C meets the x-axis, giving your answer in the form ln 3y ax b=+ , where a and b are integers to be determined. [3] (iii) Hence find the exact value of the area enclosed by the curve C, the y-axis and the tangent at the point where the curve meets the x-axis [4] O x y

3 © HCI 2022 [Turn over 5 A company sells three types of toys: boats, cars and planes. • The selling price of a toy boat is twice as much as that of a toy car. • The total selling price of 9 toy boats, 5 toy cars and 3 toy planes is $335. • The total selling price of 3 toy boats and 4 toy cars is $38 more than the total selling price of a toy boat and 2 toy planes. (i) Find the selling price of a toy plane. [3] The company decides to trial the production of a new type of toy: trains. Based on market research, the manager estimates that when the toy trains are priced at p dollars apiece, the number sold each month, D thousands, can be modelled by 0.083e pDa −=+ where a is a constant. When the toy trains were given out for free to spark public interest, the demand for the toy was 20 thousand units. (ii) Find the exact value of a. [1] (iii) Sketch the graph of D against p, stating the coordinates of any point(s) of intersection with the axes and the equation(s) of any asymptote(s). [2] (iv) Find d d D p