HCI_H2_MATH_P1

3   [Turn over 1. (a) Use the substitution 21ux to find  121 dx xx . [3] (b) Find sin 3 sin dx xx . [2] 2. Given that  1fl n 1c o s xx x    , find  f0 ,  f' 0,  f' '0 and  f ''' 0 . Hence, write down the first four non-zero terms in the Maclaurin’s series for  f. x [6] Use your result to deduce the first four non-zero terms in the Maclaurin’s series for 1c o sln 1 x x    . [1] 3. A tunnel is built to facilitate the transpor tation of goods by trains between Country X and Country Y . Due to differences in the rail systems between the two countries, two types of tracks are used – the international track with track gauge of 1435 mm and the narrow track with track gauge of 1000 mm (in rail transport, track gauge is the spacing on a railway track). It is known that the cross-section of the tunnel is a half ellipse with centre O and width MN (see diagram). The maximum height of the tunnel is 2000 mm. To standardize the volumes of the goods to be transported, the areas ABCD and EFGH are made equal. Find the width of the tunnel MN , giving your answer to the nearest mm. [4] F 1000 mm 1435 mm A B E M N H G D C O 2000 mm

4   4. A group of student councillors bought an 8 m long piece of tarpaulin canvas to build a tentage for a school event. The canvas would extend diagonally at an angle of  from the ground to a height of 3 m, where it will then stretch horizontally to the school building (see diagram for the cross-sectional view). (i) Given that the total cross-sectional area covered by the canvas is 2 mA , show that 924 9cosec cot 2A    . [2] (ii) Find, by differentiation, the largest possible value of A . [5]   5. The function f is defined by  2 f : l n 2 2 , , 2 .xx x x    (i) By considering the graph of  f,y x give a reason why 1f does not exist. [2] The function h is defined by  h: f , , .xx x a a  (ii) Find the largest possible domain of h such that 1h exists. [1] (iii) Define 1h in a similar form. [3] (iv) Find the set of values of x which satisfies the equation    11hh h hx x  . [2]     6. A curve C has parametric equations es i ntx t , ec o styt . (i) Describe the shape of C as t . [2] (ii) Find the Cartesian equation of the normal to C at the point  es i n , ec o sP    , where 0  , giving your answer in the form ym x c  . [3] The normal to C at P meets the y -axis at the point D , and the curve C meets the positive x -axis at the point E that has integral coordinates. (iii) Find the coordinates of D and E . [3] (iv) Describe the locus of the mid-point of DE as  varies. [2] 3 m School Building tentage

5   [Turn over 7. The diagram shows a vehicle ramp OBCDEF with horizontal rectangular base ODEF and vertical rectangular face OBCD . Taking the point O as the origin, th