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, the perpendicular unit vectors i , j and k are parallel to the edges OF , OD and OB respectively. The lengths of OF , OD and OB are 2h units, 3 units and h units respectively. (i) Show that 3 hOC   j k . [1] (ii) The point P divides the segment BC in the ratio 2:1 . Find OP  in terms of h . [1] (iii) A vector parallel to the normal of the plane BCEF is given as abik . By the use of a scalar product, find the value of a b . Hence find the Cartesian equation of the plane BCEF in terms of h . [4] (iv) Take 3h . Find the shortest distance from the point  1, 2, 2Q to the plane OPF. [4] 8. Mac has a 2400 000 m farm and on his farm, an area of 260 000 m is covered in weeds in June, and in September, the area increases to 269 500 m . The growth of weeds is such that the area covered in weeds increases at a monthly rate directly proportional to its area. At the same time, Mac does weeding at a constant rate of 240 0 0m per month. Let the area of the farm covered in weeds at time t (in months) be 2mA . (i) By considering a differential equation, show that ektA    , where  , k and  are constants to be determined. [5] (ii) The region covered in weeds is in the shap e of a circle. Find the monthly rate at which the radius of the region changes when the radius is 200 m . [2] (iii) Mac understands that having some weeds on the farm can be beneficial. Find the monthly rate at which Mac needs to do weeding if d 0d A t  in September. [2] (iv) Comment on the significance of d 0d A t  in the context of this question. [1] O j i k D C E F B

6   9. A researcher conducted a study on the radioisotope, Iodine-131 (I-131) which has a half- life of 8 days (i.e., the amount of I-131 is halved every 8 days). He first introduced 1000 mg of I-131 in an empty Petri dish on Day 1 and tracked the amount of I-131 in the dish. (i) State the amount of I-131 in the dish at the end of 16 days. [1] After every 16 days, i.e., on Day 17, Day 33, Day 49 etc., the researcher added 1000 mg of I-131 to the dish. (ii) Find the amount of I-131, to the nearest mg, in the dish immediately after 1000 mg of I-131 was added on Day 49. [3] (iii) Show that the amount of I-131 in the dish will never exceed 1334 mg. [2] The researcher discovered that he accidentally used a different radioisotope, Iodine-125 (I-125) on Day 1, which has a half-life of 60 days instead. He checked that he had indeed used the correct I-131 on other occasions. (iv) Find the total amount of radioisotopes I-125 and I-131 in the dish on Day 121, giving your answer correct to the nearest mg. [4] 10. (a) The equation 32 24 i 0za z a z  , where a is a constant, has a root i . (i) Briefly explain why i* may not necessarily be a root of the equation. [1] (ii) Show that 2ia . [2] (iii) Hence, find the remaining roots of the equation in exact form. [5] (b) The complex number z satisfies the equations *1 i 2z   and arg 2i 4z  . By considering izxy , find z . [5]

7   [Turn over 11. (a) The graph of  fy x is shown in the diagram above. It has asymptotes 1x and 2y . The points A , B , C and D have coordinates  0,1 ,  2, 0 , 4, 3 and  1, 0 respectively, with C and D being stationary points. On separate diagrams, sketch the graphs of (i)  f2 1yx , [3] (ii)  f'y x , [3] (iii) f.y x [3] In each case, state the coordinates of A, B , C and D whenever applicable, and the equations of any asymptotes. (b) A curve G with equation 2 2x ay x b   , where a and b are constants, has a stationary point at 14, 4  and a vertical asymptote 2x . (i) Find the values of a and b . [2] (ii) Find the range of values of x for which G is increasing and is concave downwards. [2] (iii) By sketching a suitable line on the same diagram as G, find the number of distinct real roots of the equation 32521 4 7 0xx x   . [3] x y O