HCI JC 2 H2 Maths 2012 Paper 1 Questions

HWA CHONG INSTITUTION 2012 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS 9740/01 Paper 1 12 September 2012 3 hours Additional Materials: Answer Paper List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your Centre number, index number, name and class on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notatio ns and not calculator commands. You are reminded of the need for clear presentation in your answers. At the end of the examination, place the completed cover page on top of your answer scripts and fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 6 printed pages.

[Turn over 2 1. The function f is defined for 2x by 2 53f: 2 kx xx x   , where k is a real constant. Find the range of values of k such that the function f is an increasing function for all real values of x, 2x . [5] 2. The point A represents a fixed complex number a such that arg 02 a   . The complex numbers i a and iaa are represented by the points B and C respectively. On a clearly labelled Argand diagram, show the points A, B, C and the set of points representing the complex numbers z satisfying the following: (i) iz a a , (ii) ( i )z z a a   . Write down the complex number z that gives the greatest value of ,z giving your answer in terms of a. [5] 3. It is given that xyyx , where 0x , 0y . Find d d y x in terms of x and y. Hence find the equation of the tangent to the curve xyyx which is parallel to the y-axis. [6] 4. (a) Find the comp lex number z in the form ixy , where x, y  ℝ such that 3i i1 2i z z  . [ Note: z is the conjugate of z .] [3] (b) The two roots 12, zz of the equation 2 0,z z p   where ,p are such that 1 iz a b where a, b  ℝ, 0b and 12 3.zz Find the value of .p [3]

[Turn over 3 5. (i) Use the stand ard series for ln(1 )x to find the first three ter ms of the Maclaurin's series for   1 1 ln 1 2 y x   . [3] (ii) Find the range of values of x for the above expansion to be valid. [3] (iii) Use your answer in part (i) to calculate an approximate value of 2 0 dyx . Explain why the answer obtained is not a good approximation. [2] 6. (a) One NETO logistics commander was tasked to deliver goods to an army camp in Abghan during June 2012. The commander delivered 1000 tons of goods on 1st June, 1100 tons on 2nd June, ... etc. Each subsequent day’s delivery was 100 tons more than the previous day’s delivery. After several days of delivery, due to more road attacks from Bl Paeda, the commander had to change the plan so that each subsequent day’s delivery wa s 100 tons less than the previous day’s delivery. By 15 th June, the commander managed to deliver a total of 21300 tons of goods for the past 15 days. On w hich day did the commander deliver the most g oods to the camp? How many tons of goods did he deliver on this particular day? [5] (b) Given that all terms in a geometric progression {}nu , 1, 2, 3, ...n are positive with first term a and common ratio r, where r  1, the sums H and C are defined as follows: 12 ... nH u u u    , 12 1 1 1 ... n C u u u    . (i) By expressing H and C in terms of a and r, prove that 1 n H uuC  . [3] (ii) Express the product 1 2 3 ... nu u u u in the form of f ( )n H C   , where f(n) is a function of n. [2]

[Turn over 4 7. (a) A sequence 1u , 2u , 3u , … is such that 1 1u  and  1 23 n nnu u a      , where n  and a is a constant. By considering 1nnuu  , find nu in terms of n and a. [4] (b) Prove by induction that, for every positive integer n, 3 5 3 5 3 5 3 5 3 3 1(1 3 1 ) (2 3 2 ) (3 3 3 ) ... ( 3 ) ( 1) 2n n n n             . [4] Given that 2 32 1 ( 1)4 n r nrn   , find 2 5 1 n r r   in terms of n. [2] 8. Jesse leans a 6 metre rod XY in the position as shown below . At time t seconds, the two ends X and Y are x and y metres from O respectively. (i) State an equation relating x and y . [1] As the rod slips, Jesse proposes that y decreases at a rate proportional to y . (ii) Based on Jesse’s proposal, show that 2d (36 ) ,d x k x tx  where k is a constant. [3] Initially, X is 4 metres from O. Take 2.k  (iii) Find an expression for x in terms of t and find the time taken for Y to be 3 metres from O from the instance Jesse lets go of the rod, giving your answer to 1 decimal place. [5] (iv) Sketch the graph of x against .t Hence, comment on whether Jesse’s proposed model is appropriate. Justify your answer. [3] O X Y y x 6

[Turn over 5 9. The diagram above shows a cuboid OABCDEFG with horizontal base OABC where 6 cmOA and 4 cmAB . OD, AE, BF and CG are vertical with height 10 cm. The point O is taken as the origin and vectors , and i j k are unit vectors in the directions OA, OC and OD respectively. The point P on OA and the point Q on BF are such that : 2:1OP PA  and : 3: 2BQ QF  . Point R is the midpoint of AB. (i) Show that the equation of the plane PQR is 3 3 12x y z   . [3] (ii) Find the distance from the point C to the plane PQR. [2] (iii) Find the acute angle between the plane PQR and the plane OABC. [2] (iv) Find the position vector of the point of intersection M of the line PQ and the plane OCGD. Hence write down the distance from M to the plane OABC. [3] (v) Find a vector equa tion of the plane, in the form   r u v w , when the plane PQR is reflected about the plane OABC. [2] k G R E B F D A O P C Q i j

[Turn over 6 10. (a) Find the following integrals: (i)   44cos sin dx x x ; [3] (ii) 3log (3 1) dxx . [4] (b) By using the substitution 4sinx  , show that 22 1 2 16 16 d sin 4 x x xxCxx      , where C is an arbitrary constant. [5] 11. A curve G has parametric equations 1xt t , 12yt t , where 0t  . (a)(i) Find d d y x in terms of t, and deduce that d12 d y x   . [4] (ii) Find the x-coordinate of the stationary point of G. [3] (iii) Denoting the graph of G by f( )yx , sketch the graph of f ( )'yx , stating clearly the equation of any asymptote(s) and any axial-intercept(s). [3] (b) Let P denote the point ( x, y) on the curve G. A vertical line and a horizontal line through P are drawn, intersecting the x-axis and the y-axis at ( , 0)Xx and (0, )Yy respectively. If A represents the area of the rectangle OXPY, find the rate of change of A with respect to x when 5t . [4]