NYJC H2 Maths P1

This document consists of 5 printed pages. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2014 [Turn Over NANYANG JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS 9740/01 Paper 1 16th September 2014 3 Hours Additional Materials: Cover Sheet Answer Paper List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your 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 notations and not calculator commands. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question.

2 NYJC 2014 JC2 Preliminary Examination 9740/01 1 The function f is defined by 2f: 2 ( 1 ) , , x xx x k  . (i) Determine the largest value of k for which the function 1f exists. [1] With this value of k, (ii) find 1f in a similar form. [3] (iii) show algebraically that the x-coordinate of the point of intersection of the curves   1f and fyx y x  satisfies the equation 2 31 0xx  and find the value of this x-coordinate, correct to 3 decimal places. [2] 2 The region enclosed by the curve es i nxyx where 0 2x   , the x-axis and the line 2x  is denoted by A. Find the exact area of A. [4] Find the volume of revolution when the region bounded by the curves es i nxy x , 23 1 3y xx x  and the line 2x  is rotated completely about the x- a x i s . [2] 3 For a curve with equation 23 33xx y y  , find (i) the coordinates of the point at which the tangent is parallel to the x-axis, [4] (ii) the equation(s) of the normal(s) at 1x   . [4] 4 The complex number z satisfies the relation 13 i 2z  . Illustrate, on an Argand diagram, the locus of points representing the complex number z. [2] (i) Find the greatest possible value of  arg 3 3iz . [3] (ii) Find the range of values of , where 0 2   , such that there exists a complex number w which satisfies the relations  arg 3 3iw   and  arg * 2w   , where *w is the conjugate of w. [4]

NY 5 6 YJC 2014 JC2 5 A com p when fu identica remaini (i) U s the (ii) Sh 6 In a bid path of measure denoted variable (i) U (ii) Fi (iii) Sk 2 Preliminary E pany requi r ull. The top al sides of th ing side is o se differenti e box. how that, in d to analyse f the insect. ed with re s d by the vari es are relate Using the sub ind x in term ketch the pa Examination res a box m p and the b a he isosceles of length x c iation to fin this case, y x the path of The insect spect to the iables x and ed by the dif bstitution w ms of t. ath travelled ax ax made of car d ase of the b o s triangle are cm. The hei d, in terms o 23 2 y a x a   f an insect, a ’s path was origin in t d y respectiv fferential eq etw y , find d by the inse x 3 9740/01 dboard of n ox are made e of length ght of the b of a, the val 1 1 . Hence fi an entomolo s observed f the horizon t vely. It is giv quations d d y t d y in terms o ect. x negligible t h e up of six ax cm, whe ox is y cm lue of x whi ind the rang ogist decide for 20 seco n tal and ver t ven that whe e 30 t y t    of t. ax x ax hickness to identical iso ere a is a co (see diagram ich gives a m ge of y x . es to fit a m nds. The p a tical directi o en 0, tx 0 and 2 2 d d x t  x hold 300 c osceles trian onstant and m). minimum su mathematical ath travelled ons, at tim e 1 , y = 0 an e t . [Turn cm3 of po w ngles. The t 1 2a , and urface area o l model for d by the in s e t seconds nd d 2d x t  . T Over wder two the of [7] [3] the sect , is The [5] [5] [2]

4 NYJC 2014 JC2 Preliminary Examination 9740/01 7 The complex number z is given by 2 44 3 iz  . (i) Find z in exact cartesian form x + iy , showing your workings clearly. [4] (ii) Given that  3 2*wz  , where *w is the conjugate of w, find w in the form ier  . [4] (iii) The point representing the complex number v is obtained by a counter clockwise rotation of the point representing 2z through one right angle about the point (0, 1) on the Argand diagram. By considering 2 iz  , find v in the form x + iy. [3] 8 (a) A sequence 123,,,uuu  is defined by 1 cosux a n d    1 112 s i n s i n c o s 22 nn xnn u u n n x n x     for n  where x is a constant. Prove by the method of mathematical induction that cos n nxu n for n  . [6] (b) (i) Show that 4 1 1 cos 2 N n n n  = æö ÷ç ÷ç ÷çèøå where N is a positive integer, can be written as () 2 1 11 2 nN n n= - å . [2] (ii) Use your result in b(i) and the series expansion of ()ln 1 x+ in MF15 to deduce the exact value of 1 1 cos 2n n n  ¥ = æö ÷ç ÷ç ÷çèøå . [3]

5 NYJC 2014 JC2 Preliminary Examination 9740/01 [Turn Over 9 The lines l1 and l2 meet at the point P. The line l3 is coplanar with l1 and l2 and is perpendicular to l1. Given that l1 and l2 are parallel to the vectors a and b respectively, show that l3 is parallel to the vector 2    abba a . [3] The equations of l1 and l2 are now known to be 31 1 51 0 22 t    r and 31 7 53 24 s    r respectively, where s and t are real parameters. Find the equation of the line l3, given that l3 also passes through P. [2] The line l4 has equation 10 3 34 11 u     r , where u is a real parameter. Determine if l3 and l4 are skew or intersecting. [3] The line l5 is perpendicular to both l3 and l4. Find the acute angle between l5 and the plane containing l1 and l2 . [5] 10 A curve C has parametric equations x = 1 + cos  and y = 2 sin , where 0 ≤  ≤ π. (i) Show that the equation of the tangent to C at the point with parameter  i s  2 cot cot cosecyx     . [3] (ii) The points P and Q on C have parameters 5 6   and 6  respectively. The tangent at P meets the tangent at Q at the point R. Find the y-coordinate of R. [3] (iii) The area of the region bounded by the tangent at P, the tangent at Q and the x-axis is denoted by A and the area of the region bounded by C and the x-axis is denoted by B. Find the exact value of the difference of A and B. [8] −−−−− END OF PAPER −−−−−

6 NYJC 2014 Preliminary Examination 9740/01