CJC H2 MATH P1 QP

9740/01/Prelim/2015 [Turn over CATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination MATHEMATICS 9740/01 Paper 1 26 Aug 2015 3 hours Additional Materials: List of Formulae (MF15) Graph Paper Name: ___________________________ Class: ________________ 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 an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing 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, arrange your answers in NUMERICAL ORDER. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 5 printed pages, including the cover page.

2 9740/01/Prelim/2015 [Turn over 1 A calculator is not to be used in answering this question. By considering 2wz z or otherwise, solve 432 223 1 0 0zzzz   where z , leaving the roots in exact form. [4] 2 With reference to the origin O , two points A and B have position vectors a and b respectively. The points O , A and B are not collinear. The point P divides AB in the ratio :3 : 2AP PB  . It is given that a is a unit vector, 4OB , angle π 3AOB and the foot of the perpendicular from P to the line passing through points O and A is F . Show that OF  a   , where  is a constant to be determined. [5] 3 John decided to embark on a 100-day skipping exer cise challenge. The duration of his skipping exercise each day, in seconds, follows an arithmetic progression for the odd days, and another arithmetic progression for the even days. Day Duration (s) Day 1 20 Day 2 20 Day 3 29 Day 4 30 Day 5 38 Day 6 40 Day 7 47 Day 8 50 Day 9 56 Day 10 60   (i) Find the duration of the skippi ng exercise that he does on the 75th day. [2] (ii) Find the total duration of the skipping exer cise that he does for the first 75 days. [3] (iii) After the 75 th day, John thinks that the workout is too strenuous so he decides to modify the workout such that he does 80% of the prev ious day’s duration. Find the duration of the skipping exercise that he does on the 100th day. [2] (iv) Comment on the practicali ty of this modification. [1]

3 9740/01/Prelim/2015 [Turn over 4 (i) Show that     111 2 1 13rr rr r r rr    . [1] (ii) Hence, using the method of difference, find the sum 12 2334 1 nn     . [3] (iii) Prove by mathematical induction that     1123234345 1 2 1 2 3 4nn n nn n n        . [4] (iv) Based on the results in parts (ii) and (iii), write a reasonable conjecture for the sum of the series    1 123 n r rr r r   . [1] 5 Sequence U is defined by the following recurrence relation, 11 11, 1 for all 2 nnuu u n     Z . Sequence V is defined by 2nnvu for all n Z . (i) Find the recurrence relation between 1nv  and nv . Hence show that the sequence V is a geometric progression with common ratio 1 2 . [3] (ii) Find the limit of the sequence V and that of the sequence U when n . [3] (iii) Find the sum to infinity of the sequence V . [1] (iv) Find the sum of the first n terms of the sequence U . Hence show that its sum to infinity does not exist. [3] 6 The equations of three planes 1p , 2p and 3p are 247 2578 3 xyz xyz x ay z b       respectively, where a and b are constants. (i) Find the acute angle between 1p and 2p . [2] The planes 1p and 2p meet in the line l . (ii) Find a vector equation for l . [2] The plane 3p contains the point 1, 1, 1. The three planes, 1p , 2p and 3p , have no point in common. (iii) Find the values of a and b . [3] (iv) Find the distance between l and 3p . [2]

7 8 It is gi v (i) W (ii) Fi (iii) H th (iv) G  (i) A se ta ex U (ii) It co Fi ven that f x Write down ind the bino Hence, or oth he term in x Give a reas o  1 0 fd x x is An open wa ection is a n ank is made xternal surfa Use different is given t h onstant rate ind the rate  1 1x xx  f' x . omial expan herwise, fin 2 .x [You may on why th e s not valid. ter tank, y n equilateral e of materia face area of tiation to fin hat 5y . T e of 1 3 m3/s at which th 9 2x . nsion of f x nd the Macla y refer to th e use of y o metres lo l triangle, x al of neglig i the water ta nd the value The water t s. At time t he depth is i 4 9740/01/Prelim/ x , up to an aurin series he List of Fo our answer ng, is in t h x metres on ible thickne ank, A , is 23 2Ax e of x that w tank is init i t seconds, t increasing a y /2015 nd including s for  (1 1 x   ormulae (M in part (ii) he form of n each side ess and its v 80 3 3 x  . will require ially empt y the depth o at the instan x g the term in  22 2) sinx x   MF15)]. ) to give a n an inverte d , as shown volume is 1 the least am and water of the wate r t when the d n 3x . (3 )x , up to an approxi m d prism so in the fig u 30m . Sho w mount of m is being p u r in the tan k depth is 1 m [Turn [1 [3 and includi [2 mate value [2 that its cr o ure above. T w that the t o material. [4 umped in a k is h metr m. [4 n over 1] 3] ing 2] for 2] oss- The otal 4] at a res. 4]

5 9740/01/Prelim/2015 [Turn over 9 The curve C has parametric equations 2 ,x tt 2 ,y tt  where 22 t „„. (i) Find d d y x in terms of t . What can be said about the tangents to C as 1 2t and 1 2t  ? [3] (ii) Sketch C , showing clearly the features of the curve at the points where 112, , 22t   and 2. [3] (iii) The normal to the curve at the point P where 1t  meets C again at the point Q . Use a non- calculator method to find the coordinates of Q . [4] (iv) Find a cartesian equation of C . [2] 10 The curve C has equation 2 8 1 xy x  . (i) Find the exact area of the region bounded by the curve C , the xaxis and the line 1x . [3] The region bounded by the curve C , the yaxis and the line 4y  is rotated through 2π radians about the xaxis. (ii) Using the substitution tanx  , show that the volume, V , of the solid generated is obtained by 2ππ sin d b aVp q    where ,,ab p and q are constants to be determined exactly. [5] (iii) Hence, evaluate V exactly. [3] 11 The functions f and g are defined as follows 2 f : for , 0, g : e +1 for ,x xaxx x x xx       where 01 a . (i) Sketch the graph of fyx , indicating the equations of asym ptotes and the coordinates of turning points, if any. [4] (ii) Show that the composite function fg exists, and define fg in a similar form. [4] (iii) Find the range of fg. [3] (iv) If the domain of f is further restricted to 0 x k „ , state the greatest value of k , for which the function 1f exists. Find  1f x and state the domain of 1f . [5]  THE END 