AJC H2 MATH P1

Page 2 of 7 Anderson Junior College Preliminary Examination 2011 H2 Mathematics Paper 1 Answer ALL questions. 1. Find 12cos dx xx  . [3] 2. The usual selling price of a T-box game console, a Kinect sensor and a game DVD is $499 in total. During the Great Singapore Sale, two companies, P and Q, offered the following discounts to their customers: Company Discounts given for each item Total price after the discount T-box game console Kinect sensor Game DVD P 10% 15% 10% $439.15 Q 5% 25% 20% $426.30 The employees from Company P are offered a further 5% discount on the original price of both the Kinect sensor and the Game DVD. Determine if this additional discount will make it more attractive for the employees to purchase all the 3 items from their own company than from Company Q. [4] 3. Solve the inequality      2 0xaxb xxc    , where ,,abc  and 0 cba  . Hence, find the range of values of x which satisfy       2 ln ln 0 ln ln xa xb xc x    . [5] 4. (i) By using the substitution tantx , show that  1 2 11 d tan 2 tan1s i n 2 x xcx   . [3] (ii) Find the exact volume of revolution when the region bounded by the curve 2 12 1s i ny x  , the lines 4x  , y = 2 and the y-axis is rotated completely about the line y = 2. [3]

Page 3 of 7 5. (a) Find sin 2 cos dx xx . [2] (b) The diagram below shows the curve defined by the parametric equations sinx tt , sin 2y t for 0 t  . The curve has a maximum point at A. (i) Find the equation of the tangent to the curve at the point A. [3] (ii) Find the exact area bounded by the curve, the tangent at A and the line x  . [3] 6. Given that y =  1 2s i n 2 x , show that 22 2 2 d2 d 4s i n 2dd yy yxxy x   . [3] (i) Hence find the Maclaurin’s series for y up to and including the term in x3. [3] (ii) By considering the standard series for sin x, verify that the series obtained in (i) is correct. [2] 7. The above 48 cm by 18 cm plastic sheet is cut out to form a shape represented by the shaded region. The shape is folded into a box with width x, length y and height z as y x A Base Side Side Side Side Top cover

Page 4 of 7 shown in the diagram. (i) Express the volume of the box, V, in terms of y. [2] (ii) Using differentiation, find the maximum possible volume of the box. [3] Two small robots of negligible dimensions, A and B, are initially placed on the edge of the base of the box where m = 0 cm and n = 0 cm respectively as shown in the diagram. Robot A starts to travel along the length of the base with speed 2 cm/s. One second later, Robot B starts to travel in the opposite direction along the length of the base with speed 1 cm/s. (iii) Given that the base of the box has length 20 cm and breadth 10 cm, find the rate of change of the distance between the two robots when n = 4 cm. [4] 8. The curve C has the equation 2 where axya xa  is a negative constant. (i) Show that the curve C has two stationary points for all negative values of a. [2] (ii) Sketch the curve C, showing clearly all the asymptotes, axial intercepts and turning p o i n t s . [ 3 ] (iii)Using the sketch in (ii), find the range of values of k for which   242 1xk x x  has exactly two roots. [3] A B Base of the box 2cm/s 1cm/s m n

Page 5 of 7 9. The equations of two lines are given as follows: 1 :l 01 02 , 22          r 2 :l 1 2, 2     r (i) Show that l1 and l2 are skew lines. [3] (ii) If Z is the midpoint between any point on l1 and any point on l2, show that the locus of Z is a plane. Write down the equation of this plane p in scalar product form. [3] (iii) S is a point on l1. The point S ’ is the image of point S reflected in plane p. If S ’ is also a point on l2 , find the coordinates of S. [2] 10. The sketch below shows the graph of y = f(x). The curve passes through the point A (1,0) and has a minimum point at B(0,2). The equation of the asymptotes are y = -2x+ 1 and x = 1 2 . On separate diagrams, sketch the following graphs indicating the points corresponding to A, B and asymptotes where necessary. (i) 1f 2yx  [3] (ii)  1 fy x [3] (iii) f' 2yx [3] B(0,2) 1 2x A(1,0) x y y = -2x+ 1

Page 6 of 7 11. A sequence ,....,, 321 uuu is defined by the recurrence relation 1 1 11nn nuu nn    , 2,nn and 1ua where a is a constant. (i) Use the method of mathematical induction to show that (1 )1nua n  for all positive integers n. [3] (ii) If a =1, show that 2 1 11 1 122 1 N n nnuu N     . [2] Hence, find 2 1 (2 9)(2 7) N n nn  . [3] 12. The sum of the first n terms of a series, nS , is given by 1 1 ( 1) , where is a constant and 1,n nSa a a na     Obtain an expression for the nth term of the series, nT and prove that nS is a geometric series. [3] If the sequence  nT is now grouped as follows: ( 1T ), ( 234,,TTT ), ( 5678 9,,,,TTTTT ), … where each subsequent bracket has 2 terms more than the previous bracket, find (i) the total number of terms in the first n brackets. [2] (ii) the middle term of the 11th bracket in terms of a. [2] (iii)the range of values of a for the sum to infinity of the series to exist. Hence, find the least value of n for the sum of all the terms in the first n brackets to be within 0.1% of the sum to infinity of the series when 39 20a . [4]

Page 7 of 7 13. (a) (i) The complex numbers p and q satisfy the simultaneous equations * 10 5p iq i  2 52 0p qi  . Given that Im(p) < 0, find p in the Cartesian form. [3] Hence find the values of n for which 2np is purely imaginary. [2] (ii) The complex number w is such that *arg 2 w pp    and *2ww  . Find w. [4] (b) Find the roots of the equation   6 28z  , giving your answer in the form cos iR e where 0R and   . [3] Describe the curve on which all these roots lie. [1] ----- END OF PAPER-----