MIPU3 H2 Mathematics Paper 1 2012 Question

2 Answer all the questions [100 marks] 1 The position vectors a and b are given by a = 3i – 2j + 6k and b = 4pi + 7pj – 4pk, where p > 0. Given b is a unit vector, (i) find the exact value of p, (ii) give a geometrical interpretation of |a b|, (iii) evaluate a  b. [2] [1] [2] 2 Without using a calculator, solve the inequality 2 2 2 2 1 121 xx xx   , [4] Hence solve     2 2 2 2 2 ln ln 1 1 ln ln 1 xx xx    . [3] 3 It is given that   2f x ax bx c   , where a, b and c are constants. (i) Given that the curve with equation  y f x passes through the points with coordinates  1,6 ,  1.1,0.12 and  1.5,1 , find the values of a, b and c. [3] (ii) Find the set of values of x for which  y f x is an increasing function. [2] (iii) Sketch the graph of  'y f x . [2] 4 Find the first three terms in the expansion of 4x , in ascending powers of x. State the set of values of x for which the expansion is valid. Hence, by substituting with a suitable value of x, find an approximate value for 8 as a fraction. [4] [3]

3 5(a) An educational fund is started at $2000 and the bank offers a compound interest at 2% per annum. If withdrawals of $50 are made at the beginning of each of the subsequent years, show that the amount in the fund at the beginning of the (n+1)th year is  $500 5 1.02 n . [3] Find the amount in the fund to the nearest dollar after the 30th withdrawal. Calculate the number of years this educational fund can last. [1] [2] (b) The 9th term of an arithmetic progression is 50 and the sum of the first 15 terms is 570. It is given that the sum of the first n terms is greater than 500. Find the least possible value of n. [4] 6 (i) Given that 22 21x y xy   , find dy dx in terms of x and y. [4] (ii) For the curve 22 2 1 0y x xy    , find the coordinates of each point at which the tangent is parallel to the x-axis. [4] 7 Given y = 1 when x = 0 and   21 2 dy xydx  , show that   232 32 10d y d y dyxydx dx dx        . [3] Hence, find the first four terms of the Maclaurin series for y. [4] 8 The region bounded by the curve tanyx , the x-axis and the line 3x  is rotated through 2 about the x-axis. Find the exact value of the volume of the solid formed. [4]

4 9(a) Using De Moivre’s Theorem, show that 59 cos sin cos sin 12 2 6 6ii               . [3] (b) The roots of the equation 3 27zi  are 1z , 2z and 3z . Without using a calculator, find 1z , 2z and 3z in the Cartesian form x iy , showing your working. [4] 10 The diagram below shows that graph of  ln 1 4y x x    . The roots of the equation  ln 1 4 0xx    are denoted by α and β, where α < β. (i) Find the values of α and β, each correct to 3 decimal places. [2] A sequence of real numbers 1 2 3, , , ...x x x satisfies the recurrence relation  1 ln 1 4nnxx    for 1n . (ii) Prove algebraically that, if the sequence converges, then it converges to either α or β. [3] (iii) By considering the maximum point of  ln 1 4y x x    , show that 1 2nnxx  . [3] α β 1 x y

5 11 Water is flowing into a rectangular tank, with a horizontal base, at a constant rate. Water is flowing out of the tank at a rate which is proportional to the depth of water in the tank. At time t seconds, the depth of water in the tank is x metres. The depth of the water remains constant when it is 0.5 m. Show that )12(  xkdt dx , where k is a positive constant. Initially, the depth of water in the tank is 1.5 m and is decreasing at a rate of 0.02m/s. Find the time at which the depth of water is 1.01 m. [3] [5] 12 The diagram shows the graph of  y f x . On separate diagrams, sketch the graphs of (i)  y f x , [3] (ii)   2y f x , [3] In each case, state the equation of any asymptote(s), the coordinates of any intercept(s) with the axes and any stationary point(s). x = 1 2 - 2 -1 0 y x (2, 4) (0.5, 0.5)

6 13 The plane p passes through the points with coordinates  0, 2,1 ,  2, 3,2 and  4,5, 2 . (i) Find the vector equation of the plane p, in scalar product form. [4] The line l1 has equation 1 4 1 2 1 4 x y z   and the line l2 has equation 11 23 x y z k  , where k is a constant. It is given that l1 and l2 intersect. (ii) Find the value of k. [5] (iii) Explain why l1 does not lie in p. [2] (iv) Find the coordinates of the point at which l1 intersects p. [3] (v) Find the acute angle between l2 and p. [2]