2019 PU3 MI H2 Math Prelim (with ans)

This document consists of 24 printed pages. [Turn over 2019 Preliminary Exams Pre-University 3 MATHEMATICS 9758/01 Paper 1 3 September 2019 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your admission number, name and class on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. Give your answers in the spaces provided in the question paper. 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. The use of an approved graphing calculator is expected, where appropriate. 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. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 100. Qn No. Q 1Q 2Q 3Q 4Q 5Q 6Q 7Q 8Q 9 Q 10 Q 11 * T otal Score Max Score 6 6 6 8 8 9 10 10 12 12 13 100 CANDIDATE NAME CLASS ADMISSION NUMBER www.KiasuExamPaper.com 376

2 © Millennia Institute 9758/01/PU3/Prelim/19 1 The nth term of a sequence 123, , , ...uuu is given by 2 1 ! nun n . (i) Show that 1 121 ! nn nuu n n     .[ 2] (ii) Hence find 2 2 121 ! n r rr r     .[ 3] (iii) State, with a reason, if the series in part (ii) converges. [1] 2 It is given that   2 3 for 0 2, f 1 2 for 2 4, 6 xx x xx        and that   ff 4xx  for all real values of x. (i) Sketch the graph of fyx for 31 0x  .[ 3] (ii) Find the volume of revolution when the region bounded by the graph of fyx , the lines 1x  , 2x  and the x-axis is rotated completely about the x-axis. [3] 3( i) Find  2d cosd xx .[ 2] (ii) Hence find 32sin dxx x .[ 4] 4( a) The curve C has equation fyx . (i) Given that  2 34f 1 xxx x   , sketch the curve C, showing the equations of the asymptotes and the coordinates of any turning points and an y points of intersection with the axes. [3] (ii) Hence, state the range of values of x where f0 x  .[ 2] (b) The curve with equation gyx is transformed by a stretch with scale factor 2 parallel to the x-axis, followed by a translation of 1 unit in the negative x-direction and followed by a translation of 1 unit in the positive y-direction. The resulting curve has equation 2 34 1 xxy x   . Find g x .[ 3] www.KiasuExamPaper.com 377

3 © Millennia Institute 9758/01/PU3/Prelim/19 [Turn over 5( i) On the same axes, sketch the graphs of 1y xa  and yb x a  , where a and b are positive constants and 1ab a , stating clearly any axial intercepts and equations of any asymptotes. [3] (ii) Given that the solution to the inequality 1 bx axa   is 1 12 x or 1x  , find the values of a and b.[ 4] (iii) Using the values of a and b found in part (ii), write down the solution to the inequality 1 bx axa   .[ 1] 6( i) Given that  sin 2fe ax x     where a is a constant, find f0 , f0 and f0 in terms of a. Hence write down the first two non-zero terms in the Maclaurin series for f x . Give the coefficients in terms of e. [5] (ii) The first two non-zero terms in the Maclaurin series for f x are equal to the first two non-zero terms in the series expansion of 2 1 bx , where b is a constant. By using appropriate expansions from the List of Formulae (MF26), find the possible values of a and b in terms of e. [4] 7( a) Find the exact value of 1 2 0 d2 x xx .[ 3] (b) The expression  2 2 2 x x can be written in the form  2 2 2 BCA x x    . (i) Find the values of A, B and C.[ 3] (ii) Show that  1 2 2 0 d ln 2 2 x xp q x   , where p and q are constants to be found. [4] www.KiasuExamPaper.com 378

4 © Millennia Institute 9758/01/PU3/Prelim/19 8( a) Referred to the origin O, the point Q has position vector q such that q 312 22 ij k . (i) Find the acute angle between q and the y-axis. [2] It is given that a vector m is perpendicular to the xy-plane and its magnitude is 1. (ii) With reference to the xy-plane, explain the geometrical meaning of qm and state its value. [2] (b) Referred to the origin O, the point R has position vector r given by ra b , where  is a positive constant and a and b are non-zero vectors. It is known that c is a non-zero vector that is not parallel to a or b. Given that  ca bc , show that r is parallel to c.[ 2] It is also given that a is a unit vector that is perpendicular to b and 2b . By considering rr , show that 241k c , where k is a non-zero constant. [4] 9 The function f is defined by 2 f: 1 2 e xx 6 , x   . (i) Show that f does not have an inverse. [2] (ii) The domain of f is further restricted to xk , state the largest value of k for which the function 1f  exists. [1] In the rest of the question, the domain of f is x   , xk , with the value of k found in part (ii). (iii) Find  1f x .[ 3] The function g has an inverse such that the range of 1g is given by 1, 3. (iv) Explain clearly why the composite function gf exists. [2] It is given that the composite function gf is defined by gf xx  . (v) State the domain and range of gf. [2] (vi) By considering gf 2 , find the exact value of  1g2  .[ 2] www.KiasuExamPaper.com 379

5 © Millennia Institute 9758/01/PU3/Prelim/19 [Turn over 10 As a raindrop falls due to gravity, its mass decreases with ti me due to evaporation. The rate of change of the mass of a raindrop, m grams, with respect to time, t seconds, is a constant c. (i) (a) Write down a differential equation relating m, t and c. State, with a reason, whether c is a positive or negative constant. [2] (b) Initially the mass of a raindrop is 0.05 grams and, after a further 60 s, the mass of the raindrop is 0.004 grams. Find m in terms of t.[ 3] In recent years, scientists are looking for alternative sourcesof sustainable energy to meet our energy needs. One approach aims to extract kinetic energy from rain to harvest energy. In order to test for the viability of this approach, scientists need to find the maximum kinetic energy that can be harvested from a falling raindrop. It is known that the kinetic energy K, in millijoules, of an object falling from rest is given by 22 2 mg tK  , where m grams is the mass of the object at time t seconds and g is a positive constant known as the gravitational acceleration. In the rest of the question, use your answer in part (i)(b) as the mass of a raindrop at time t. (ii) (a) Show by differentiation that the maximum value of the kinetic energy of a raindrop falling from rest is 2pg , where p is a constant to be found . Give your answer correct to 2 decimal places. [5] (b) It is given that 10g  . The kinetic energy of a raindrop falling from rest is 1 millijoules when it hits the surface of the ground. Given that it takes more than one minute for a raindrop to fall from rest to the surface of the ground, find the time taken for a raindrop to hit the surface of the ground. [2] www.KiasuExamPaper.com 380

6 © Millennia Institute 9758/01/PU3/Prelim/19 11 In this question, the distance is measured in metres and time is in seconds. A radio-controlled airplane takes off from ground level and is assumed to be travelling at a steady speed in a straight line. The position vector of the airplane t seconds after it takes off is given by r = a + tb, where a refers to the position vector of the point where it departs and b is known as its velocity vector. (i)