Name: Inde Number: Class: DUNMAN HIGH SCHOOL Preliminary Eamination Year 6 MATHEMATICS (Higher ) 970/0 Paper 7 September 05 Additional Materials: Answer Paper Graph paper List of Formulae (MF5) 3 hours READ THESE INSTRUCTIONS FIRST Write your Name, Inde Number 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-eact numerical answers correct to 3 significant figures, or decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are epected 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 eamination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. For teachers use: Qn Q Q Q3 Q Q5 Q6 Q7 Q8 Q9 Q0 Q Q Total Score Ma Score 5 6 6 6 6 8 8 9 9 3 3 00 DHS 05 This question paper consists of 6 printed pages (including this cover page).
A graphic calculator is not to be used in answering this question. (i) Find the value of ( + i), showing clearly how you obtain your answer. [] (ii) Given that + i is a root of the equation z z + ( a + bi) = 0, find the values of the real numbers a and b. [] (iii) For these values of a and b, solve the equation in part (ii). [] Using partial fractions, find + + ( )( ) 6 d. [6] 3 A curve C has parametric equations = θ cos θ, y = θ sin θ, for 0 θ π. (i) Sketch the curve C. [] (ii) The point P on the curve C has parameter p and the point Q has coordinates ( π,0). The origin is denoted by O. Given that p is increasing at a constant rate of 0. units per second, find the rate of decrease of the area of triangle OPQ when p = 3 π. [] The comple number z is given by i π 6 ( 3)e. (i) Given that w = ( + i) z, find w and arg w in eact form. [] n (ii) Without using a calculator, find the smallest positive integer n such that w is purely n imaginary. State the modulus of w when n takes this value. [] DHS 05 Year 6 H Mathematics Preliminary Eamination Paper
3 5 y O The diagram shows the graph of y = f( ). The graph has a minimum point at (, ) and a maimum point at (, 7). It intersects the aes at =, = and y = 3. The equations of the asymptotes are y = and = 3. (i) Sketch the graph of y =, f ( ) giving the coordinates of any stationary points, points of intersection with the aes and the equations of any asymptotes. [3] (ii) Solve the inequality ( ) f < 0. [3] 6 (i) By using the Maclaurin series for e and cos, find the Maclaurin series for g( ), where g( ) = e cos, up to and including the term in. [3] (ii) Use your answer in part (i) to give an approimation for g( ) d in terms of a, and 0 evaluate this approimation in the case where a = giving your answer correct to 3 e, 5 significant figures. [3] a e 3 (iii) Use your calculator to find an accurate value for g( ) d, up to 5 significant figures. 0 Why is the approimation in part (ii) not very good? [] DHS 05 Year 6 H Mathematics Preliminary Eamination Paper
7 Relative to the origin O, two points A and B have position vectors a and b respectively. It is given that a =, b = and 3a b = (37). (i) By considering the scalar product (3a b) i (3a b), show that aib = and give the geometrical meaning of ai b. [] (ii) Give the geometrical meaning of ( a b) b and find its eact value. [3] (iii) Write down, in terms of a and b, a vector equation of the line that passes through O and bisects the angle AOB. [] 8 A curve C has equation 3 + y y = 80. (i) Show that d y 3 + y =. d y [] (ii) Show that the curve C has no stationary points. [3] (iii) The normal to the curve at the point P(6, ) meets the curve again at the point Q. Find the coordinates of Q. [] 9 y R R O The diagram shows the curve with equation y sin ( ) = for 0. (i) Find the area of the region R bounded by the curve, the lines y = π, y = π and the 6 y-ais. [] (ii) Find the volume of revolution when the region R is rotated through π radians about the -ais. [3] (iii) Without using a calculator, find the eact area of the region R bounded by the curve, the lines =, = 3 and the -ais. [] DHS 05 Year 6 H Mathematics Preliminary Eamination Paper
0 (a) Show that the substitution to the form w 5 = y reduces the differential equation dy y y y d = + d d = + w aw b where a and b are to be determined. Hence obtain the general solution in the form y = f ( ). [5] (b) A certain species of bird with a population of size n thousand at time t months satisfies the differential equation d n t = e. dt Find the general solution of this differential equation. [] Sketch three members of the family of solution curves, given that n = 30 when t = 0. [], Functions f and g are defined by f : ln( 3 ) for R, >, g : for,. ( )(5 ) < R (i) Describe fully a sequence of transformations which would transform the curve y = ln onto the curve of y = f ( ). [3] (ii) Sketch the graph of y = g( ). [] (iii) Find the eact range of fg. [] (iv) If the domain of g is further restricted to k, state with a reason the least value of k for which the function 3 g eists. [] In the rest of the question, the domain of g is defined as R, k, where k is the value found in part (iv). (v) Find g ( ). [] 6 (vi) If h is a function such that gh is well-defined and the point (, ) α lies on the graph of y = gh( ), find the value of h( α ). [] 5 DHS 05 Year 6 H Mathematics Preliminary Eamination Paper
6 H G E C F B k O i j A The diagram shows a cuboid with rectangular base OABC and top EFGH, where OA = units, OC = 3 units and OE = units. The point O is taken as the origin and unit vectors i, j and k, are taken along OA, OC and OE respectively. (i) Find the cartesian equation of the plane p which contains the points A, C and E. [3] (ii) Find the acute angle between p and the base OABC. [] The line l, passing through O, is perpendicular to p and intersects the plane containing B, C, G and H at the point T. (iii) Find the position vector of the point T and deduce the perpendicular distance from T to p. [5] (iv) A point Q lies on the line passing through C and T such that its distance from p is twice that of the distance from T to p. Find the possible position vectors of the point Q. [3] DHS 05 Year 6 H Mathematics Preliminary Eamination Paper