EJC H2 2020 Prelim P1

EUNOIA JUNIOR COLLEGE JC2 Preliminary Examinaton 2020 General Certificate of Education Advanced Level Higher 2 CANDIDATE NAME CLASS INDEX NO. MATHEMATICS Paper 1 [100 marks] 9758/01 28 August 2020 3 hours Candidates answer on the Question Paper Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name, civics group and question number 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 paper clips, highlighters, glue or correction fluid. Answer all 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. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 28 printed pages (including this cover page) and 2 blank pages. For markers’ use: Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Total

1 The curve C has equation 22 ( 3)x y y x   . Find the coordinates of the points on C at which the tangents are parallel to the y-axis. [4] 2 State the derivative of 2tan x . Hence, or otherwise, find 3 2 2sec dx x x . [4] 3 [It is given that a sphere of radius r has surface area 24 r and volume 34 3 r .] Air is being blown into a spherical balloon at a constant rate of 12 3cm per minute. Initially there is no air in the balloon. (i) Find the rate at which the radius of the balloon is changing when the radius of the balloon is 5 cm. [2] (ii) Find the rate at which the surface area of the balloon is changing 10 minutes after air is blown into the balloon. [4] 4 (i) On the same axes, sketch the graphs of 1 aby x  and y b x a , where a and b are positive constants such that 01 ab   . It is given that the two graphs intersect exactly twice. [3] (ii) Hence, solve the inequality 1 axa x  . [3] 5 You are given that e d.ee x xxFx   (i) Use the substitution exy to find another expression for F. [2] (ii) By considering e (e e ) (e e )x x x x xAB     where and AB are real constants to be determined, find an expression for F. [3] (iii) Show algebraically that your answers to parts (i) and (ii) differ by a constant. [1] 6 The functions f and g are defined by   2f: 3xa xa  for , ,x x a 2 g : 1 22 aaxx    for 0x where a is a positive constant. (i) Define, in a similar form, the inverse function 1f  and show that   2f xx . [3] (ii) Hence, find 21 (2 )f k a for positive integer k, giving your answer in terms of a. [2] (iii) Show that the composite function fg exists and find the range of fg . [3]

2020 JC2 H2 Mathematics Preliminary Examination 7 The sum, nS , of the first n terms of a sequence 1 2 3, , ,u u u is given by  3n nS a bn c   , where a, b and c are constants and 1n . (i) Given that 1 2 35, 9 and 33u u u   , find a, b and c. [4] (ii) Show that  1 3n nu A B  , where A and B are constants to be determined. [3] (iii) Using your answer in part (ii), find 1 2 n r r u    in terms of n. (You need not simplify your answer.) [3] 8 Do not use a calculator in answering this question. (a) The complex number 1z is given by 1i . (i) Given that 1z is a root of the equation   2 1 3 i 0z az b     , find the values of the real numbers a and b. [3] (ii) Using these values of a and b, find the second root of this equation in exact form. [2] (b) The complex numbers 1w and 2w are given by 2 2i and 3i respectively. (i) Find the modulus and argument of 12ww in exact form. [3] (ii) Hence, or otherwise, show that 7 1 3cos12 22   [2] 9 (a) (i) Given x and y are related by the differential equation 2 d d yx xy kx for k , show that  lnkxy x  is a solution of the differential equation where  is an arbitrary constant. [2] (ii) Hence, show that   11e , e k is a stationary point of the curve  lnkxy x  . [2] (b) It is g iven that x and y are related by the differential equation 22d d yy x x yx   and that 0y when 2x . (i) By substituting 22v x y , show that the differential equation can be written as d 2d v vx  . [2] (ii) Find v in terms of x and hence show that   2 fyx where  f x is to be determined. [4]

10 The diagram shows the graph of 2 1 1y x  when x > 0. (i) Evaluate 1 2 1 d1 k k xx   for 0k  , leaving your answer in terms of k. [2] (ii) By considering appropriate rectangles on the interval  ,1kk  for the curve 2 1 1y x  , show that     11 2 2 11 tan 1 tan 111 kk kk      for k  . [2] (iii) Use the identity   tan tantan 1 tan tan ABAB AB   to show that 1 1 1tan tan tan 1 xyxy xy      , where 0.xy [2] (iv) By considering parts (ii) and (iii), prove by the method of differences that   1 2 2 11 11 tan 2111 nn kk n nkk      . [4] 11 (a) The angular diameter of an object is the angle the object makes (subtends) as seen by an observer. As shown in the diagram below,  denotes the angular diameter (measured in radians) of a circle whose plane is perpendicular to the line between the point of view (point P) and the centre of said circle . D denotes the distance from point P to the centre of the circle and d denotes the diameter of the circle. (i) Show that, if  is sufficiently small, Dd  . [2] (0,1) O x y D d P 

2020 JC2 H2 Mathematics Preliminary Examination The equation in (i) is often used in astronomy to estimate the diameters of stars from the angular diameter, assuming their shapes to be approximately circular. (ii) If the angular diameter of a star is measured to be 0.00873 rad and the distance of the star from the earth is 129.46 10 km, estimate the diameter of the star. [1] (b) An astronaut A is at a large distance x km from the surface of the earth. The radius of the earth is assumed to be a constant R km. The furthest point on the earth’s surface that the astronaut can see is a point P such that AP = y km and the angle OAP =  , where O is the centre of the earth (see diagram). (i) Show that 1 221 Ryx x   . [3] (ii) It is given that R is small compared to x. Show that, if R x  , 23tan 1.5      . [4] (iii) It is also given that 0.0345  rad and the astronaut A is 180,000 km from the surface of the earth, find  and hence estimate the radius of the earth. Leave your answer to the nearest km. [3] R R P y A O x

12 One of the highlights of the grand opening of the Grand Egyptian Museum in Cairo is a tightrope walking contest. For this contest, as shown in the diagram, a glass pyramid is constructed beside the museum building, with a rectangular base OABC and vertex V. Points ( , , ) x y z are defined relative to O (0, 0, 0) , where units are metres. As the ground is uneven, the pyramid is tilted slightly with A, C and V at (8, 1, 1)  , ( 1, 9, 1)  and (5, 4, 14) respectively. (i) Find a cartesian equation of the plane containing the triangular face ABV. [2] The contestant walks on rope R1 which is firmly secured to the starting point 29 15 1,,2 2 2 Sa  on the museum building such that 0a . The taut rope R1 penetrates through the glass face ABV of the pyramid, and le