2024 NYJC Prelim Exam H2MATH Paper 1

2024 Nanyang Junior College Preliminary Examination H2 Mathematics Paper 1 1 Find the complex numbers v and w which satisfy the following simultaneous equations. 2 i 2 0vw− + = i3vw= + + Give your answers in the form iab+ , where a and b are real numbers. [4] 2 The path traced by a moving particle is a curve C, given by sinx a a=− and cosy a a =− , where a is a positive constant and 02 π . (a) Show that the gradient of C at a point with parameter  , is 1cot 2  . [2] (b) Find the equation of the tangent to C at the point where 3  = and show that this tangent passes through the point 1 , 23 aa  . [4] 3 (a) Given that ( ) 21sinyx −= , show that ( ) 2 2 2 dd1 2. dd yyxx xx − − = [2] (b) Hence obtain the Maclaurin expansion of y in terms of x, up to and including the term in 4x . [3] (c) By using 1 2x= , find an approximation for  in surd form. Find the percentage error of this approximation and comment on its accuracy. [3] 4 It is given that f ( ) 3cos sinx x x=+ . (a) Write f ( )x as ( )cosRx − , where R and α are constants to be found. [2] (b) Find the exact value of 26 0 1 df ( ) xx       . [2] (c) Find the exact value of 12 0 1 df (2 ) xx   . [3] 5 (a) Find 2 41 d 44 x x xx − ++  , giving your answer in simplest form. [4] (b) Find the value of 1 2 0 41 d 44 x x xx − ++  , expressing your answer in the form p ln q + r, where p, q and r are exact constants in simplest form to be found. [3]

2 NYJC 2024 JC2 Preliminary Examination 9758/01 6 (a) An arithmetic series has first term a and common difference d, where d  0. The 1st, 6th and 14th terms of this series are the 1st, 2nd and 3rd terms of a geometric series. Find d in terms of a. [3] (b) A geometric series has first term b, where b > 0, and common ratio 0.5. (i) Find the sum to infinity of this series in terms of b. [1] (ii) Find the smallest possible value of n for which the sum of the first n terms of the series differs from the sum of the first 2n terms of the series by less than 0.004b. [4] 7 An isosceles triangle ABC, with AB = AC, is inscribed in a fixed circle of radius 1 unit and centre O. It is given that angle BOC = 2, where  is acute. Using calculus, show that the area of the triangle ABC is a maximum when it is equilateral. State the maximum value of this area exactly. [8] 8 A curve C has equation 1 ,y ax b xa= + + − where a and b are positive real constants such that 1a and .xa (a) Sketch C on the axes below stating the equations of any asymptotes and the coordinates of the point where C crosses the y-axis. [4] (b) On the same axes, sketch the graph of ( ) ( ) 22 22 ,x a y a b r− + − − = where r is a positive constant, such that it intersects C at more than 2 points. ( ) 222y r x a a b= − − + + [2] x y

NYJC 2024 JC2 Preliminary Examination 9758/01 (c) By stating the values of a and b, solve the inequality ( ) 212 1 16 2 5. 2xx x+ +  − − +− [3] 9 A curve R has equation 2 lny x x= for x > 0. R crosses the x-axis at point A and has a turning point at B. (a) State the coordinates of A. [1] (b) Find the exact coordinates of B. [3] (c) Find the exact area of the region bounded by R and the line segment AB. [5] 10 The functions f and g are defined by f : 2 1, , 1g : , , , x x x ax ax x xbx a b + − − where a and b are positive constants. (a) Find fg(1) in terms of a and b. You do not need to simplify your answer. [2] (b) Explain why the function gf does not exist. [1] (c) Find 1g ( ).x− [2] (d) Hence, or otherwise, find 2g ( ).x [1] (e) Given that a > 1 and b = 1, describe fully a sequence of transformations which transforms the curve ( )gyx= onto the curve 1 .y x= [4] 11 Ethylene is a gaseous plant hormone that plays an important role in inducing the ripening process for bananas. As a banana matures, ethylene is produced as a signal to induce fruit ripening. The amount of ethylene that a banana produces can be used to determine its ripeness. Scientists set up an experiment to measure the amount of ethylene produced by a banana. T he amount of ethylene, x parts per million (ppm), produced by a banana is modelled such that, at any time t hours, the rate of increase in the amount of ethylene is (160 )kx − , for some positive constant k. Initially no ethylene is present and it is found that there are 40 ppm of ethylene at time 12t= . (a) Write down and solve a differential equation involving x and t. Find the time it takes for the amount of ethylene to reach 100 ppm. [4] To reduce the amount of ethylene produced, ethylene absorbers can be used. The scientists repeat the experiment by adding ethylene absorbers at a rate of d ppm per hour. Initially, no ethylene is present and the rate of increase in the amount of ethylene, (160 )kx − , is the same as that found in part (a). (b) Write down and solve a differential equation, giving x in terms of t and d. [6] (c) Find the value of d for which the amount of ethylene will be kept at a maximum of 10 ppm in the long run. [2]

4 NYJC 2024 JC2 Preliminary Examination 9758/01 12 In a room, there is a standing mirror and a light source at D below the ceiling. The three points A, B and C on the mirror , and D have position vectors 23= − +a i j k , 6=− +b j k , 26=− − +c i j k and 58= − +d i j k respectively with respect to a fixed point of reference. (a) Show that the mirror can be modelled as a plane with cartesian equation 3 3 2 15x y z− + = . [2] (b) Find the exact position vector of point F, the foot of perpendicular from D to the mirror. [3] A light ray is sent in the direction of 6 2 13 − −− from D and hits the mirror at point E. (c) Find the acute angle between the light ray and the mirror. [2] (d) Show that the coordinates of E is 32, 2, 2  − . [2] (e) The light ray from D is reflected in the mirror. The reflected ray and the light ray lie in the same plane. The angle between the light ray and the mirror is the same as the angle between the reflected ray and the mirror. Find a vector equation of the reflected ray. [3] Credit: Image taken from https://www.fortytwo.sg/theola-standing-mirror-birch.html Ceiling D Light ray Standing mirror A B C