RVHS 9758 2023 Prelim P1

1 ©RIVER VALLEY HIGH SCHOOL 9758/01/2023 RIVER VALLEY HIGH SCHOOL 2023 JC2 Preliminary Examination Higher 2 NAME CLASS INDEX NUMBER MATHEMATICS Paper 1 Candidates answer on the Question Paper Additional Materials: List of Formulae (MF26) 9758/01 15 Sep 2023 3 hours READ THESE INSTRUCTIONS FIRST This document consists of 25 printed pages and 3 blank pages. For examiner’s use only Question number Mark 1 2 3 4 5 6 7 8 9 10 11 12 Total Calculator Model: Write your class, index number and name 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. Write 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.

2 ©RIVER VALLEY HIGH SCHOOL 9758/01/2023 1 (i) Without using a calculator, solve 2 2 4 41 012 xx xx −+ <+− . [3] (ii) Hence solve ( ) 21 21 21 012 2 x xx + + − ≤+− . [3] 2 (i) The complex number iza= + , where a is a real constant, is such that the modulus of ( ) 1 *1 i z z − + is 1 2. Find the value of a where 2a< . [3] (ii) It is given that a is the value found in part (i). The complex number w has argument 4 π. Find the 3 smallest positive integer of n such that the complex number 1 * n z zw −  is purely imaginary with the imaginary part negative. [3] 3 A sequence 1 23, , , ...uuu is such that 1 nu n= for 1n≥ . (i) Show that for 2n≥ , 11 32n nn Au uu nn −+ −+= − , where A is a constant to be found. [2] (ii) Hence find 3 2 1N n nn= −∑ . (You need not express the answer as a single fraction.) [2] (iii) Explain why 3 2 1N n nn= −∑ converges. [1] (iv) Use your result in part (ii) to find ( )( )( )3 1 21 N n n nn= −−∑ . [2] 4 It is known that 0x is one of the roots of the equation 432 46 0x x x ax b− + − += , where a and b are real. (i) Show algebraically that 0 *x is also a root. [1] (ii) Given that 0 2ix = − , find the values of a and b and the other roots. [4] (iii) Hence, solve 43 2 6 4 10by ay y y− + − += . [2]

3 ©RIVER VALLEY HIGH SCHOOL 9758/01/2023 5 OACB is a square with origin O. Points A and B are such that OA= a    and OB= b   . M and N are midpoints of OB and AC respectively. S is on AC such that : 1:4AS AC = . MS produced and BN produced meet at point T. (i) Find the position vector OS    in terms of a and b. [2] (ii) Find vector equations of the line MS and the line BN. [3] (iii) By finding the position vector of T , deduce that T , A and O are collinear points. [2] A O C B - N M - -- S T

4 ©RIVER VALLEY HIGH SCHOOL 9758/01/2023 6 A function f is said to be self-inverse if 1f () f () .xx −= Show that f given by f : ,where , ,ax bx x xaxa + ∈≠− for some constants ,,ab ∈  is a self-inverse function. [2] State a geometrical relationship between 1f () a n d f () .yx y x −= = [1] The graph below shows the function g defined for x∈  . The equation of its asymptotes are y = 0 and y = e and it passes through the points (a, c) and (0, d ). Determine if gf exists. If it exists, give its domain and range. [4] 7 (a) An arithmetic series has non-zero first term a and exactly n terms, where n is odd. Given that the sum of the first and middle terms equals the last term, find the middle term in terms of a. [3] (b) A convergent geometric series with positive terms has first term 33p and common ratio 2 p q ,where 0q≠ . (i) If the nth term is m , show that ( ) ( )ln ln 3 ln 2 lnm nA p B n q= ++ +− , where A and B are constants to be determined. [2] (ii) Suppose 2qp= , find the range of values of p. [3] 8 The graph of a function, ( )fyx= , cuts the y-axis at 1 and has a gradient of 1− at the same point. (i) Given that ( ) ( ) ( ) 21 f '' f ' 0x xxx− −= , find the first three non-zero terms in the Maclaurin series for ( )f x . [4] (ii) Hence, find the series expansion of ( ) ( ) f ' f x x up to and including the term in 2x . [2] (iii) Deduce the series expansion of ( )ln f x up to and including the term in 2x . [2] x y y = e 0 (0, d) (a, c)

5 ©RIVER VALLEY HIGH SCHOOL 9758/01/2023 9 The curve C has equation 2 14ax bxy xc +−= + . It is given that the equations of the asymptotes of C are 3x= and 5yx= + . (i) Determine the values of a, b and c. [3] (ii) Sketch the graph of C indicating clearly the equations of the asymptotes, coordinates of axial intercepts and stationary points. [3] (iii) On the same axes, sketch the graph of ( ) ( ) 2 2 2 3 81x ym − −− = , where m is a positive real constant, indicating clearly the equations of the asymptotes [2] (iv) Hence state the range of values for m for which the equation ( ) 22 2 2 3 6 10 13 x xx mx −  −+−= − has roots. [1] 10 The curve C has parametric equations 24xt= + , 2yt= , 0t≥ . (i) Find the equation of the tangent to curve C at point P ( ) 224, pp+ . [3] (ii) The tangent at P meets the x -axis at point Q . Point M is the midpoint of the line segment PQ. Find the cartesian equation of the curve traced out by M as p varies. [3] (iii) Show that the area of the region bounded by C, the x-axis and the line x = 3 is given by 3 2 d 4 b a t t t+ ⌠ ⌡ , where a and b are constants to be determined. Hence eva luate the integral to find the exact area of the region. [4] 11 The Law of Universal Gravitation states that the force F causing any two bodies A and B with constant masses to be attracted toward each other, can be expressed as 2 kF r= , where r is the distance between them and k is a positive constant. Body A is situated at the point ( )1, 0 while body B is in motion along the path described by the equation 2 2 14 x y+= . The bodies A and B have constant masses, and F denotes the gravitational force between them. (i) Find an expression for F in terms of k and x. [2] (ii) Using differentiation, find the maximum value of F as x varies, leaving your answer in terms of k. [5] (iii) S