MI 9758 2024 Promo PU1

© Millennia Institute 9758/01/PU1/EOY/24 [Turn over 2024 End-of-Year Examination Pre-University 1 MATHEMATICS 9758/01 Paper 1 14 October 2024 QUESTION PAPER 2 hours 15 minutes Additional Materials: Printed Answer Booklet List of Formulae (MF27) READ THESE INSTRUCTIONS FIRST Answer all the questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. 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 must show all necessary working clearly. 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 75. This document consists of 6 printed pages. H CANDIDATE NAME CLASS ADMISSION NUMBER

2 © Millennia Institute 9758/01/PU1/EOY/24 1 The sum, nS , of the first n terms of the sequence 1 2 3, , , ...u u u is given by 2 nS n kn=+ , where k is a non-zero constant. (i) Find nu in terms of n and k. [2] (ii) Hence determine if the sequence is an arithmetic progression. [2] 2 (i) Without using a calculator, solve the inequality 2 32 24 xx x −+ + . [4] (ii) Hence solve the inequality 2 32 24 xx x ++ − . [2] 3 (i) A curve C has equation 32y ax bx cx d= + + + , where a, b, c and d are constants. The curve C intersects the y-axis at the point (0, 7) and passes through the points ( 2, 11)−− , (1, 1) and (3, 19) . Find the values of a, b, c and d. [4] (ii) C first undergoes a reflection about the y-axis and subsequently a translation of 1 unit in the positive y-direction. Find the equation of the resulting curve. [2] 4 Referred to the origin O, the points A, B and C have position vectors a, b and c. C lies on AB such that : 1:3AC AB = . (i) Find c in terms of a and b. [2] (ii) Show that the area of triangle OAC is given by k ab , where k is a constant to be determined. [2] It is now given that a and b are unit vectors. (iii) Using the result in part (i), find the value of ( )2 −a b c . [3]

3 © Millennia Institute 9758/01/PU1/EOY/24 [Turn over 5 (a) Describe a sequence of 2 transformations that transforms the curve C with equation 2 2 19 x y+= onto the curve D with equation ( ) 2 221xy− + = . [2] (b) The diagram shows the curve with equation f ( )yx= . The curve passes through the origin and the point (2, 0) and has a minimum point at (3, 4)− . The asymptotes of the curve are the lines 1, 0xy== and 2y= . Sketch the graph of 1 f ( )y x= , indicating the equations of any asymptotes and the coordinates of any points where it crosses the axes and of any turning points, where applicable. [3] (c) The graph of g( )yx= is a semi-circle with its centre at the point ( , 0)a , where 0a . The diagram shows the graph of ( )gyx= , which intersects the axes at the points ( ,0)b− , ( ,0)b and (0, )c , where ab− . Sketch the graph of g( )yx= , labelling the coordinates of the centre and the radius of the semi-circle, in terms of a and/or b. [3] y x O 2 0y= O y x b b− c f ( )yx= ( )gyx=

4 © Millennia Institute 9758/01/PU1/EOY/24 6 The curve C has equation 2x bx by xa −+= − , where a and b are constants. C has a vertical asymptote at 1x= and a turning point at (0, 2)− . (i) Determine the values of a and b. [2] Use the values of a and b found in part (i) for the following parts. (ii) Sketch C, stating the coordinates of any points of intersection with the axes and of any turning points and the equations of the asymptotes. [4] (iii) By adding a suitable curve to your diagram in part (ii), solve the inequality 2 ln(5 ) 2x bx b xxa −+ − − − . [3] 7 The functions f and g are defined by 2f : 5 ( 2) , for , 0, g : e + 3, for .x x x x x xx − −    (i) Show that the composite function fg exists. [2] (ii) Find an expression for ( )fg x and state its domain. [2] (iii) Find the range of fg. [2] (iv) Show that 1f − does not exist. [1] (v) If the domain of f is restricted to ),k  , find the smallest value of k such that 1f − exists. [1] For the rest of the question, the domain of f is now restricted to ),k  , with the value of k found in part (v). (vi) Find ( ) 1f x− and state the domain of 1f − . [3]

5 © Millennia Institute 9758/01/PU1/EOY/24 [Turn over 8 An arithmetic series has first term a and common difference d, where a is non-zero. The 8th, 4th and 2nd term of the arithmetic series are the first three consecutive terms of a geometric series respectively. It is also given that the terms of the arithmetic series are increasing. (i) Show that ad= . [4] It is given that the sum of the first 20 terms of the arithmetic series is 630. (ii) Find the value of a. [2] (iii) Hence find the exact sum of the first 10 terms of the geometric series. [3] (iv) Give a reason why the geometric series converges and find the sum to infinity of this series. [2] 9 To construct a special tentage, 3 steel poles OR, AQ and BP are secured to the horizontal ground such that each of the poles are perpendicular to the ground . The point O on the ground is taken as the origin and the unit vectors i, j and k are parallel to OA, OB and OR respectively, with units in metres. It is given that 12OR= , 9AQ= , 6BP= , 20OA= and 15OB= (see diagram). (i) Find PR and PQ . [2] (ii) The canvas roof PQR, which is of negligible thickness, is part of the plane p. Using your answers in part (i), show that the equation of p is given by 3 8 240 20   = r . [2] i j k O A Q P B R 15 m 20 m 12 m 9 m 6 m

6 © Millennia Institute 9758/01/PU1/EOY/24 (iii) Find the acute angle between the canvas roof and the plane OAB. [2] (iv) A cable is to be installed close to the tentage. This cable is part of the line L with vector equation 2 20 8 13 , where 12 4  −       = + −        −    r . Show that this cable is not parallel to the edge PR of the roof and does not meet th e edge PR. [4] (v) A lamp is hung vertically downwards from the canvas roof using a cable such that it is at the point M with coordinates 20 , 5, 83   . To secure the lamp, a second cable is connected from the lamp to the point N on the roof. It is a ssumed that the cables are laid in straight lines and have negligible thickness. For the shortest length to be used for the second cable, show that 2525 473 s ON t   =  , where s and t are constants to be found. [3] End of Paper