CJC 9758 2024 Promo Qns

9758/01/J1PROMO/2024 [Turn Over CATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC1 Promotional Examination CANDIDATE NAME CLASS INDEX NUMBER MATHEMATICS 9758/01 Paper 1 7 Oct 2024 3 hours Candidates answer on the Question Paper. Additional Materials: Li st of Formulae (MF27) READ THESE INSTRUCTIONS FIRST Write your class, index number and name on the work you hand in. Write in dark blue or black pen. You may use a 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 signifi cant 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. Question 1 2 3 4 5 6 7 8 9 10 11 12 Total Marks Total 4 5 5 6 6 7 7 10 12 12 13 13 100 This document consists of 28 printed pages, including this cover page.

2 9758/01/J1PROMO/2024 1 A curve C has equation  lnyp x q x r x  , where p , q and r are constants. Given that C crosses the x -axis at the points where 1x , 2x and 5x , find the values of p , q and r , giving your answers correct to 3 decimal places. [4] 2 By expressing   37 132 x xx   as a single simplified fraction, solve exactly the inequality   37 132 x xx   . [3] Hence, solve exactly the inequality   3e 7 1 e3 e2 x xx    . [2] 3 It is given that 1 11 111 n r rr n  . (a) Find  1 5 1 1 n r rr    . [3] (b) Give a reason why the series in part (a) is convergent and state the value of 5 1 1r rr    . [2] 4 A curve C has equation 2 44 1 xxy x   . (a) Sketch C , stating clearly the equations of any asym ptotes, coordinates of turning points and points of intersection with the axes. [3] (b) By drawing a suitable graph on the same diagram in part (a), find the range of values of b , where 0b , such that the equation  22 2 2 14 421 4 1 xxx bx      has no real roots. [3] 5 Differentiate each of the following expressions with respect to x . (a) 31tan 2 x  [3] (b) ln 1 x x    [3]

3 9758/01/J1PROMO/2024 [Turn Over 6 It is given that  sin e 1xy . (a) Show that 2 2 2 dd edd xyy yxx  . [3] (b) Hence, find the Maclaurin series of y, up to and including the term in 2x . [2] (c) Using the Maclaurin series found in part (b), determine the series expansion of  1s i n e 1x , up to and including the term in 2x . [2] 7 (a) Three non-zero vectors p , q and r are such that 3p×q =r×p . Find a linear relationship between p , q and r . [3] (b) Referred to the origin O , the points A and B have position vectors a and b . It is given that a and b are non-parallel unit vectors. The point B divides AC in the ratio 1:3 . The point D lies on OB produced such that :1 :OB OD m , where ,2mm . Given that AB  is perpendicular to CD   , find the numerical value of m . [4] 8 (a) An arithmetic sequence has first term a and common difference 0.5, where a is an integer. Find the smallest value of a such that the sum of the first 36 terms is at least 500. [3] (b) A sequence 012, , , uuu  is given by 0 400u  and 11.01nnuu x  for 1n , where x is an integer. (i) Show that    1.01 400 100 1.01 1nn nux . [3] (ii) Given that 16x and 1.01 kyu , where 01 6y , find the value of k and y . [4]

4 9758/01/J1PROMO/2024 9 (a) Express 248 4xx in the form of  2 A xB where A and B are constants to be determined. [1] Hence or otherwise, state a sequence of tran sformations that would transform the curve with equation 2 3exy onto the curve with equation 248 43e 10xxy  . [3] (b) The diagram shows the curve  fy x . The curve has a turning point at 6, 5 and crosses the x -axis at  4, 0 and  0, 0 . The lines 2y and 3x are the asymptotes to the curve. On separate diagrams, sketch the graphs of (i)  3fyx , [2] (ii)  1 ,fy x [3] (iii) f ' ,y x [3] labelling clearly the equation(s) of any asymptote(s), coordinates of any axial intercept(s) and turning point(s) where applicable. y 3x 2y (6, 5) (4 , 0 ) O x fyx

5 9758/01/J1PROMO/2024 [Turn Over 10 A cone-shaped cup is made of pape r of negligible thickness to hold 320π cm of liquid. The open inverted cone has radius r cm and height h cm as shown in the diagram above. The external surface area of the cup is denoted by 2 cmA . The manufacturer wants to reduce the cost of production by minimizing the value of A . (a) Find h in terms of r . [1] (b) By considering 2A or otherwise, show that 23 3 d 3600π 2d AAr rr   . [3] (c) Find the exact value of r that gives the minimum value of A , proving that A is a minimum. Find also the ratio of the radius to the height, r h , giving your answer in terms of 2k , where k is a constant to be determined. [5] The manufacturer decides to make paper cups at minimum external surface area using the ratio r h found in part (c). (d) The cup is being filled completely with water. However, there is a small hole at the bottom of the cup that causes water to leak out at the rate of 33 cm per second. Find the rate of decrease of the depth of the water at the instant when the depth is 2 cm. [3] [It is given that the volume of a circular cone with base radius r and height h is 21 π3 rh and the curved surface area is πrl , where l is the slant height of the cone.] r h

6 9758/01/J1PROMO/2024 11 A function h is self-inverse if     1hhx x for all x in the domain of h . The functions f and g are defined by  1f2 , f o r , 2 2xx x x      2 31 f o r 1 , g= 12 f o r1 . xx x xx    (a) Sketch the graph of fy x . With the aid of your graph, explain why f has an inverse. [2] (b) Show that f is self-inverse and find   2f x . [4] (c) Hence, or otherwise, evaluate   2025f4 . [2] (d) Find an expression for gf and state its domain. [3] (e) Find the range of gf . [2] 12 The diagram below shows a triangul ar base pyramid with vertices, A , B , C and D . With reference to the origin O, the points A , B , C and D are 23ij k , 24ij k , 45ij k and 25ik respectively. Let the plane containing points A , B and C be represented by  . (a) Show that the equation of  can be expressed as 9 7 12    r , where  is a constant to be determined. [3] (b) Find the acute angle between line BD and  . [2] (c) Find the position vector of the foot of perpendicular from the point D to  . [4] (d) Find the area of triangle ABC . Hence find the volume of the pyramid. [4] [Volume of pyramid 1 base area height3  ] A B C D

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