CJC 2025 Prelim P1 QP

9758/01/J2PRELIM/2025 [Turn Over CATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination CANDIDATE NAME CLASS INDEX NUMBER MATHEMATICS 9758/01 Paper 1 02 Sep 2025 3 hours 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 must 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. This document consists of 5 printed pages and 1 blank page.

2 9758/01/J2PRELIM/2025 1 The diagram above shows the curve C with equation     2 2 19 x p y q r     , where p, q and r are constants. The vertices are  5, 3 and  1, 3 and the asymptotes are 3 15 2 2y x  and 3 3 .2 2y x  (a) Find the values of p, q and r. [3] (b) The curve D has equation     2 2 3 3x m y m    , where m is a positive constant. Find the range of values of m for which curves C and D do not intersect. [2] 2 The nth term of a sequence is given by 2 nu an bn c   . The first three terms of this sequence of numbers are 2, 6 and 12. (a) Find the values of a, b and c. [3] (b) Given that 1 1 1 1 1 N n nu N    , find 8 1 n nu    . [3] 3 A curve C has parametric equations 2 π πsin , 1 2sin , . 2 2x t y t t      (a) Sketch C, stating clearly the coordinates of the endpoints and the coordinates of the y-intercept(s). [2] (b) Find the exact cartesian equation of l, the normal to C at the point 1 , 24      . [4] x y O (5, 3) (1, 3)

3 9758/01/J2PRELIM/2025 [Turn Over 4 It is given that   ππ π f π 22 for 0 sin for 2 x xx x x x       and that    f πf 2x x  for all real values of x. (a) Sketch the graph of  fy x for π π 4x   . You do not need to label the coordinates of any stationary point(s). [3] (b) Find the exact area bounded by the curve  fy x , the x-axis and the lines 0x and 3 .2 πx [4] 5 A sequence of negative real numbers 1 2 3, , ,u u u  is given by 1 1u  and 1 2 6 1 n n n uu u     , for 1n  . (a) Given that nu l as n , find the value of l. [3] (b) Sketch the graph of 2 6 1 xy x x     , stating the equations of the asymptotes and the coordinates of the points where it crosses the axes. [3] (c) Hence show that 1n nu u  if nu l . [2] 6 Relative to the origin O, the position vectors of points A, B and C are a, b and c respectively. It is given that a is a unit vector and b is perpendicular to b a . (a) Show that 0 1 b . [3] It is given further that 1 2b and C lies on AB such that : 2 :1AC CB  . (b) Find the value of a c and state the geometrical interpretation of a c . [4] (c) Find the value of   a b a c . [2] 7 It is given that the curve  fy x satisfies the equation   1πln tan e4 xy   . (a) Show that   2 d1 e e 0 d x x y yx   . [2] (b) Find the Maclaurin series for y, up to and including the term in 2x . [5] (c) Deduce the series expansion for   1π tan e4e 1 x x   , up to and including the term in 2x . [4]

4 9758/01/J2PRELIM/2025 8 The plane p passes through the points A, B and C with coordinates  1 ,0,2 ,  2, 1,3 and  4, 1 ,0  respectively. (a) Show that a cartesian equation of p is 2 3x y z   . [3] The line l has equation 2 2 0 1 , 5 1                        r  . (b) Find the acute angle between l and p. [2] It is given that a variable point R lies on p and is at a distance of 22 from the point Q with coordinates  3, 4, 2 . (c) Find the foot of perpendicular from Q to p. [4] (d) Hence describe geometrically the path traced by R. [2] 9 A Koi pond farmer introduces a circular patch of duckweed at the center of his circular pond to enhance water purification and provide supplemental nutrition for his Koi , as shown in the figure below. The duckweed grows over time t days, forming a continuous circular patch on the pond's surface. It is a ssumed that the duckweed forms a thin, floating layer with negligible thickness . The rate of increase of the area covered by duckweed is directly proportional to the area of the pond that remains uncovered by the duckweed. (a) Given that the radius of the pond is 10 m, show that the radius, r m of the circular patch of duckweed satisfies the differential equation   2100d , 0d 2 k rr kt r    . [2] (b) The initial radius of the circular patch of duckweed is 1 m. Solve the differential equation, expressing r in the form e ktr P Q   , where P and Q are constants to be determined. [6] (c) Sketch the graph of r against t. [3] Circular pond Circular patch of duckweed Centre of both pond and duckweed 10 r

5 9758/01/J2PRELIM/2025 [Turn Over 10 Mr Huat is saving for a car that he plans to buy in the future. He needs to save a minimum of $150,000. A savings plan allows him to deposit $6,000 into a savings account on the first day of every month. At the end of each month, the total amount in the savings account (including interest) is increased by 3%. Mr Huat makes an initial deposit on 1 January 2026. (a) Show that the total amount, correct to 2 decimal places, in the savings account at the end of December 2026 is $87,706.74. [2] (b) Find the month and year in which the total amount in the savings account will first exceed $150,000. Explain whether this occurs on the first or last day of the month. [6] Mr Huat can also choose to save regularly in a different savings account which offers no interest. He makes an initial deposit on 1 January 2026 of $6,000. On the first day of each subsequent month, he deposits $d more than he deposited in the previous month. (c) Find, in terms of d, the total amount in the savings account at the end of December 2026. [2] (d) Hence, find the minimum value of d such that Mr Huat can buy the car by end of December 2026. Give your answer correct to the nearest integer. [2] 11 (a) (i) Show that    2 2 3 1 3 1 x x x x   can be expressed as 21 3 1 A Bx C x x x    , where A, B and C are constants to be determined. [2] (ii) Hence, find    2 2 3 d 1 3 1 x x x x x      . [4] (b) The region R is bounded by the curve    2 2 3 1 3 1 xy x x x     , the line 4x and the x-axis. (i) Sketch the graph of    2 2 3 1 3 1 xy x x x     for 0x , giving the coordinates of any intercepts with the axes and the equations of any asymptotes. Shade the region R on your sketch. [3] (ii) Find the exact area of R. [3] (iii) Find the volume of the solid generated when R is rotated through 2π radians about the x-axis. [2]

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