CJC 2026 Prelim P1 QP

9758/01/J2PRELIM/2026 [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 01 Sep 2026 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (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/2026 1 The curve  fy x undergoes in succession, the following transformations: A: a translation in the positive x direction by 1 unit B: a scaling parallel to the x axis by factor 2 C: a reflection in the y axis The equation of the resulting curve is 2 2y x  . Obtain the equation of the original curve  fy x . [3] 2 (a) Differentiate  1tan 2 cos3 x with respect to x . [2] (b) Hence, find the exact value of   9 2 18 sin 3 d 2 1 4cos 3 x x x    . [3] 3 The curve C has equation . (a) Sketch C, stating the coordinates of any points of intersection with the axes. [3] (b) State the equations of the lines of symmetry of C. [1] (c) State the range of values of x for which C exists. [1] 4 The diagram below shows the graph of  fy x . It intersects the x-axis at the points (a, 0) and (d, 0) and the y-axis at the point (0, e). It has a minimum point at the point (b, c) and a horizontal asymptote with equation y = p. On separate diagrams, draw sketches of the graphs of (a)  fy x , [2] (b)   1 fy x , [3] (c)  f 'y x . [2] State in each sketch, in terms of a, b, c, d, e and p, the equations of the asymptotes, the coordinates of the points where the graphs cross the axes, and the turning points, where applicable.   2 21 19 25x y  

3 9758/01/J2PRELIM/2026 [Turn Over 5 The region P is bounded by the curve     2 5 9 4y x   and the line 4x . (a) Find the exact area of the region P, leaving your answer in the form 2k where k is a constant to be determined. [4] The region Q is bounded by the curve     2 5 9 4y x   , the line 5x and the y-axis. (b) Find the volume of solid of revolution formed when the region Q is rotated 2 radians about the y-axis, giving your answer correct to 2 decimal places. [3] 6 A curve C has equation 2 2x xy x m   , where 2m . (a) Find, in terms of m, the equations of asymptotes of C. [2] (b) Given that C has a turning point at (1,1), show that m = 3. Hence, sketch C, labelling clearly the equations of asymptotes, and the coordinates of the turning points and the points where C crosses the axes. [3] (c) By inserting a suitable graph, solve 2 2 92 4xx x    for 3x . [3] 7 Fig. 1 shows a right circular cone with a base radius of 8 cm and a height of H cm. A smaller right circular cone with a base radius of r cm is removed from the cone in Fig. 1 along a plane parallel to the base, resulting in a frustrum as shown in Fig. 2. The variables r and h are related such that 9r h  where 0 8 r  . 8 cm r cm h cm Fig. 2 Fig. 1 8 cm H cm

4 9758/01/J2PRELIM/2026 (a) Show that the volume of the frustum is   21 9 8 643V r r r    cm3. [4] (b) Using differentiation, find the maximum volume of the frustum, leaving your answer in terms of  . [5] [The volume of a cone of base radius r and height h is given by 213V r h  .] 8 The curve D is defined by the parametric equations 3x t t  and 4 2y t t  where t  . (a) What can be said about the tangent at 0t  ? [2] (b) Find the equation of the normal to D at the point  3 4 2 ,P p p p p  , leaving your answer in the form y mx c  . (You do not need to simplify your answer.) [3] The point M is the midpoint of OP where O is the origin. (c) Find a cartesian equation of the curve traced by M as p varies, leaving your answer in the form 4 2 3 0Ax Bx y Cy   where A, B and C are integers to be determined. [4] 9 The sum of the first n terms of a sequence 1 2 3, , ,u u u  is given by 3nS n n  . (a) Show that 23 3nu n n  , 1n . [2] (b) Find a recurrence relation for this sequence, giving the answer in the form  1 fn nu u  . [3] (c) Using   2 1 1 2 16 n r nr n n     , find 2 5 n r r u   simplifying the final answer in the form 3 2an bn cn d   , where a, b, c and d are constants to be determined. [5] 10 (a) A curve C has equation 2cos 2y x y   , 0y . (i) Show that   dcos 2 sin dyx y y x x  . [2] (ii) Find the Maclaurin series for y in ascending powers of x , up to and including the term in 2x . [4] (b) In triangle ABC, 6 cmBC  , radiansABC   and radians3BAC        . A B C 6 cm

5 9758/01/J2PRELIM/2026 [Turn Over Given that  is a sufficiently small angle, show that 2AC p q r     where p, q and r are exact constants to be determined. [5] 11 The function f is defined by   2 f : , , , x x k x x k    where k is a positive constant. (a) Find  1f x and state its domain. [4] (b) Sketch on the same diagram, the graphs of  fy x and  1fy x  , giving the coordinates of the end points. Your sketch should indicate clearly the geometrical relationship between the graphs of f and 1f  . [3] The function g is defined by g : xx e , x  (c) Explain why the composite function gf exists. [1] (d) Find the range of gf. [2] (e) Without first finding any inverse and composite functions, determine  1f 1g  in terms of k. [2] 12 Scientists are interested in the rate of release of a particular drug from a capsule. The capsule initially contains 100 milligrams (mg) of the drug. When 3 hours have elapsed, half of the drug has been released. Let M be the mass, in mg, of drug released into a human body t hours after the capsule is consumed. A team of tertiary students, Team A, proposes that the rate of change of M is directly proportional to the time elapsed after the consumption of the capsule. (a) Write down a differential equation for this model and solve it for M in terms of t. [4] A team of medical students, Team B, proposes a different model which they think is an improvement to the model proposed by Team A . In this model, the rate of change of M is directly proportional to the product of the mass of drug remaining inside the capsule and the reciprocal of 1t given by  100d d 1 k MM t t   where k is a positive constant. Given that when 3 hours have elapsed, the drug is being released at the rate of 25 ln 22 mg/h and half of the drug has been released. (b) Solve the differential equation proposed by Team B, expressing M in terms of t. [7] (c) Using the answer in (b), sketch the graph of M against t. [2] (d) Explain whether the model proposed by Team B is an improvement of the model proposed by Team A. [1]

6 9758/01/J2PRELIM/2026 BLANK PAGE