JPJC 2026 Prelim P1 Qn

Name:____________________________________ Class:_____________ JURONG PIONEER JUNIOR COLLEGE JC2 Preliminary Examination 2026 MATHEMATICS 9758/01 Higher 2 2 September 2026 Paper 1 3 hours Additional materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST This document consists of 6 printed pages and 2 blank pages. [Turn over Answer all 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 by [ ] at the end of each question or part question.

2 112 cm 6 cm 1 The curve C has equation 1 2cos , 0 2y x x     . Sketch the graph of C, stating the exact coordinates of the turning point(s) and where the curve meets the axes. [3] Hence find the set of values of k for which the equation 1 2cos x k  has at least 3 distinct roots in the interval 0 2 .x   [2] 2 Water is draining from an inverted right circular conical container with its vertex at the bottom. The container has a height of 12 cm and a base radius of 6 cm. Suppose water drains at a constant rate of 2 cm3/min, f ind the rate at which the water depth h is decreasing w hen h = 4 cm. [4] [It is given that the volume of a cone of base radius r and height h is 21 π3 r h .] 3 The diagram shows the region R which is bounded by the curve 225y x  , the lines 5y x  and 4x . (i) Calculate the area of R. [2] (ii) Find the volume of the solid of revolution obtained when R is rotated completely about the y-axis. [3]

3 4 A solid cylinder has radius r cm, height h cm and total surface area 1500 cm 2. Find the maximum volume of the cylinder in the form 10a  cm3, where a is a constant to be determined. [5] 5 (i) A cartesian equation of a plane is given by ax by cz d   , where a, b, c and d are constants. (a) State the significance of the vector a b c         in relation to the plane. [1] (b) If 2 2 2 1a b c   and 0d  , state what d represents. [1] (ii) (a) The non -zero vectors u and v are such that ( ) ( )   u v u v 0 . Determine the geometric relationship between u and v. [3] (b) If u and v are unit vectors which are perpendicular, state the magnitude of ( ) ( )  u v u v . [1] 6 It is given that 2 3i  is a root of the equation 3 2 7 0z az z b    , where a and b are real constants. (a) Find the values of a and b and the other two roots of the equation. [5] (b) Represent all the three roots on an Argand diagram. [2] (c) Hence find the area of triangle PQR, where P, Q and R are points that represent the roots. [1] 7 (a) A sequence 1 2 3, , , ...v v v is such that 120 17 14n nv v   , where n > 0 and 1 3v  . (i) Find the value of 15v , giving your answer to 3 decimal places. [1] (ii) Given that as n , nv L , find the exact value of L. [2] (b) It is given that 2 1 1 ( 1)(2 1)6 n r r n n n     . Find in terms of n, the sum of the series (i) 2 2 2 21 2 3 ........ 4 n    , [1] (ii) 2 2 2( 1) ( 2) ........ 4n n n     . [3] By using your answer to (b)(i), find in terms of n, the sum of the series 2 2 2 21 3 5 ........ (2 1) n     . [2] [Turn over

4 8 (a) Find (i) 2cos 2 d ,x x [2] (ii) 2 2 e d , e 2 x x x x  [2] (iii)   1 2tan dx x x  . [4] (b) By using the substitution 24cosx  , find the exact value of 3 2 1 d 4 x x x  . [5] 9 The curve C is defined by the parametric equations 2 3 2 1 9 49 23 4 , 3 2 , for 0 1, 2 2 18 3x t t y t t t t t          . (i) Find d d y x in terms of t. [2] (ii) Find the coordinates of the stationary point Q. Hence write down the equation of the tangent to C at Q. [3] (iii) The tangent at Q cuts C again at the point R. Find the equation of the normal to C at R. [4] (iv) The normal at R cuts the y-axis at the point N. Find the area of the triangle QRN. [2] 10 A line 1L has cartesian equation given by 4 , 1x z y   . Another line 2L is parallel to 2 1 3         and passes through the point (4, 3, )A c , where c is a constant. (a) Find the acute angle between 1L and 2L . [3] (b) Determine the possible values of c if 1L and 2L are skew. [3] (c) Show that the perpendicular distance from A to 1L is   21 82 c  . [2] A plane  has equation given by 1 0 0 1         r . (d) Find the value of c if A is equidistant from 1L and  . [3]

5 11 (i) The graphs of f ( )y x and f ( )y x  , where  df ( ) f ( )dx x x   , are shown in Figure 1 and Figure 2 respectively. Figure 1 Figure 2 On separate diagrams, sketch the graphs of (a) f ( )y x , [3] (b) 1 f ( )y x , [4] showing clearly the coordinates of any axial intercepts, stationary points, and the equations of any asymptotes. (ii) Each of the following two sequences of transformations can be used to transform the graph of g( )y x to the graph of g(3 1)y x  : Sequence A: A translation of a unit(s) in the negative direction of the x-axis, followed by a scaling parallel to the x-axis by a scale factor of b. Sequence B: A scaling parallel to the x-axis by a scale factor of c, followed by a translation of d unit(s) in the negative direction of the x-axis. Determine the values of a, b, c and d. [4] [Turn over x y O x y 4 O 1

6 12 The population growth of a certain species of fish in a farm is studied. The number of fish, t days after the study begins, is denoted by x (in thousands). It is found that the birth rate per day is twice of x and the death rate per day is proportional to 2x . (i) If there is no change in the population of fish when its population hits 8000, show that the differential equation relating x and t is given by 2d 2d 4 x x xt   . [2] (ii) The owner of the farm decides to sell 1750 fish daily. Modify the differential equation in part (i) to model the new situation. Hence, show that the resulting differential equation can be written as 2 2d 1 ( 4)d 4 x x at      , where a is a constant to be determined. [2] (iii) It is known that the initial population of the fish is 15 000. Solve the differential equation found in part (ii), expressing x in terms of t. [5] (iv) Hence sketch the part of the curve which is relevant in this context. Explain what happens to the number of fish in the farm in the long run. [3]

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