ASRJC Prelims 2025 H2 Math P1 Questions

ANDERSON SERANGOON JUNIOR COLLEGE H2 MATHEMATICS Preliminary Examination Paper 1 9758/01 3 hours Additional Material(s): Printed Answer Booklet 25 Aug 2025 CANDIDATE NAME CLASS / READ THESE INSTRUCTIONS FIRST 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 in brackets [ ] at the end of each question or part question.

2 O B C A r This document consists of 6 printed pages and 2 blank pages. 1 A cubic curve passes through the points (2, 3) and ( 3, 22)−− . Find the equation of the curve if it has a stationary point at (1, 6)− . [4] 2 (a) Show that 3cos3 4cos 3cosθ θθ= − . [3] (b) Hence, or otherwise, evaluate 6 0 sin 2 cos3 d π θ θθ∫ exactly. [4] 3 (a) Given that θ is small, show that 21sec 1 2θθ≈+ . [2] (b) The diagram below shows a circle, centre O and radius r , with points A and B on the circumference such that radians AOB θ∠= , where θ is small. AC is a tangent to the circle at A and OBC is a straight line. (i) Show that the length of chord AB can be approximated by rθ . [2] (ii) Hence show that the perimeter of triangle ABC can be approximated by ( )r abθθ + , where a and b are constants to be determined. [4] 4 The Folium of Descartes is a curve, defined by the equation 33 3x y axy+= , where a is a real constant. It is given that a ≠ 0 for this question. (a) Show that 2 2 d d y ay x x y ax −= − . [2] (b) The point 33 ,22 aa  lies on this curve. Show that the equation of the normal at this point on the curve is independent of the value of a. [2] (c) Given that the curve has a stationary point at (,)pa qa , where p and q are positive constants, find the exact values of p and q. You need not determine the nature of this stationary point. [4]

3 [Turn Over 5 The curve C has equation given by y = 21 3 18 93 xx x +− + . (a) Without the use of a calculator, find the range of values that y can take. [4] (b) Sketch the graph of C indicating clearly its asymptotes and stationary points. [3] (c) State a sequence of transformations that will transform the curve C to the curve with equation y = 21 3(2 1) 18(2 1) 9(2 1) 3 xx x + −− − −+ . [2] 6 It is given that 1 74 9 2 5 ( 1)( 2) 2 1 2 n r r rr r n n= + = −−++ + +∑ . (a) Find 2 3 7 73n r r rr= − −∑ giving your answers in terms of n. [4] (b) Show algebraically that 3( 1) ( 1)( 2)r rr r+>+ + for all positive integers r. [2] (c) Hence show that 3 1 749 ( 1) 2 n r r r= + <+∑ . [3]

4 7 Curve C is a circle with radius 2 and center at the origin with equation 22 4xy+= . Line L has the equation 32yx= + . The diagram below shows the shaded region A which is enclosed between C and L. (a) Use the substitution 2sinx θ= to show that the area of region A can be written in the expression 24cos d a b cθ θ −∫ , where a, b and c are exact constants to be determined. Hence evaluate this area exactly. [6] (b) The region B is bounded by the curve C, the line L and the line 2x= . Find the volume generated when region B is rotated through 2π radians about the y-axis. Give your answer to two decimal places. [3] 8 (a) (i) The ninth, fifth and second terms of an arithmetic progression are successive terms of a geometric progression with first term a and common ratio r, where 1r ≠ and 0a> . Find the value of r and deduce that the geometric series is convergent. [3] (ii) Using the value of r found in (i), find the least value of n such that the sum of all the terms after the nth term of the geometric progression is less than 1% of its sum of the first n terms. [3] (b) The sum, ,nS of the first n terms of another sequence is given by ( 1)ln 2qn nSn += , where q is a constant. Prove that the sequence follows an arithmetic progression. [4] y x

5 [Turn Over 9 The line  has the equation 1222 xa z y b −− =−= , where a and b are real constants. T he plane π has the equation 3 4 10xy z−+ = . (a) Given that the line  and the plane π do not intersect, show that 10 3a≠ and 5 2b= . [5] (b) It is given that 1a= and 3b= . Find the point of intersection between the line  and the plane π. [3] (c) It is given now that 1a= , 5 2b= . (i) Find the distance between the line ℓ and the plane π. [2] (ii) Determine whether the line  and the origin O lie on the same side of the plane π. State an equation of the other plane that is equidistant from line  and parallel to plane π. [3] 10 The function f is defined by f : x → 12 2x x− , 02 x<< . It is given that 1f− exists. (a) Define 1f− in a similar form. [3] (b) Sketch the graphs of f( )yx= and 1f ()yx −= on the same diagram. [2] (c) The region R is bounded by the curve 1f ( ),yx −= yx= and the y-axis. Find the exact area of R. [3] (d) Another function g is defined by 2 33 for 3,2 3 11g: 1 for 3 5.3 xxx xx x  +≤ −→  − <≤ Show that the composite function gf exists and find the range of gf. [2]

6 11 In a chemical reaction, compounds X and Y react to form a product. Let x and y denote the concentrations, in mol per kilolitres (mol/kL), of X and Y respectively, at time t minutes after the start of the experiment. The initial concentrations of X and Y are given by 0x and 0y mol/kL. (a) In a particular experiment where Y is present in excess, the reaction rate can be modeled as a pseudo-first-order reaction, leading to the differential equation d d x axt =− , where a is a positive constant. (i) Solve this differential equation, expressing x in terms of t, 0x and a. [3] (ii) The half-life of the reaction, denoted by 0.5t , is defined as the time taken for the concentration of X to decrease to half its initial value. Show that 0.5 ln 2t a= . [2] (b) In another experiment conducted by a chemist, the rate of the reaction is directly proportional to the product of the concentration of X and the square of the concentration of Y. It is also known that at any instance during the reaction, every 1 mol of X reacts with every 2 mol of Y, giving the equation 00 2( )yy xx−= − . The initial concentrations of X and Y are 1 mol/kL and 4 mol/kL respectively. (i) Show that 2d ( 1)d x bx xt = −+ , where b is a positive constant. [2] (ii) It is given that the concentration of X is 0.5 mol/kL at the instance 1 min after the start of the experiment. Find the concentration of X at the instance 2 min after the start of the experiment, giving your answer to 3 significant figures. [6]

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