2024 JC2 H2 Math prelim - MI

1 © Millennia Institute 9758/01/PU3/Prelim/24 2024 Preliminary Examination Pre-University 3 MATHEMATICS 9758/01 Paper 1 9 September 2024 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your admission number, name and class on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. A nswer all the questions. Give your answers in the spaces provided in the Question Paper. 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 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. T he number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 100. This document consists of 7 printed pages. Qn No. Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 * Total Score Max Score 5 4 7 8 9 9 11 11 11 13 12 100 CANDIDATE NAME ADMISSION NUMBER CLASS

2 © Millennia Institute 9758/01/PU3/Prelim/24 1 (i) A quadratic curve passes through the point ( )1, 4−− and has its turning point at ( )2,5 . Find the equation of the curve. [4] (ii) Given instead that a cubic curve passes through the same point ( )1, 4−− and has the same turning point as stated in part (i). Explain whether it is possible to obtain a unique equation of the curve based on given information. [1] 2 Use the substitution 5xu= to find ( ) 25 sin 5 dxx x∫ . [4] 3 A line l has equation 4eyx= , 0x≥ and curve C has equation 2 2e xyx= , 0x≥ . Both l and C intersect at the points with coordinates (0, 0) and 4(2, 2e ) as shown in the diagram below. (a) The region R is bounded by the line l , curve C, and the lines 1x= and 3x= . Find, correct to 4 significant figures, the area of region R. [3] (b) The region S is bounded by the line l and curve C. Show that the volume V of the solid formed when S is rotated 2π radians about the x -axis i s ( ) 8e6V ABπ= + , where A and B are exact constants to be determined. [4] x y O

3 © Millennia Institute 9758/01/PU3/Prelim/24 [Turn over 4 The equation of a curve is ( ) 2 21 .xy y x++ = (i) Find the equations of the two tangents which are parallel to the y -axis. [4] (ii) The normal to the curve at the point A ( )4, 5− meets the curve again at the point B. Find the coordinates of point B. [4] 5 The diagram below shows the curve of f( )yx= . The curve cuts the axes at ( )6,0− , ( )1, 0 , and ( )0, 2− . It has a minimum point at ( )2, 6−− . There is a horizontal asymptote at 3y= and a vertical asymptote at 4x= . On separate diagrams, sketch the following graphs, stating the equations of any asymptotes, the coordinates of any turning points and axial intercepts . (i) ( )fyx= − [3] (ii) ( )f'yx= [3] (iii) ( ) 1 fy x= [3]

4 © Millennia Institute 9758/01/PU3/Prelim/24 6 (a) The sum, nS , of the first n terms of a sequence 123, , ,...uu u is given by 22nS n An= −+ for 1n≥ , where A is a non-zero constant. (i) Find an expression for nu in terms of n and A. [2] (ii) Hence, determine if the sequence is an arithmetic progression. [2] (iii) Describe how the sequence of sums 123, , ,...SS S behaves when 20A= . [1] (b) A geometric progression has first term 7 and common ratio r. The sum of the first 15 terms of the progression is 28. (i) Show that 15 4 30rr − += . Explain why the common ratio cannot be 1 even though 1r = is a root of this equation. [2] (ii) Given that 1r < , find the sum to infinity, giving your answer correct to 2 decimal places. [2] 7 Functions f and g are defined respectively by 2 2 ( 2)f : , , 1, 1 1 xx xx x x − ∈ ≠− ≠− 1 , , 0, 4g: xx xx x− ∈ >≠ (i) Show that the composite function fg exists. [2] (ii) Find the range of fg. [2] (iii) Explain why f does not have an inverse. [2] (iv) If the domain of f is further restricted to xk≥ , state the least value of k for which the function 1f− exists. [1] (v) For this restricted domain, find 1f () x− and state the domain of 1f− . [4]

5 © Millennia Institute 9758/01/PU3/Prelim/24 [Turn over 8 Do not use a calculator in answering this question. (a) The complex numbers z and w satisfy the following equations. It is known that w is not purely imaginary. 4 1 6i * 2 3 8i zw wz += + −= − Find z and w, giving your answers in the form iab+ , where a and b are real numbers. [5] (b) Two complex numbers are 1 2 3 6iz = −+ and i32 8ez π− = . (i) Find 1 3 2 z z in the form ( )cos isinr θθ+ , where 0r > and πθπ−<≤ . [4] (ii) It is known that 1z , 2z and ( ) 3 2z are roots of a polynomial equation of degree n with real coefficients. Explain why the smallest possible value of n is 5. [2] 9 (a) (i) Verify that ( ) ( ) ( ) 11 1 ! 1! 1! 1rr r r−= + −+ . [1] (ii) Hence find an expression for ( ) ( )1 1 1! 1 n r rr= −+∑ . [3] (b) It is given that ( )( )2 1 12 1 6 n r nn nr = ++=∑ . (i) Show that ( ) ( ) 2 2 1 41 21 3 n r nn r = − −=∑ . [3] (ii) Hence find an expression for ( ) 22213 15 ... 4 1 m+ ++ − in terms of m. [4]

6 © Millennia Institute 9758/01/PU3/Prelim/24 10 The diagram below shows a curve C with parametric equations given by 2 cos 2 , 3 sin 2 , for 0. 2xy πθθ θθ θ= = −≤≤ The area bounded by curve C and the x-axis is a vineyard owned by John in front of his house where he used to grow grapes. He decided to install a Wi-Fi-enabled surveillance camera which moves automatically along the boundary of the vineyard in a clockwise direction along the curve C starting from point O and ending at point Q before moving in an anti -clockwise direction along the curve C back to point O. At any point, the camera is located at a point P with parameter θ on the curve C . The camera is orientated such that the field of view spans from point P to points O and Q exactly as shown. It is assumed that the camera is at O initially. (i) Show that the area of triangle OPQ, 2 unitsA , is given by ( )3 sin 22A π θθ= . [1] (ii) Using differentiation, find the value of for 0 2 πθθ −≤≤ that would maximise A and explain why A is a maximum for that value of θ. Hence find this value of A. [5] (iii) The image captured shows a good view of the vineyard when the camera is positioned such that OP PQ= . Find the coordinates of the position of the camera at this instant. [3] (iv) John decides to apply fertilisers to a certain area of the vineyard to observe its effectiveness. This area is enclosed by the curve C , the line 1x= and the x-axis, where 1x≥ . Find the approximate value of this area. [4] John’s Home Q P O y x

7 © Millennia Institute 9758/01/PU3/Prelim/24 [Turn over 11 Organisers of an airshow are setting up the venue and performing safety checks before the event. Points ( ),,xyz are defined relative to the entrance at ( )0, 0, 0 , where units are in metres. A spectator area of length 50 metres and width 40 metres i s created on the horizontal ground, with a transparent rectangular flat shield erected to protect the spectators. You may assume that the shield is of negligible thickness. Support poles me