MI 9758 2025 Prelim P1

©Millennia Institute 9758/01/PU3/25 2025 Preliminary Examination Pre-University 3 MATHEMATICS 9758/01 Paper 1 August/September 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 are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 7 printed pages and 1 blank page. H CANDIDATE NAME ADMIN NUMBER CLASS

2 ©Millennia Institute 9758/01/PU3/25 1 In this question you may use expansions from the List of Formulae (MF27). It is given that 1f ( ) 3 sin = − for 0  . Show that when  is sufficiently small 2f ( ) p q r   + + where p q and r are exact constants to be determined. [4] 2 The diagram below shows the curve with equation f( )yx= where 2 f ( ) 2 x px qx x ++= − for some constants p and q. The curve passes through the point with coordinates ( )0, 1.5− and has asymptotes 1yx=− and xr= for some constant r. (i) Write down the value of r and show that 3p=− and q = 3. [3] (ii) Solve the inequality 7f ( ) 2x  . [1] 3 (a) Find 2 d 4 x x x − . [2] (b) Find ( ) 2 ln 2 dx x x . [3] (c) Find 2 sin 2 d1 cos x xx+ . [2] y x O f( )yx=

3 ©Millennia Institute 9758/01/PU3/25 [Turn over 4 Three different blends of coffee beans – Blend A Blend B and Blend C – are sold at both M Café and I Café. The usual selling prices of the blends are the same at both cafés. The total cost of buying one package each of Blend A Blend B and Blend C is $176. During a holiday sale the two cafés offered the following discounts: Café Discounts given for each package Total price after the discount Blend A Blend B Blend C M Café 15% 10% 5% $161.80 I Café 7% 5% 2% $169.56 (i) Find the usual selling price of each package of Blend A Blend B and Blend C coffee beans respectively. [3] Loyal customers of I Café receive an additional discount of x% on the usual selling price of the packages for Blend A and Blend B only. (This loyalty discount is calculated as x% of the usual price of each package