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 and is subtracted from the sale prices above.) I Café wants the total cost for its loyal customers to be less than the total cost at M Café when buying one of each package of the three blends. (ii) Form an inequality and find the smallest integer value of x. [3] 5 (i) Using the substitution show that ( ) ( ) 3 322 2 1d 3 3 333 x x x x C x = + − + + + . [4] (ii) Hence find the exact value of 3 3 2 1 3 d 3 x x x− + leaving your answer in the form where a and b are integers to be determined. [3] 2 3ux=+ 3ab+

4 ©Millennia Institute 9758/01/PU3/25 6 A company produces customised badges. The diagram below shows the design of one such badge which consists of a rectangle and two identical semicircles. It is g iven that 2 cmAG x= cmFG y= cmBD DF x== and the outer perimeter ABCDEFGA of the badge is 25 cm. (i) Show that the area of the badge S cm2 is given by the formula 22 325 2 4S x x x = − − . [3] (ii) Use calculus to find the exact value of x which gives the maximum value of S. [4] D E A B F G 2x C x y

5 ©Millennia Institute 9758/01/PU3/25 [Turn over 7 Epidemiologists are modelling the spread of an infectious disease in a town. The following assumptions are made: • At Week 0 there are 500 new infections. • In each subsequent week the number of new infections is 60% the number of new infections in the previous week due to ongoing community transmission. Let nu denote the number of new infections in Week n for 0, nn . (i) Write down the value 1u and verify that 2 180u = . [2] (ii) If nS denotes the total cumulative number of new infections after n weeks show that ( ) 11250 1 0.6 n nS +=− . [2] The local healthcare facility has a capacity of 1240 beds. You may assume that each infected person occupies one bed and that none are discharged during the period being considered. (iii) Find the smallest value of n such that the total number of infections up to and including Week n first exceeds 1240. [1] The healthcare facility later increases its capacity to 1280 beds. (iv) Based on the given model comment on whether this new capacity is sufficient to accommodate all infected persons over a prolonged period of time. [2] 8 It is given that 12sine xy − = . (i) Show that 2 d12 d yxy x−= . [2] (ii) By further differentiation of the result in part (i) find the Maclaurin series for y up to and including the term in 2x . [4] (iii) Using your result from part (ii) find an approximate value for 10.1 2sin 0 e d xxx −  giving your answer to 4 significant figures. [2]

6 ©Millennia Institute 9758/01/PU3/25 9 The coordinates of point A and B are (1 2 3) and (17 10 19) respectively. The line l1 passes through both points A and B and is parallel to the vector where k is a constant. (i) Show that k = 2. [2] The line l2 has equation 32 40 61         =+          r where  is a parameter. (ii) Determine whether the lines l1 and l2 intersect. If they do find the position vector of their point of intersection; otherwise explain why they do not intersect. [4] The plane p has equation 1 0 16 1  = r . (iii) Let  be the acute angle between l1 and p. Find the exact value of sin and cos . [3] (iv) Find the coordinates of point F the foot of the perpendicular of the point B to the plane p. [3] 10 The curves C1 and C2 are given by the equations 2 1 14y x= + and 2 5 xy= respectively. (i) Find the coordinates of the intersection points for C1 and C2. [2] The region R is bounded by C1 and C2. (ii) Find the exact area of R. [5] (iii) Find the exact volume generated when R is rotated through π radians about the y-axis. [5] 1 2 k  

7 ©Millennia Institute 9758/01/PU3/25 [Turn over 11 Referred to the origin O the points A and B have position vectors a and b respectively where a and b are non-zero and non-parallel vectors. The point C lies on OA such that : 1: 2OC CA = . The point D lies on OB such that : 3 :1OB DB = . (i) Find the position vectors OC and OD giving your answers in terms of a and b. [2] (ii) Show that the point E where the lines AD and BC meet has position vector 14 77 +ab . [5] (iii) Show that the area of triangle OAE can be written as k ab where k is a constant to be found. [3] It is further given that b is a unit vector. (iv) Give the geometrical meaning of ab . [1] (v) If 1 2=a b a find the perpendicular distance of point A to OB leaving your answer in terms of a . [3] 12 A tank initially contains 100 litres of pure water. A brine solution with a salt concentration of 0.2 kg per litre is pumped into the tank continuously at a constant rate of 5 litres per minute. At the same time a well-stirred mixture of the tank’s contents is continuously drained fr