MSHS 2026 Prelim Math P2 Solution

This document consists of 24 printed pages. [Turn over For Examiners’ Use Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 / 10 / 10 / 9 / 11 / 10 / 10 / 11 / 9 / 10 SUBTOTAL Statement Presentation Units Rounding Off Class/ Index Number Centre Number/ ‘O’ Level Index Number Name / / MARIS STELLA HIGH SCHOOL PRELIMINARY EXAMINATION SECONDARY FOUR MATHEMATICS 4052/2 Paper 2 19 August 2026 Candidates answer on the Question Paper. 2 hours 15 minutes READ THESE INSTRUCTIONS FIRST Write your class, index number and name 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. Answer all the questions. The number of marks is given in brackets [ ] at the end of each question or part question. If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks. The total of the marks for this paper is 90. The use of an approved scientific calculator is expected, where appropriate. If the degree of accuracy is not specified in the question and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For  , use either your calculator value or 3.142. 90

2 Mathematical Formulae Compound interest Total amount = 1 100 n rP+ Mensuration Curved surface area of a cone = rl Surface area of a sphere = 24 r Volume of a cone = 21 3 rh Volume of a sphere = 34 3 r Area of triangle ABC = 1 sin2 ab C Arc length = r , where  is in radians Sector area = 21 2 r  , where  is in radians Trigonometry A a sin = B b sin = C c sin 2 2 2 2 cosa b c bc A= + − Statistics Mean =   f fx Standard deviation = 22       −     f fx f fx

3 1 (a) It is given that 2bca bc += − . (i) Find a when b = 5 and c = – 3. 2(5) 3 5 ( 3)a −= −− 7 8= (or 0.875) Answer a =................................... [1] (ii) Rearrange the formula to make b the subject. 2ab ac b c− = + ( 2)a b c ac− = + 2 c acb a += − Answer b = .................................... [2] (b) Solve the inequality (3 1) ( 2) 143 xx−+ − . 3(3 1) 4( 2) 12xx− − +  9 3 4 8 12xx− − −  5 23x 34 5x Answer ....................................... [2]

4 (c) Simplify 1 4 2 1436 y x −   . 11 4 14 22 14 4 36 36 yx xy −     =       7 2 6x y= Answer ....................................... [2] (d) Simplify 2 22 4 12 5 15 2 18 x ax x a xa − − + − . 2 2 2 2 2 4 12 5 15 4 ( 3 ) 5( 3 ) 2 18 2( 9 ) x ax x a x x a x a x a x a − − + − − − =−− (4 5)( 3 ) 2( 3 )( 3 ) x x a x a x a −−= −+ 45 2( 3 ) x xa −= + Answer ....................................... [3]

5 2 (a) A is the point (−2, 4) and B is the point (4, 1). (i) Find the length of the line AB. AB = ( ) ( ) 22 2 4 4 1− − + − = 45 = 6.7082 = 6.71 units (to 3 s.f.) Answer …........................................ units [1] (ii) Find the equation of the line AB. Gradient = 4 1 1 2 4 2 − =−−− [M1] Sub x = 2− , y = 4, 4 = ( )1 22−− + c c = 3 Equation of AB is 1 32yx=− + Answer ………......................................... [2] (iii) The line AB cuts the x-axis at P and the y-axis at Q. Find the area of triangle OPQ where O is the origin. P = (6, 0) and Q = (0, 3) Area of triangle OPQ = 1 362 = 9 units2 Answer…......................................... units2 [2]

6 (b) A company manufactures three sizes of glass which are geometrically similar. The height of the small, medium and large glasses are h cm, 8 cm and 10 cm respectively. (i) The top of the large glass has a circumference of 30 cm. Find the circumference of the top of the medium glass. Circumference of the top of the medium glass 8 3010 24 = = cm Answer…......................................... cm [1] (ii) The surface area of the small glass is 9 16 of the surface area of the medium glass. Show that h = 6. Answer 2 2 9 16 8 3 48 6 ( ) ss mm Ah Ah h h h shown =  = = = [2] Small Medium Large 8 10 h

7 (iii) Calculate the ratio of the volume of the small glass to the volume of the large glass. 3 3 6 10 27 125 : 27 :125 ss ll s l s l sl Vh Vh V V V V VV =  = = = Answer …................. : ....................... [2]

8 3 60 students were each asked how many hours of screen time they had in a day. The results are shown in the table. Screen time (t, hours) 02 t 24 t 46 t 68 t 8 10t 10 12t Frequency 4 7 15 19 11 4 (a) Find the interval that contains the median screen time. 68 t Answer .............. ≤ t < .............. [1] (b) Calculate an estimate of the mean daily screen time. Mean= 4(1) 7(3) 15(5) 19(7) 11(9) 4(11) 60 + + + + + = 376 60 = 4615 hours or 6.27 hours Answer ................................ hours [1] (c) Calculate an estimate of the standard deviation of the screen time. Standard deviation= 22 2 2 2 2 24(1) 7(3) 15(5) 19(7) 11(9) 4(11) 4 660 15 + + + + + − =2.5552(5s.f) = 2.56 hours (3s.f) Answer ................................ hours [1]

9 (d) Three students are selected at random from the survey. Find the probability that two of them have less than 6 hours of screen time and the other has at least 8 hours of screen time. Probability= 26 25 15 360 59 58     = 975 6844 (or 0.142) Answer ....................................... [3] (e) 60 adults were also asked how many hours of screen time they had in day. The results are summarised in the table. Mean 5.23 Standard deviation 1.89 Make two comparisons between the number of hours of screen time by the adults and by the students. 1. Students spent more hours of screen time as their mean hours of screen time is higher than the adults’. 2. The number of hours of screen time is more consistent for adults as the standard deviation is lower than the students’. 1. ……………………………………………………………………………………… ………………………………………………………………………………………... 2. ……………………………………………………………………………………… ………………………………………………………………………………….…......[2] (f) Another 10 adults were asked how many hours of screen time they had in a day. The data was added to the study. The estimated mean hours of screen time by the adults is now 5.4 hours. Explain what this tells you about the number of hours of screen time of the adults added to the study. The mean number of hours of screen time of the 10 adults is more than 5.23 (or 5.4) hours. (accept higher than the original 60 adults’ mean) .………………………………………………………………………………………. ………………………………………………………………………………….…......[1]

10 4 \ (a) Complete the table of values for 321 15y x x= − + . [1] x –2 –1 0 1 2 3 4 5 y –4.6 –0.2 1 0.2 – 1.4 –2.6 –2.2 1 (b) On the grid provided, draw the graph of 321 15y x x= − + for 25 x−   . [3]