2026 GMS P2 MS

1 2026 Sec 4E/5NA Preliminary Exam Math Paper 2 (Mark Scheme)

2 1 (a) (i) Solve the inequalities 321 1 5 4x−−  +  . Solutions Mark Remarks 321 1 5 4x−−  +  3224 4x−−   8 3 2 16x−  −  6 3 18x−   26 x−   Total marks 3 (ii) Hence, find the smallest positive value of 221x + . Solutions Mark Remarks For 26 x−   , 0x= ( ) 2 2 0 1 1+= Total marks 2 (b) Simplify 0 3 26 12 7 21 a b bc ca  . Solutions Mark Remarks 3 26 21 7 12 ba c bc 4 13 2 a c= Total marks 2 (c) Simplify 1 69 3 327 pq r − −   . Solutions Mark Remarks 1 3 6 9 3 27 r p q −   1 3 23 27 rp q= 23 3 rp q= Total marks 2

3 (d) Write as a single fraction in its simplest form 733 21 xxx+− −− . Solutions Mark Remarks ( )( ) ( ) ( ) ( )( ) 3 2 1 7 1 3 2 21 x x x x x xx − − + − − − −− ( )( ) 229 6 3 7 7 6 3 21 x x x x x xx − − + − − += −− ( )( ) 10 13 21 x xx −= −− Total marks 3

4 2 Penny is a property agent. She handles the sale of landed homes (L), private condominiums (C) and HDB flats (F) in the east and central parts of Singapore. The matrix T shows the number of transactions of each property type that Penny handled in a particular month. L C F 4 8 East 5 2 9 Central x=T The commission earned by Penny is $13 000 for landed homes, $8000 for private condominiums and $4000 for HDB flats. (a) Represent the commission in a 31 matrix C. Solutions Mark Remarks C 13000 8000 4000  = Total marks 1 (b) Find, in terms of x, the matrix P = TC. Solutions Mark Remarks P 1300048 80005 2 9 4000x  =   4 13000 8000 8 4000 5 13000 2 8000 9 4000 x +  + =  +  +  84000 8000 117000 x+= Total marks 2 (c) State what the first element of matrix P represents. Solutions Mark Remarks The total commission for east area earned by Penny. Total marks 1 (d) Penny earns $15 000 more from selling the properties in the east than those in the central area. Find the value of x. Solutions Mark Remarks 84000 8000 15000 117000x+ − = 6x= Total marks 1

5 (e) Penny decreases her commission for properties in the east by 20% and increases it for properties in the central area by 15%. Calculate the percentage change in her total commission earned. Give your answer to 1 decimal place. Solutions Mark Remarks 0.8 132000 105600= 1.15 117000 134550= 105600 134550 240150+= 8850 100%249000 3.554%= (to 3 dp) 3.6%= (to 1 dp) Total marks 2

6 3 (a) ( )4.5,0A and ( )0,9B are points on a straight line. (i) Find the equation of the straight line, AB. Solutions Mark Remarks Gradient 9 24.5= =−− 29yx=− + Total marks 2 (ii) M is a point on AB such that it is equidistant from the x-axis and the y-axis. Show that coordinates of M are ( )3,3 . Solutions Mark Remarks Therefore, 29yx=− + M lies on the line yx= and 29yx=− + Solving, 3x= and 3y= ( )3,3M Total marks 2 O x x x y

7 (iii) 0 3NM =  . Prove that triangle ANM is similar to triangle AOB. Solutions Mark Remarks 0 9OB =  Since 3OB NM=  , OB is parallel to NM. ON OM MN=+  30 33ON    =−        3 0ON =  MAN BAO = (common) MNA BOA = (corresponding angles, OB // NM) Triangle ANM is similar to triangle AOB. (AA Test) Total marks 4 (b) The ratio of the height of two square pyramids is 1 : 4. The ratio of the length of their square bases is 2 : 3. Find the ratio of the volume of the smaller pyramid to that of the larger pyramid. Solutions Mark Remarks V ol of smaller pyramid : V ol of larger pyramid 222 1 3 4:33= 1:9= Total marks 2

8 4 The diagram shows a container made up of a triangular prism and a cuboid with dimensions ( )5 x− cm by ( )5 x− cm by x cm. The height of the triangular prism is 2x cm. The volume of the solid, y cm3 is given by 322 20 50y x x x= − + . (a) Explain why 05 x . Solutions Mark Remarks When x = 0, the height of the container will be zero. When x = 5, the width of the container will be zero. When 5x , the width of the container will be negative. Therefore, 05 x . Total marks 2 (b) Complete the table of values for 322 20 50y x x x= − + . x 0.5 1 1.5 2 2.5 3 3.5 4 4.5 y 20.25 32 36.75 36 31.25 24 8 2.25 Solutions Mark Remarks y = 15.75 Total marks 1 (c) On the opposite grid, draw the graph of 322 20 50y x x x= − + for 0.5 4.5x . (d) Use your graph to find the smallest value of x when the volume of the solid is 28 cm3. Solutions Mark Remarks x = 0.8 Total marks 1

9 322 20 50y x x x= − +

10 (e) Water is poured into the container. It takes 1t seconds to fill the triangular prism and 2t seconds to fill the cuboid. Diagram 1 and Diagram 2 show how the height of the water changes with respect to time. Which diagram shows the correct representation of the change in height of water with respect to time taken? Explain and justify with mathematical reasoning. Solutions Mark Remarks The height of the water in the inverted triangular prism increases at a decreasing rate followed by a constant increase in height in the cuboid. In Diagram 1, the graph shows the height of water increasing at a decreasing rate from 0 to t1 seconds and constant increase in height from t1 to t2 seconds. Diagram 2, the graph shows the height of water increasing at an increasing rate from 0 to t1 seconds and constant increase in height from t1 to t2 seconds. Diagram 1 is correct. Total marks 2 Height (cm) 2x 3x 0 Time (s) 1 1 Diagram 1 Height 2x 3x Time (s) 1 1 Diagram 2 0