2026 Pcss Math Prelim P2 MS for student

1 1 (a) Find all the prime numbers that satisfy the inequality 113 62 x −−   . 113 62 18 1 , 1 3 17 , 4 17 4 : 2, 3 [A1] x xx xx x Answer −−   −  − −  −   −   Answer ………………………… [3] (b) Simplify 43 53 69 5 20 xy x z zy  43 53 43 53 7 26 69 5 20 6 20 59 8 3 xy x z zy xy y z x z y xz  = = Answer ………………………… [2] (c) Simplify 1 34 12 927 r s −   1 34 12 9 1 9 3 3 3 27 27 3 r s s r s r −   =  = Answer ………………………… [2] 2026 4e5n Math Prelim Paper 2 Marking Scheme

2 (d) Write as a single fraction in its simplest form 2 73 125 4 2 5xx −+−− . 2 2 2 2 2 2 73 125 4 2 5 7 3 25 4 + (5 2)(5 2) (5 2) 25 4 7 3(5 2) 25 4= (5 2)(5 2) 7 15 6 25 4= (5 2)(5 2) 25 15 9 (5 2)(5 2) xx x x x x x xx xx xx xx xx xx −+−− −=+ − + − − + + + − −+ + + + − −+ ++= −+ 2 (a) In the diagram, ABCD is a square. The coordinates of A and D are (2, 3) and (5, 2) respectively. Find the area of square ABCD. Length of AD = 22(5 2 2 3) ( )− + − = 10 Area of square ABCD 2 10 10 10 units = = B C A 0 (2, 3) D (5, 2) y x

3 Answer …………………………units2 [3] (b) The points M(0, 2k), N(1, k) and P(5, 6− ) lie on the same straight line. (i) Show that the value k is 2. Answer Gradient of MN = gradient of MP 2 6 2 0 5 0 1 26 51 2 6 5 3 6 =2 k k k kk kk k k +− =−− + =−− += = [2] (ii) Find the equation of MN. Gradient of MN = -2 2 24 2 4 y x c ck yx = − + == = − + Answer ………………………… [2]

4 3 (a) The diagram below shows a circle of radius 2 cm inscribed in a regular hexagon. (i) Show that the length of one side of the hexagon is 2.31cm Answer Looking at the triangle drawn – tan 30 = x 2 x = 2 tan 30 = 1.155cm  Length of each side = 1.1552 = 2.31cm [2] (ii) Find the area of the shaded region. Shaded area =2( Area of  − Area of sector) = 2 2 1 602( 2 2.31 2 ) 2 360 0.431 cm =   −   = Answer …………………………cm2 [3] 2cm m

5 (b) Two regular pentagons ABCDE and PQRST are joined together to form the shape shown below. Find angle AEC. AECB is a quadrilateral. EAB = ABC = (5 − 2) 180 5 = 108 AEC = ECB 2AEC = 360− 2(108) 2AEC =144  AEC =72 Answer ………..………………..o [3]

6 (c) The sides CD and AB are produced to meet at Z. Show that triangle BZC is an isosceles triangle. Answer ∠ABC=108∘ CBZ = 180 − 108 =72 BCZ = 180 − 108 =72 Since angle CBZ = angle BCZ, triangle BZC is an isosceles triangle.

7 4 The table below shows the distribution of three types of apartments in three blocks of an Executive Condominium. This information can be represented by the matrix A= 25 20 60 35 45 30 30 50 10    . The areas of the 3-room, 4-room and 5-room apartments are 75 m², 95 m² and 120 m² respectively. (i) Represent the areas of the apartments in a 31 column matrix C. Answer C = 75 95 120    (ii) Evaluate the matrix M = AC AC = 25 20 60 35 45 30 30 50 10    75 95 120    10975 10500 8200 AC  =  Answer M = [1] 3-room 4-room 5-room Block 1 25 20 60 Block 2 35 45 30 Block 3 30 50 10

8 (iii) State what the elements of M represent. Answer The elements in AB represent the total floor areas of the apartments in blocks 1, 2 and 3 respectively. (iv) Estate management fees are charged based on the floor area of the apartments in the three blocks per quarter. The estate management fee rates is $15/m 2 per quarter. Find the total amount of estate management fee collected in a year for the Executive Condominium. Total amount collected in a year 4 15 (10975 10500 8200) $1780500 =   + + = Answer $………………………… [3]

9 (v) The total estate management fee collected in one year for the Executive Condominium is used in the following areas. Areas Amount Day-to-day operating expenses 25% of the total amount collected Cleaning 1 5 of the total amount collected Security guards $2350 per month for each security guard. The estate has 2 security guards. Landscaping $50 000 per year Electricity for common areas $120 000 per year The remaining amount is transferred to the sinking fund, which is used for future major repairs and replacement of shared facilities. Calculate the amount transferred to the sinking fund. Day-to-day operating expenses =25% × 1 780 500 = $445 125 Cleaning = 1 5 × 1 780 500 = $356100 Security guards : Annual cost = 2 × 2350 × 12 = $56 400 [M2 for finding either day to day operating expenses, cleaning or security guards] Landscaping = $50 000 Electricity for common areas: Annual cost= $120 000 Total expenditure= $445 125 + 356100 + 56 400 + 50 000 + 120 000 = $102 7625 Amount transferred to the sinking fund 1 780 500 − 102 7625 [𝑀1] = $752 875 [A1]

10 5 _ 114  _ 37  _ T _ E _ D _ C _ B _ A The points A, B, C, D and E lie on the circle. AD is the diameter of the circle, DT is a tangent to the circle, AET is a straight line and AD is parallel to BC. Angle BCD = 114 o and angle DAE = 37 o. (a) Find angle EBC. Give a reason for each step of your working. 90 [right angle triangle in semi-circle] A = 180 37 90 53 [angles sum of a triangle ] 180 114 66 [angles in opposite segme nt] EBC =180 66 53 o o o o o oo o o o o AED EB DAB =  − − =  = − =  − − 61 [interior angles] o= (b) Emily claims that quadrilateral 𝐴𝐸𝐷𝐵 is a parallelogram. Do you agree with her claim? Give a reason for each step of your working. 37 66 = 103 180 By interior angles, is not parallel to , so quadrilateral B is not a parallogra m.[A1] oo o o EAB EAB AED AB ED AED  = +  +  