MGS 2025 EMATH PRELIM P1 MS

Class Index Number Name : Mark Scheme This question paper consists of 26 printed pages and 2 blank pages. METHODIST GIRLS’ SCHOOL Founded in 1887 PRELIMINARY EXAMINATION 2025 Secondary 4 Wednesday MATHEMATICS 4052/01 20 August 2025 Paper 1 2 h 15 min Candidates answer on the Question Paper. INSTRUCTIONS TO CANDIDATES Write your name, class and index number in the spaces at the top of this page. 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. 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 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, unless the question requires the answer in terms of The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 90. 90

Page 2 of 28 Methodist Girls’ School Mathematics Paper 1 Sec 4 Preliminary Examination 2025 Mathematical Formulae Compound Interest Total amount = Mensuration Curved surface area of a cone = Surface area of a sphere = 4 Volume of a cone = Volume of a sphere = Area of a triangle = absin C Arc length = r θ , where θ is in radians Sector area = , where θ is in radians Trigonometry a 2 = b 2 + c 2 – 2bc cos A Statistics Mean = Standard deviation = ( ) 100 r 1+ n P rlπ 2rπ h r2 3 1π 3 3 4 rπ 2 1 θ2 2 1 r C c B b A a sin sin sin = = ∑ ∑ f fx 22         − ∑ ∑ ∑ ∑ f fx f fx

Page 3 of 28 Methodist Girls’ School Mathematics Paper 1 Sec 4 Preliminary Examination 2025 1 Simplify ( ) 1 6 32 3 274 3 aa rr − ÷ , giving your answer in positive index. ( ) 1 6 32 22 33 23 2 2 4 274 43 33 4 33 4 9 aa aa r r rr ar xra r a −− − ÷= ÷ = = Answer………………………………. [2] 2 3 3148 4mm−− ×= Use the laws of indices to find the value of m . Show your working. ( ) ( ) 3 31 2 3 33 1 2 2 69 3 2 11 9 2 48 4 22 2 22 22 11 9 2 11 11 1 mm mm mm m m m m −− −− −+ − − ×= ×= = = −= = = Answer m = …………………………. [2] M1 for 3a2 A1 A1 M1 for expressing terms in base 2

Page 4 of 28 Methodist Girls’ School Mathematics Paper 1 Sec 4 Preliminary Examination 2025 3 Two identical containers are filled with a mixture of rose syrup and evaporated milk in the ratio of 5 : 3 and 4 : 7. The contents of the containers are poured into a big bowl and mixed thoroughly. Find the ratio of the rose syrup to the evaporated milk in the big bowl. 1st Container 2nd Container 5 : 3 4 : 7 55 : 33 32 : 56 Rose Syrup : Evaporated milk = 87 : 89 Answer …………… : ……………. [2] 4 Simplify ( )( )3 12 3 52 36 mm m−+ −+ . ( )( ) ( ) 2 2 2 26 9 2 3 5 23 12 3 52 36 6 12 14 6 5 2 6 12 19 8 6 m mm mmm m mm m mm + − −+ −−+ −+= + −+ −= +−= Answer………………………………. [2] A1 M1 for both ratios A1 M1 for expansion of (3m-1)(2m+3)

Page 5 of 28 Methodist Girls’ School Mathematics Paper 1 Sec 4 Preliminary Examination 2025 5 A map is drawn to a scale of 1 : 400 000. (a) Find the actual distance, in km, represented by 10.5 cm on the map 1 cm : 4 km 10.5 cm : 42 km Answer…………………………… km [1] (b) A forest covers an area of 64 km 2. Find, in cm2, the area representing the forest on the map. 1 cm : 4 km 1 cm 2 : 16 km2 4 cm2 : 64 km2 Answer…………………………… cm2 [1] A1 A1

Page 6 of 28 Methodist Girls’ School Mathematics Paper 1 Sec 4 Preliminary Examination 2025 6 (a) Written as a product of its prime factors, 4240 2 3 5= ×× and 32750 2 5 11=×× . (i) Find the smallest positive integer n for which 2750n is a whole number. 24 2 3 24 2 24 2 3 2750 2 5 11 2 5 11 2 5 11 2 5 11 2 5 11 2 5 11 110 n n n =×× ×× =×× ××= ×× =×× = Answer………………………………. [2] (ii) Find the smallest positive integer k for which 240k is a multiple of 2750. 43 43 43 2 4 LCM of 240 and 2750 =2 3 5 11 240 2 3 5 11 2 3 5 11 5 11 2752 35 k k ×× × = ×× × ×× ×= =×=×× Answer………………………………. [1] (b) 18 12 36Ya b c=×× where a, b and c are prime numbers. Explain why Y is a perfect cube. Answer The indices of the prime factors a, b and c are multiples of 3. [1] A1 A1 M1 for perfect square

Page 7 of 28 Methodist Girls’ School Mathematics Paper 1 Sec 4 Preliminary Examination 2025 7 The graph shows the Category A Certificate of Entitlement (COE) price for cars up to 1600cc for the months of April, May and June in 2025. State one aspect of the graph that maybe misleading and explain how this may lead to a misinterpretation of the graph. Answer By starting the vertical axis at $90000 and not at zero, it exaggerates small differences in COE prices, making the COE price of May look dramatically more expensive than April or June, even though the actual difference is relatively small. [2] Category A COE (Certificate of Entitlement) Price April May June June May April

Page 8 of 28 Methodist Girls’ School Mathematics Paper 1 Sec 4 Preliminary Examination 2025 8 The interior angles of a hexagon are in the ratio 3 : 5 : 6 : 7 : 7 : 8. Find the smallest exterior angle of the hexagon. ( ) 8size of largest interior angle =180 6 2 36 160 smallest exterior angle = 180 160 20 °−× = ° °− ° = ° Answer……………………………… o [2] A1 M1

Page 9 of 28 Methodist Girls’ School Mathematics Paper 1 Sec 4 Preliminary Examination 2025 9 P and Q are similar pentagonal pyramids. The ratio of surface areas of the two pyramids, P and Q, is 81 : 121. (a) Find the ratio of the height of pyramid P to that of pyramid Q. Height : Height 81 : 121 9 :11PQ = = Answer……………… : ……………. [1] (b) Both pyramids are completely filled with sand. Find the mass of the sand, in kg, in pyramid Q if the mass of sand in the pyramid P is 192 g. ( ) 3 3 3 3 0.192 9 M 11 11 0.192M 9 0.35055 0.351 kg 3sf Q Q = ×= = = Answer…………………………… kg [2] 10 Factorise completely. (a) ( ) ( )( ) 225 5 51 51 1 xx xx −=− = −+ Answer………………………………. [2] (b) ( ) ( ) ( )( ) 22 2 2 xq yq yp xp qx y px y qpxy − −+ = −+ − = +− Answer………………………………. [2] A1 A1 M1 M1 A1 M1 A1 P Q

Page 10 of 28 Methodist Girls’ School Mathematics Paper 1 Sec 4 Preliminary Examination 2025 11 Simplify 2 34 1 32 x x xx +−+ ++ . ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) 2 3 43 4 1 32 1 1 2 3 2 ( 4) 12 36 4 12 22 12 21 12 2 2 xx x xx x x x xx xx xx xx x xx x xx x ++−= −+ ++ + + + +−+= ++ +−−= ++ += ++ += ++ = + Answer………………………………. [4] A1 M1 for factorisation of x2 + 3x + 2 M1 for making single fraction M1 for simplifying numerator