NHHS 2023 Prelim Math P1 (Solution)

Page 1 of 28 Name SOLUTION ( ) Class PRELIMINARY EXAMINATION 2023 Subject : Mathematics Paper : 4052/01 Level : Secondary Four Express Date : 17 August 2023 Duration : 2 hours 15 min READ THESE INSTRUCTIONS FIRST Write your name, index number and class 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. Gives answers in degrees to one decimal place. For π, use either your calculator value or 3.142. This paper consists of 286 printed pages and 2 blank pages. U P A NAN HUA HIGH SCHOOL For Examiner’s Use

Nan Hua High School Preliminary Examination 2023 Mathematics 4052/01 Page 2 of 28 Compound interest Total amount = n rP       1001 Mensuration Curved surface area of a cone = rl Surface area of a sphere = 24 r Volume of a cone = hr 2 3 1  Volume of a sphere = 3 3 4 r Area of triangle ABC = Cabsin2 1 Arc length = r , where  is in radians Sector area = 2 2 1 r , where  is in radians Trigonometry C c B b A a sinsinsin  2 2 2 2 cosa b c bc A   Statistics Mean = f fx   Standard deviation = 22          f fx f fx

Nan Hua High School Preliminary Examination 2023 Mathematics 4052/01 Page 3 of 28 Answer all questions. 1 11 11 113 3 3 3x    Find the value of .x 11 11 11 11 1 11 12 3 3 3 3 3(3 ) 3 3 x        12x  Answer x ………………………………….. [1] 2 Charles invested with a bank that pays interest of 3.5% per annum compounded half- yearly. He received $3500 as interest at the end of 3 years. Find the original amount that Charles invested. Give your answer to the nearest dollar. Let P be the original amount. 6 6 6 6 6 3.53500 1 200 1.0175 1.0175 3500 (1.0175 1) 3500 3500 1.0175 1 31905 P P P P P P P                Answer $ …………………………………...... [2] 2023 Prelim Exam Math Paper 1 - MARKING SCHEME

Nan Hua High School Preliminary Examination 2023 Mathematics 4052/01 Page 4 of 28 3 Ken draws this graph to show his annual electricity cost for his 5-room flat for year 2020, 2021 and 2022. State one aspect of the graph that may be misleading and explain how this may lead to a misinterpretation of the graph. Answer Comparing scale The vertical axis is not labelled at the start OR Start value not indicated on the vertical axis. It is not possible to determine the exact costs for the three years or that it is not possible to find the increase in costs as it is not possible to compare the years without a scale. OR Comparing height or area It is not clear as to whether the height of the lightbulb or the area of the circle should be used to determine the annual electricity bill. Or The size of the picture not only increases with height, but also in the width. When using area, the cost looks 4 times more in 2022 as compared to 2020. If using the height of the lightbulbs, the cost in 2022 looks twice as much in 2020. [2]

Nan Hua High School Preliminary Examination 2023 Mathematics 4052/01 Page 5 of 28 4 The marked price of a 65-inch Ultra HD 4K TV is $6000. Mr Lim buys the TV on hire purchase scheme with down payment of 20% on the marked price. He then pays monthly instalments over 4 years at a simple interest rate of 1.37% per annum. Calculate the total amount that Mr Lim pays for the television. 4800 1.37 4Interest = 100 $263.04    Total amount 263.04 6000 $6263.04    Answer $ ………………………………...... [2] 5 The table shows the time, to the nearest hours, taken by 12 students to complete a project. Time t (hours) Frequency 30 35t  1 35 40t  2 40 45t  5 45 50t  4 (a) Calculate an estimate for the mean time taken. 1 32.5 2 37.5 5 42.5 4 47.5Mean time = 12 510 12 42.5 hours          Answer …………………………… ……. hours [1] (b) Calculate an estimate for the standard deviation of the time taken. 22 2 2 2 Standard deviation 1 32.5 2 37.5 5 42.5 4 47.5 510 12 12 4.5644 4.56 hours (3 s.f.)                Answer …………………………… ……. hours [1]

Nan Hua High School Preliminary Examination 2023 Mathematics 4052/01 Page 6 of 28 6 The average mass of haemoglobin in a red blood cell is about 31 picograms. (a) Express 31 picograms in grams, giving your answer in standard form. [1 picogram = 1210 grams] 12 11 30 picograms 31 10 grams =3.1 10 grams      Answer …………………………… …….. g [1] (b) There are 65 10 red blood cells in 1 cubic millimetre of blood. Calculate the mass, in grams, of haemoglobin in 1 cubic millimetre of blood. Give your answer in standard form. Average amount 11 63.1 10 5 10    5 4 15.5 10 1.55 10 g       Answer …………………………… …….. g [1]

Nan Hua High School Preliminary Examination 2023 Mathematics 4052/01 Page 7 of 28 7 In the diagram, C is the point (0, 5) and A is a point on the y-axis. The line AB meets the horizontal line through C at point B. The equation of line AB is 2 9.y x  (a) State the equation of the line parallel to line AB, passing through (0, 7). 2 7y x  Answer …………………………… …….. [1] (b) Find the coordinates of point A and point B. Coordinates of A =(0, 9). When 5,y  5 2 9 2 14 7 x x x      Coordinates of B = ( 7, 5).  Answer A (……………… , ………… ….. ) B (……………… , ………… ….. ) [2] ( 0 , 5 ) O O

Nan Hua High School Preliminary Examination 2023 Mathematics 4052/01 Page 8 of 28 8 The ratio of the number of employees in company P to that in company Q was 3: 4. 8 employees left company Q to join company P. The number of employees in company P now is 4 more than company Q. Find the original number of employees in company P. Let the number of employees in be 3 Let the number of employees in be 4 3 8 (4 8) 4 12 Original number in 36 P x Q x x x x P       Alternative solution:   Let no of employees in each company be and Ratio of : 4 : 3 4 3 8 8 4 48 8 43 123 36 P Q Q P Q P P Q P P P P             Answer …..…………………………… …… [3]

Nan Hua High School Preliminary Examination 2023 Mathematics 4052/01 Page 9 of 28 9 Straight lines AC, AD, BE, BD, CE intersect to form the figure in the diagram. Find the value of x. 35 21 (Exterior of ) 56 28 (Exterior of ) GFC FGC x               28 56 34 180 ( sum of ) = 62 x x        Alternative: 180 28 21 ( sum of ) = 131 180 131 (adjacent on a straight line) = 49 180 35 34 ( sum of ) = 111 180 111 (adjacent on a straight line) = 69 180 69 49 ( sum of ) = 62 a c b d x                                    Answer x = …………..…………………… [2]

Nan Hua High School Preliminary Examination 2023 Mathematics 4052/01 Page 10 of 28 10 (a) Factorise completely 3 1 3 .fg g f   3 1 3 3 1 3 (3 1) (1 3 ) (3 1) (3 1) ( 1)(3 1) fg g f fg g f g f f g f f g f                   Answer …………………………… ……...... [2]