NHHS 2022 Math Prelim P1 (Solution)

This paper consists of 19 printed pages. Page 1 of 19 Name ( ) Class PRELIMINARY EXAMINATION 2022 Subject : Mathematics Paper : 4048/01 Level : Secondary Four Express Date : 17 August 2022 Duration : 2 hours READ THESE INSTRUCTIONS FIRST Write your name, class and index number on all the work you hand in. Write in dark blue or black pen. You may use a HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue, correction fluid or correction tape. Answer all 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 80. NAN HUA HIGH SCHOOL For Examiner’s Use

Nan Hua High School Preliminary Examination 2022 Mathematics 4048/01 Page 2 of 19 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 2022 Mathematics 4048/01 Page 3 of 19 Answer all questions. 1 The number of people living in a town is given as 60 000, correct to 2 significant figures. Write down values for the smallest and largest possible number of people who could be in the town. 2 (a) Use prime factors to explain why 135 200 is a perfect cube. (b) The lowest common multiple of x and 135 is 42 3 5 7   . Find the smallest possible value of x.

Nan Hua High School Preliminary Examination 2022 Mathematics 4048/01 Page 4 of 19 3 Write the following numbers in order of size, starting with the smallest. 2 3 10   , 21 90− , 90% , 0.23− , 0.3 4 Factorise 2 3 6 15 20 8 25 4 a a ab b aa − + − − completely.

Nan Hua High School Preliminary Examination 2022 Mathematics 4048/01 Page 5 of 19 5 Rearrange the formula 1 1 1 23a b c−= to make b the subject. 6 Sketch the graph of ( )( )32y x x= − + on the axes below. Indicate clearly the coordinates of the points where the graph crosses the axes and the turning point of the curve.

Nan Hua High School Preliminary Examination 2022 Mathematics 4048/01 Page 6 of 19 7 The scale of a map is 8 cm : 2 km . (a) Write this scale in the form 1 : n . (b) The actual area of a lake is 290 000 m . Calculate the area, in square centimetres, of the lake on the map. 8 3.8 is the mean of 5 positive numbers a, b, c, d and e. The sum of their squares is 360. Each of the numbers is now multiplied by 2. Find the new standard deviation.

Nan Hua High School Preliminary Examination 2022 Mathematics 4048/01 Page 7 of 19 9 One solution of the equation ( ) 21 15k x kx+ + = is 3x=− . (a) Find the value of k. (b) Find the second possible value of x. (b) Find angle BCD .

Nan Hua High School Preliminary Examination 2022 Mathematics 4048/01 Page 8 of 19 12 (a) The first four terms in a sequence of numbers are given below. 2 1 2 2 2 3 2 4 27 3 12 4 17 5 22 T T T T =+ =+ =+ =+ Explain why the value of nT must be odd for all values of n. (b) The product of the first n terms of a sequence is given by 223nn+ . Find the 12th term of this sequence.

Nan Hua High School Preliminary Examination 2022 Mathematics 4048/01 Page 9 of 19 13 The thickness of a layer of ice in a water body is 0.00205 m . (a) Write 0.00205 in standard form. 30.00205 2.05 10 −= (b) The ice covers an area of 521.60 10 m . Assuming that all the ice melts and ignoring the expansion of volume when water freezes, calculate the volume of water, in litres. Answer .……………………….………l [2] (c) A thunderstorm occurs and rainwater is falling at an average rate of 15 10 − litres per second over the water body. Calculate the percentage increase in the volume of water in the water body after two hours.

Nan Hua High School Preliminary Examination 2022 Mathematics 4048/01 Page 10 of 19 14 Given that the coordinates of A is ( )2, 3− , 5 7AB −= and 2.5 3.5FE k = − . (a) Find the value of k if ABFE is a parallelogram. 5 2.5 7 3.5 2 AB EF k k = −−   =       = (b) Find the coordinates of B. 52 73 3 4 is ( 3, 4) AB OB OA OB OB B =− −   =−    −    −= −