NHHS 2023 EM Prelims P2 (Solution)

Page 1 of 24 Name ( ) Class PRELIMINARY EXAMINATION 2023 Subject : Mathematics Paper : 4052/02 Level : Secondary Four Express Date : 21 August 2023 Duration : 2 hours 15 minutes 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. Give answers in degrees to one decimal place. For π, use either your calculator value or 3.142. For Examiner’s Use Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 U TOTAL P A This paper consists of 22 printed pages and 2 blank pages. NAN HUA HIGH SCHOOL

Nan Hua High School Preliminary Examination 2023 Mathematics 4052/02 Page 2 of 25 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 4052/02 Page 3 of 25 1 (a) (i) Solve the inequalities 4 1 3 5 9 .2 4 5 x x x     4 1 3 5 9 2 4 5 4 1 3 3 5 9 and2 4 4 5 2 4 1 3 15 20 36 8 2 3 5 36 5 2 2 1 75 5 1 27 or 7.2 0.45 5 x x x x x x x x x x x x x x x x x x x                        Answer ……………… …………... [2] (ii) Represent the solutions of the inequalities on the number line below. [1] (iii) State the smallest integer which satisfies the inequalities. 7 Answer ……………… …………... [1]

Nan Hua High School Preliminary Examination 2023 Mathematics 4052/02 Page 4 of 25 (b) Simplify 3 0 2 3 5 1 2 10 a ab ab b         , leaving your answer in positive index form. 3 0 2 3 5 2 3 5 33 3 1 2 10 8 10 4 4 or 5 5 a ab ab b b a a a b a a b b                 Answer ……………… …………... [3] (c) Express 2 10 1 3 21 30 2 x x x x     as a single fraction in its simplest form.            2 10 1 3 21 30 2 10 1 3 2 5 2 10 3 5 3 2 5 25 2 3 2 5 x x x x x x x x x x x x x x x                  Answer ……………… …………... [3]

Nan Hua High School Preliminary Examination 2023 Mathematics 4052/02 Page 5 of 25 (d) (i) y is inversely proportional to the square of x. Which of these diagrams represents the graph of y against x? Answer Diagram …… …………... [1] (ii) Find the percentage change in y when x is increased by 150%.     2 0 0 0 2 0 0 2 0 2 0 0 0 0 0 0 when , when 2.5 2.5 6.25 1 4 or 6.25 25 1 6.25% change 100% 84% ky x x x y y ky x x x ky x k x y y y y y                   Answer ……………… …………% [3] 2

Nan Hua High School Preliminary Examination 2023 Mathematics 4052/02 Page 6 of 25 2 At a store, 20 red apples cost $x. The same number of green apples in the store cost $4 more. (a) Write an expression, in terms of x, for the number of red apples that can be bought with $2.40. 48 20 or 2.4x x  Answer ……………… …………... [1] (b) Write an expression, in terms of x, for the number of green apples that can be bought with $2.40. 48 20 or 2.44 4x x   Answer ……………… …………... [1] (c) $4.80 can buy a total of 10 apples at the store. Write down an equation in x to represent this information and show that it reduces to 25 28 96 0.x x   Answer     2 2 2 48 48 104 48 4 48 10 4 48 192 48 10 40 10 56 192 0 5 28 96 0 x x x x x x x x x x x x x x                 [3]

Nan Hua High School Preliminary Examination 2023 Mathematics 4052/02 Page 7 of 25 (d) Solve the equation 25 28 96 0.x x      25 28 96 0 5 12 8 0 2.4 or 8 x x x x x                 2 2 5 28 96 0 28 28 4 5 96 2 5 2.4 or 8 x x x           Answer x = ….. ……… . or …… ……… [2] (e) Find the cost of 30 green apples in store A. rej 2.4 as price 0 8 cost of 20 green apples $12 12cost of 30 green apples $ 3 2 $18 x x         Answer $……………… …………... [2]

Nan Hua High School Preliminary Examination 2023 Mathematics 4052/02 Page 8 of 25 3 The cumulative frequency curve shows the distribution of the daily temperature of city X in the first 80 days of the year 2021. (i) Use the curve to estimate (a) the median temperature, Median = 13.5°C Answer ……………………… .. °C [1] (b) the interquartile range of the temperatures, Interquartile range = 16.5 – 11 = 5.5°C Answer … …………………… °C [2] (c) the 65th percentile, 65th percentile = 15.4°C Answer ……………………… . °C [1] 0 10 20 30 40 50 60 70 80 10 11 12 13 14 15 16 17 18 19 Temperature (°C) Cumulative frequency

Nan Hua High School Preliminary Examination 2023 Mathematics 4052/02 Page 9 of 25 (d) the percentage of days where the temperature was at least 17°C. 80 66required % 100% 17.5%80    Answer ……………………… .. % [1] (ii) The temperatures of the first 80 days of City X for the year 2022 were also recorded. The median temperature for the year 2022 is smaller than that of the year 2021. The interquartile range of the temperatures for the year 2022 is the same as that of the year 2021. Describe how the cumulative frequency curve for the year 2022 may differ from the curve in the year 2021. ……………………………………………………… …………… ………………… ………… ………………………………………………………………………… ……… ...……… [1] (iii) The temperatures of the first 80 days of City Y in the year 2021 were also recorded. The box-and-whisker plot shows the distribution of the temperatures. Make two comments comparing the temperatures of both cities. …………………………………………………………………………… ……………… … …………………………………………………………………………… ……………… … …………………………………………………………………………… ……………… … …………………………………………………………………………… ……………… … ………………………………………………………………………… ……………… … [2] 10 11 12 13 14 15 16 17 18 19 Temperature (°C) The median temperature of City Y is 14.5°C which is higher than that of City X. The temperature in City Y is generally warmer/higher than in City X. The interquartile range of temperatures of City Y is 4°C which is lower than that of City X. There is a narrower/smaller spread of temperatures in City Y than in city X. The curve of the year 2022 will be to the left of the year 2021.

Nan Hua High School Preliminary Examination 2023 Mathematics 4052/02 Page 10 of 25 4 (a) Complete the table of values for   3 1 4 1.7 3 x xy    Give your answer correct to 1 decimal place. x – 3 – 2 – 1 0 1 2 3 4 5 y – 4.1 – 0.2 1.2 0.9 – 0.3 – 1.5 – 1.9 – 0.5 3.5 [1] (b) On the grid, draw the graph of   3 1 4 17 3 x xy    for 3 5.x   [3] -3 -2 -1 0 1 2 3 4 5 y – – – –1 –2 –3 –4 –5 x 1 2 3 4