NHHS 2022 Math PRELIMS P2 (QP)

This paper consists of 28 printed pages and 3 blank pages. Page 1 of 28 Name ( ) Class PRELIMINARY EXAMINATION 2022 Subject : Mathematics Paper : 4048/02 Level : Secondary Four Express Date : 18 August 2022 Duration : 2 hours 30 minutes 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 an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue, or correction fluid. 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 100. For Examiner’s Use Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 TOTAL NAN HUA HIGH SCHOOL

Nan Hua High School Preliminary Examination 2022 Mathematics 4048/02 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 2022 Mathematics 4048/02 Page 3 of 28 Answer all questions. 1 (a) Simplify 22 3 3 12 .5 25 ac b ab  Answer …………………………... [3] (b) n is a positive integer. Show that, for all n , 22(3 2) (3 2)nn+ − − is a multiple of 3. Answer [2]

Nan Hua High School Preliminary Examination 2022 Mathematics 4048/02 Page 4 of 28 (c) Solve the equation 27 1.4 3 2 x xx +=−− Answer x = …..………. or …………… [4] (d) Use the quadratic formula to solve the equation. 23 7 4 0xx− − = Answer x = …..………. or …………… [2]

Nan Hua High School Preliminary Examination 2022 Mathematics 4048/02 Page 5 of 28 2 A box of chocolate contains 7 dark chocolates and 5 milk chocolates. Sufyan takes a chocolate, selected at random, from the box and eats it. LeLe then takes a chocolate, selected at random, from the box. (i) Draw a tree diagram to show the probabilities of the possible outcomes. Answer [2] (ii) Find, as a fraction in its simplest form, the probability that (a) Sufyan and LeLe both picked dark chocolates, Answer …………………………... [1] (b) LeLe picked a milk chocolate, Answer …………………………... [2] (c) at least one milk chocolate was chosen. Answer …………………………... [2]

Nan Hua High School Preliminary Examination 2022 Mathematics 4048/02 Page 6 of 28 3 In the diagram, and .OA OB==ab 2 3 , 3 , and 2 .OB BC OD OA AX XB= = = (i) Express, as simply as possible, in terms of a and b, (a) ,BA Answer BA= …………………. [1] (b) ,OX Answer OX = …………………. [1] (c) .CD Answer CD= …………………. [1] (ii) It is given that CM : MD is 10 : 9 . (a) Express , as simply as possible, in terms of and OM a b. Answer OM = …………………. [1]

Nan Hua High School Preliminary Examination 2022 Mathematics 4048/02 Page 7 of 28 (b) Show that O, X and M are collinear. Answer [2] (c) Given that MY is parallel to OD, express in terms of and .MY ab Answer MY =…………………. [1] (iii) Find the ratio of (a) area of ,area of XMY XOA   Answer …………………………. [1] (b) area of .area of OBX OYA   Answer …………………………. [2]

Nan Hua High School Preliminary Examination 2022 Mathematics 4048/02 Page 8 of 28 4 In the diagram, OAPB is a sector of circle with centre O and radius 9 cm. Angle AOB = 1.4 radians. C is the centre of the circle enclosed inside the sector. OCP is a straight line and the circle touches the sector at points P, Q and R. (i) Show that the radius of the enclosed circle is 3.526 cm, correct to 3 decimal places. Answer [5]

Nan Hua High School Preliminary Examination 2022 Mathematics 4048/02 Page 9 of 28 (ii) Calculate the area of the shaded region. Answer ………………………… cm2 [4]

Nan Hua High School Preliminary Examination 2022 Mathematics 4048/02 Page 10 of 28 5 The diagram shows a right circular cone cut from a solid steel sphere. Point O is the centre of the sphere with radius 12 cm. (a) Given the circumference of the base of the cone is 50 cm, show that the height of the cone is 21.0 cm, corrected to 3 significant figures. Answer [3] (b) Find the curved surface area of the cone. Answer ……………………. cm2 [2]