PLMGS 4E 5NA Math P2 Prelim 2017 FINAL

Name: ___________________________ ( ) Class: _______ Centre Number S Index Number MATHEMATICS 4048/02 Paper 2 21 August 2017 Additional Materials: Answer Paper (10 sheets) 2 hours 30 minutes Graph Paper (1 sheet) READ THESE INSTRUCTIONS FIRST Write your Centre number, index number, name and class on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a 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 a 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 . At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 100. This paper consists of 13 pages, including this cover page. Paya Lebar Methodist Girls’ School (Secondary) Preliminary Examination 2017 Secondary 4 Express / 5 Normal Academic

2 [Turn over Mathematical Formulae Compound interest Total amount = n rP       1001 Mensuration Curved surface area of a cone = rl Surface area of a sphere = 4r2 Volume of a cone = 3 1 r2h Volume of a sphere = 3 4 r3 Area of triangle ABC = 2 1 ab sin C Arc length = r, where  is in radians Sector area = 2 1 r2, where  is in radians Trigonometry C c B b A a sinsinsin  a2 = b2 + c2 – 2bc cos A Statistics Mean = f fx   Standard deviation = 22          f fx f fx

3 [Turn over Answer all the questions. 1 (a) (i) Factorise 254016 22  xyyx . [1] (ii) If kxyyx  254016 22 , express y in terms of x and k, where k > 0. [2] (b) Solve the equation 2 15 2 3 436 xx  . [2] (c) Simplify 2 2 412 28 xx x   . [2] (d) Given that 311  yx , find the value of yxyx yxyx   2 232 . [2]

4 [Turn over R S T Q O P 2 (a) Each exterior angle of an n-sided regular polygon is 16 greater than each exterior angle of another 3n-sided regular polygon. (i) Find n. [2] (ii) Calculate an interior angle of the 3n-sided regular polygon. [2] (b) In the diagram, PQRS is a rhombus and QS is a diagonal. RS is produced to point T such that S is the midpoint of RT. PS and QT intersect at O. (i) Show that triangles QRS and PST are congruent. [3] (ii) Explain why PQST is a parallelogram. [2] (iii) Show that triangles POQ and RQT are similar. [2]

5 [Turn over 3 Toothpicks are used to form a series of patterns as shown. 1st 2nd 3rd 4th nth pattern Total number of toothpicks used Total number of regions formed within the pattern Number of points at which 2 or more toothpicks meet T R P 1 5 1 5 2 14 4 11 3 27 9 19 4 44 16 29     6 a b c (a) Write down the value of a, of b and of c. [3] (b) Explain why the number 99 cannot appear in column R. [1] (c) Express P in terms of T and R. [1] (d) Show that the nth term of the pattern, Tn, is given by nnTn 32 2  . [1] (e) Tp + 1 and Tp are consecutive terms in the pattern. (i) Find and simplify an expression, in terms of p, for Tp + 1 − Tp. [2] (ii) Explain why two consecutive terms of the pattern is not a multiple of 4. [1] (f) Hence or otherwise, find the number of points at which 2 or more toothpicks meet in the 11th pattern. [1] 1st 2nd 3rd 4th

6 [Turn over 4 (a) The diagram shows a regular hexagon ABCDEF enclosed exactly by a circle of centre O and radius r cm. (i) Show that angle AOC is 2 3 radians. [1] (ii) Find the shaded area in terms of r. [4] (b) The diagram shows a circle with centre O. A, B, C, D, E and F are points on the circle. Show that angle ABC + angle CDE + angle EFA = 360. [3] B A O E F D C

7 [Turn over 5 A, B and C are three points on level ground. Angle BAC = 38° and AC = 50 m. T is the top of a vertical flagpole TB. (a) The angle of elevation of the top of the flagpole from A is 24°. The angle of elevation of the top of the flagpole from D, 6 m from A, is 32°. Calculate the height of the flagpole. [3] (b) Find the length BC. [2] (c) Another flagpole EF is to be positioned vertically on AC such that its base, E, is nearest to B. Find the length AE. [2] 38° 50 m

8 [Turn over 6 In the diagram, OADE is a trapezium and FC is parallel to OB. It is given that OF = 4a and OB = 7b. OF : OE = 3 : 7 and 7OA = 5OB. (a) Express, as simply as possible, in terms of a and/or b, (i) ,FC [1] (ii) ,CB [1] (iii) ,EC [1] (iv) .ED [1] (b) What two facts can be deduced about E, C and B? [2] (c) Find the ratio of (i) area of triangle EFC : area of triangle EOB, [1] (ii) area of triangle EDC : area of triangle ABC, [1] (iii) area of quadrilateral EDCF : area of quadrilateral OBCF. [2] A B O C D E F

9 [Turn over 7 Sticker printing machines print stickers at a constant rate. (a) Machine A prints 35 stickers in one hour. Machine B prints y stickers in one hour. Machine A and machine B print a total of 38 stickers in 30 minutes. Find y. [2] (b) (i) Machine C prints x stickers in one hour. Write an expression, in terms of x, for the number of hours taken by Machine C to print 37 stickers. [1] (ii) Compared to machine C, Machine D prints 4 more stickers in one hour. Write an expression, in terms of x, for the number of hours taken by Machine D to print 37 stickers. [1] (iii) The difference in the timing taken by machine C and machine D, to each print 37 stickers is 15 minutes. Write an equation in x to represent this information and show that it reduces to x2 + 4x – 592 = 0. [3] (iv) Solve the equation x2 + 4x – 592 = 0. [3] (v) Find the time taken by Machine C to print 37 stickers. Give your answer in hours and minutes, correct to the nearest minute. [1]

10 [Turn over 8 Answer the whole of this question on a sheet of graph paper. The variables x and y are connected by the equation .5232       xxy Some corresponding values of x and y, are given in the table below. x 0.5 1 1.5 2 2.5 3 3.5 4 y 4 0 0 1 2.4 4 p 7.5 (a) Calculate the value of p. [1] (b) Using a scale of 4 cm to represent 1 unit, draw a horizontal x-axis for 0  x  4. Using a scale of 2 cm to represent 1 unit