2022 XMS EMATH P2 PRELIM

XINMIN SECONDARY SCHOOL SEKOLAH MENENGAH XINMIN Preliminary Examination 2022 CANDIDATE NAME CLASS INDEX NUMBER MATHEMATICS Secondary 4 Express/5 Normal (Academic) Setter: Ms Pang Hui Chin Vetter: Ms Low Yan Jin Moderator: Ms Yap Bee Leng 4048/02 25 August 2022 2 hours 30 minutes Candidates answer on the Question Paper READ THESE INSTRUCTIONS FIRST Write your name, register number and class in the spaces at the top of this page. 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  . At the end of the paper, 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. Errors Qn No. Errors Qn No. Accuracy Simplification Brackets Units Geometry Marks Awarded Presentation Marks Penalised This document consists of 25 printed pages and 1 blank page. [Turn over For Examiner’s Use 100 Parent’s/Guardian’s Signature:

2 Mathematical Formulae Compound Interest Total amount = 1 100 n rP+ Mensuration Curved surface area of a cone = rl Surface area of a sphere = 24 r Volume of a cone = 21 3 rh Volume of a sphere = 34 3 r Area of triangle ABC = 1 sin2 ab C Arc length = r , where  is in radians Sector area = 21 2 r  , where  is in radians Trigonometry sin sin sin a b c ABC== 2 2 2 2 cosa b c bc A= + − Statistics Mean = fx f   Standard deviation = 22fx fx ff −  

3 BLANK PAGE

4 1 (a) Express 2 21 (2 3) 2 3 x xx −−− as a single fraction in its simplest form. Answer …………………………….. [2] (b) It is given that 22() 3 h r pV  −= . (i) Find the value of V when 4.5h= , 7r = and 3.4p=− . Answer V = …………………………. [1] (ii) Express r in terms of V, h and p. Answer …………………………….. [2]

5 [Turn over (c) (i) Simplify 2 3 6 729 x   . Answer …………………………….. [1] (ii) Given that 3 10 2 4 20 q r x px xyy = , where p, q and r are constants, find the value of p, q and r. Answer p = ………………………….. q = ………………………….. r = ………………………….. [3]

6 2 The diagram shows a circle PQRS, with centre O. SOP is a straight line. When produced, the line SP and the tangent to the circle at Q meet at T. Angle PRQ = 35 . (a) Giving reason(s) for each answer, (i) show that angle SOQ 110= , [3] (ii) calculate the angle OTQ. [2] 35 O P S R Q T V

7 [Turn over (b) Given that the radius of the circle is 6 cm and PR = 10.4 cm, find (i) the length of OT, Answer ………………………….. cm [2] (ii) the exact value of cos RPT . Answer …………………………….. [2]

8 3 (a) The point X is ( )12, 5− and the point Y is ( )4,7− . (i) Express XY as a column vector. Answer …………………………….. [1] (ii) The point Z is such that XZ : XY = 3 : 4 and Z is on the line XY. Find the coordinates of Z. Answer …………………………….. [2]

9 [Turn over (b) In the diagram, ABCD is a parallelogram. AG= a , AF = b and 2AG GB= . The diagonals AC and BD intersect at E. F is the midpoint of AE. (i) Express, as simply as possible, in terms of a and/or b, DF . Answer …………………………….. [1] (ii) Show that triangles AGF and CDF are similar. Answer ………………………………………………………………….. …………………………………………………………………………… …………………………………………………………………………… …………………………………………………………………………… [2] (iii) Find (a) area of area of DFE DEC   , Answer …………………………….. [1] (b) area of area of AFG DFE   . Answer …………………………….. [2] A B C D F E G a b

10 4 The diagram shows a circle with centre B and two congruent semicircles with centres A and C inscribed in a large semicircle with centre O. The circle with centre B has a radius of (2r + 1) cm. The semicircles with centres A and C have a radius of (12 – 6r) cm. (a) (i) Write down an expression, in terms of r, for the length of BC. Answer ………………………….. cm [1] (ii) Show that the length of BO is ( )23 14 cmr− . [1] (b) Hence, form an equation in r and show that it reduces to 26 19 14 0rr− + = . [3] Answer B A O C Y X 12 6 r− 21r+