2022 XMS AMATH P2 PRELIM QP

This document consists of 21 printed pages and 1 blank page. [Turn over XINMIN SECONDARY SCHOOL SEKOLAH MENENGAH XINMIN Preliminary Examinations 2022 CANDIDATE NAME CLASS INDEX NUMBER ADDITIONAL MATHEMATICS Paper 2 Secondary 4 Express/ 5 Normal Academic Setter : Mr Johnson Chua Vetter : Mrs Loh Si Lan Moderator : Ms Pang Hui Chin Candidates answer on the Question Paper. No Additional Materials are required. 4049/02 30 August 2022 2 hour 15 minutes 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 the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an approved scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. 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 90. Errors Qn No. Errors Qn No. Accuracy Simplification Brackets Units Geometry Marks Awarded Presentation Marks Penalised For Examiner’s Use 90 Parent’s/Guardian’s Signature:

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the quadratic equation ax 2 + bx + c = 0, 2 4 2 b b acx a −  −= Binomial Expansion ( ) 1 2 2 12 n n n n n r r nn n na b a a b a b a b b r − − −     + = + + + + + +           , where n is a positive integer and ( ) ! ( 1)...( 1) ! ! ! n n n n n r r n r r r  − − +== − 2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B  sin A sin B tan ( A ± B ) = tan tan 1 tan tan AB AB  sin 2A = 2 sin A cos A cos 2A = cos2 A – sin2 A = 2cos2 A – 1 = 1 – 2 sin2 A tan 2A = 2 2 tan 1 tan A A− Formulae for  ABC sin sin sin a b c A B C== a 2 = b 2 + c 2 − 2bc cos A  = 2 1 bc sin A

3 [Turn over 1 (a) Solve the equation lg( 3) 1 lg( 2)xx+ = − − . [3] (b) Given that 2log pa= , log 8q b= and 2cp q = , express c in terms of a and b. [3]

4 2 Given that 4 14 xy x = − , show that ( ) 3 d 4 8 d 14 yx x x −= − . [4]

5 [Turn over 3 The graph cos 2y a x b=+ , where a and b are positive integers, has a maximum value of 8 and a minimum value of 2− . (a) Show that a = 5 and b = 3. [2] (b) Hence, sketch the graph cos 2y a x b=+ for 3π0 2x . [3]

6 4 (a) By considering the general term of 9 kx x  + , where k is a positive constant, explain why every term is dependent of x. [2] (b) Given that the coefficients of x3 and x5 are the same, find the value of k. [3]

7 [Turn over 4 (c) Using the value of k found in (b), find the coefficient of x7 in the expansion of ( ) 9 615 kxx x −+  . [3]

8 5 The diagram above shows a structure. The structure has three fixed points O, A and D such that OA = 18 cm, OD = 30 cm, and angle AOD = 90 . The lines AB and DC are perpendicular to OC which makes an angle  with the line OD. The angle θ can vary in such a way that B lies between the points O and C. (a) Show that AB + BC + CD = (12sin 48cos )+ cm. [3] (b) Express AB + BC + CD in the form sin( )R + and 0 90   . [3] O C B A D  18 cm 30 cm

9 [Turn over 5 (c) Find the maximum value of AB + BC + CD and the corresponding area of the structure. [3]

10 6 (a) The straight line 5yx− =− intersects the curve 222 2 11x y xy− = + at the points A and B. Find the coordinates of A and B. [3] (b) Find the exact values of the constant k for which the line ( 1)y k x=+ is a tangent to the curve 2 63y x x k= + + . [3]