Dunman Sec 2025 Prelims 4EXP Add Math Paper 2 QP

This document consists of 17 printed pages and 1 blank page. DUNMAN SECONDARY SCHOOL CANDIDATE NAME CLASS INDEX NUMBER PRELIMINARY EXAMINATION 2025 SECONDARY 4 EXPRESS ADDITIONAL MATHEMATICS 4049/02 Paper 2 25 August 2025 2 hours 15 minutes Candidates answer on the Question Paper. 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 a pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all 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. The number of marks is given in brackets [ ] at the end of each question or part question. The total marks for this paper is 90. Calculator Model Total Marks

2 4049/02/PRELIM/4EXP/2025 1. ALGEBRA Quadratic Equation For the equation 2 0ax bx c+ + = , 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 rn r r n r r − − + == − 2. TRIGONOMETRY Identities 22sin cos 1AA+= 22sec 1 tanAA=+ 22cosec 1 cotAA=+ ( )sin sin cos cos sinA B A B A B =  ( )cos cos cos sin sinA B A B A B= ( ) tan tantan 1 tan tan ABAB AB = sin 2 2sin cosA A A= 2 2 2 2cos 2 cos sin 2cos 1 1 2sinA A A A A= − = − = − 2 2 tantan 2 1 tan AA A= − Formula for  ABC sin sin sin a b c A B C== 2 2 2 2 cosa b c bc A= + −  = 1 sin2 bc A

3 4049/02/PRELIM/4EXP/2025 [Turn over 1 (a) A ball is thrown vertically upwards. Its height h m, above the ground at time t seconds after being thrown is given by the formula 27 20 5h t t= + − . Find the maximum height attained by the ball and the time at which this occurs. [4] (b) Find the set of values of the constant k for which the curve 2 6y x x k= + + lies entirely above the line ( )1y k x=− . [4]

4 4049/02/PRELIM/4EXP/2025 2 A and B are acute angles, such that 3cos 5A= and 12tan 5B= . Find the exact values of (a) tan 2A , [2] (b) 3sin 2B  + , [3] (c) cos 2 B . [3]

5 4049/02/PRELIM/4EXP/2025 [Turn over 3 In the diagram, PQS lies on a circle and PCBQ lies on another circle. PQ is a common chord of the two circles. BPS is a straight line. QC is a tangent to the circle PQS at Q. QC and BP intersect at M. (a) Prove that BC is parallel to QS. [3] (b) Prove that triangle PQM is similar to triangle QSM. [2] (c) Hence show that SMPMQM =2 . [2] P Q B C M S

6 4049/02/PRELIM/4EXP/2025 4 (a) Show that ( ) 2d cot cosec d xxx =− . [4] (b) Hence find the value of 23 4 cot d xx   − . [4]

7 4049/02/PRELIM/4EXP/2025 [Turn over 5 (a) Given that the constant term in the binomial expansion of 8 3 kx x  − is 7, find the value of the positive constant k. [4] (b) Using the value of k found in part (a), show that there is no constant term in the expansion of ( ) 8 4 31 kxx x +−  . [4]

8 4049/02/PRELIM/4EXP/2025 6 The diagram below shows an isosceles triangle ABC, where AB = AC = 2 cm, and a square BCDE. M is the midpoint of BC. Angle BAM = θ and can vary. (a) Show that the total area of ABEDC is 16 sin2 θ + 4 sin θ cos θ cm2. [3] (b) Show that 16 sin2 θ + 4 sin θ cos θ = 2 sin 2θ − 8 cos 2θ + 8 . [2]

9 4049/02/PRELIM/4EXP/2025 [Turn over (c) Express 2 sin 2θ − 8 cos 2θ in the form R sin (2θ − α) , where R > 0 and 0° < α < 90° . [3]

10 4049/02/PRELIM/4EXP/2025 7 (a) Prove the identity x x xx xx tan1 tan1 cossin21 sincos 22 + −=+ − . [4] (b) Solve the equation xxx xx tan3 2 cossin21 sincos 22 =+ − for 02 x  . [4]