AISS 2025 AMath Paper 2 Prelim QP

_____________________________________________________________________________ This document consists of 19 printed pages. AISS PRELIM/4E/4049/P2/2025 [Turn over AHMAD IBRAHIM SECONDARY SCHOOL GCE O-LEVEL PRELIMINARY EXAMINATION 2025 SECONDARY 4 EXPRESS Name: Class: Register No.: ADDITIONAL MATHEMATICS Paper 2 Candidates answer on the Question Paper. 4049/01 13 August 2025 2 hours 15 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. Give non-exact numerical answers 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 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 90. For Examiner’s Use /90

2 AISS PRELIM/4E/4049/P2/2025 Mathematical Formulae 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 n n 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 r n r r  − − +== − 2. TRIGONOMETRY Identities 1cossin 22 =+ AA AA 22 tan1sec += 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= − Formulae 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 AISS PRELIM/4E/4049/02/2025 [Turn over 1 A curve has the equation sin 2 2 cos 2 xy x= − . (a) Show that the gradient function can be expressed in the form ( ) 2 cos 2 2 2 cos 2 kx x − − , where k is a constant. [3] (b) Find the acute angle between the tangent to the curve a t 12x = and the line 0y= . [3]

4 AISS PRELIM/4E/4049/02/2025 2 (a) Factorise 33 27xk+ as a product of a linear and a quadratic factor. [2] (b) Hence solve ( )( ) 3 27 3 10x x x+ = + + , expressing non-integer roots in surd form. [3] (c) Find the value of k given that 33 27xk+ leaves a remainder of 351 when divided by 2x− . [2]

5 AISS PRELIM/4E/4049/02/2025 [Turn over 3 The diagram shows a L-shaped rod ABC where AB and BC have length 18 m and 8 m respectively and angle ABC is 90°. The rod is hinged to a wall at A so as to rotate in a vertical plane. The rod AB makes an acute angle θ with the vertical wall surface OA. (a) Given that G is a point directly below C, show that cos sinOG p q =+ , where p and q are constants to be found. [2] (b) Express OG in the form ( )cosR − where 0R and 0 90   . [3] (c) Find the length of OG and the corresponding value of θ if G is at maximum displacement from O. [3]

6 AISS PRELIM/4E/4049/02/2025 4 A circle 1C has equation 22 6 4 12x y x y+ − + = . (a) Find the radius and the coordinates of the centre of 1C . [3] (b) Find the equation of the tangent to the circle at the point P (7, 5)− . [3] (c) Another circle 2C has centre ( 8, 4)− and radius 7 cm. Find the shortest distance between the 2 circles. [2]

7 AISS PRELIM/4E/4049/02/2025 [Turn over 5 (a) Prove the identity ( )( ) 3 3 sin cos 1 sin cos tan 1cos A A A A AA −+ =− . [4] (b) Hence solve ( )( ) 3sin cos 1 sin cos 2cos 0A A A A A− + + = exactly, for A−   radians. [4]

8 AISS PRELIM/4E/4049/02/2025 6 The diagram shows the graph of 2 4y x ax= + + and cbxxy ++= 2 . The graph of 2 4y x ax= + + touches the x-axis at P. Points M and O are the -interceptsx of the graph of cbxxy ++= 2 . The origin O is the mid-point of MP. (a) Find the values of a, b and c. [4]

9 AISS PRELIM/4E/4049/02/2025 [Turn over (b) The graph of 2 4y x ax= + + and cbxxy ++= 2 intersects at N. Find the coordinates of N. [2] (c) The graph 2y px qx r= + + has its turning point at N and passes through point P. Find the values of p, q and r, where r > 0. [3]

10 AISS PRELIM/4E/4049/02/2025 7 A particle P, travels in a straight line, so that its displacement, s m, from O at time t seconds, is modelled by 321 533s t t= − − . (a) Find the value of t when particle P returns to its initial position. [2] (b) Find the minimum velocity of particle P. [3] (c) Another particle, Q, travels in a straight line from O such that its velocity, v m/s, at time t seconds after passing O is given by 1624 t v e e − −=−   . Find the value of t at which the particle Q is instantaneously at rest. [2]