AISS 2025 AMath Paper 1 Prelim QP

_____________________________________________________________________________ This document consists of 19 printed pages. AISS PRELIM/4E/4049/P1/2025 [Turn over AHMAD IBRAHIM SECONDARY SCHOOL GCE O-LEVEL PRELIMINARY EXAMINATION 2025 SECONDARY 4 EXPRESS Name: Class: Register No.: ADDITIONAL MATHEMATICS Paper 1 Candidates answer on the Question Paper. 4049/01 12 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/P1/2025 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/4049/01/2025 [Turn over 1 It is given that P, Q and R are the angles of a triangle. (a) Show that cos P = – cos (Q + R). [2] (b) Given that Q = 45 and R = 60, find cos P in the form 1 4 ( a – b ), where a and b are integers. [3]

4 AISS PRELIM/4E/4049/01/2025 2 Baking powder is poured onto a flat surface at a constant rate of 312 cm s − and formed a right circular cone. The radius of the cone is always 1 18 of its height. Find the rate of change of the radius of the cone after 3 seconds of pouring. [5]

5 AISS PRELIM/4E/4049/01/2025 [Turn over 3 (a) Determine the set of values of m for which the equatio n 26242 2 −=++ mxmxx has real roots. [4] (b) Hence state what can be deduced about the curve 2)1(2 += xy and the line .26 −= xy Justify your statement. [2]

6 AISS PRELIM/4E/4049/01/2025 4 (a) Show that ( )( )d ln cos tand xxx =− . [2] (b) Differentiate tanxx with respect to x. [2] (c) Using the results from part (a) and (b), find 2sec dx x x and hence show that 24 0 1sec d ln 2 42x x x  =− . [4]

7 AISS PRELIM/4E/4049/01/2025 [Turn over 5 (a) In the expansion of ( )2 n x+ , where n is a positive integer, the coefficient of x2 is twice the coefficient of x. Find the value of n. [3] (b) Find the value of the term that is independent of x in the expansion of 15 4 12 4x x  − . [4]

8 AISS PRELIM/4E/4049/01/2025 6 The diagram shows a circle passing through the points P, Q and R. The point Q lies on the line RB. AB is a tangent to the circle at P. The points S and T lie on PR and PQ respectively. Given that AB is parallel to ST, prove that (a) triangle PST is similar to triangle PQR, [3] (b) PQ × PT = PR × PS, [2]

9 AISS PRELIM/4E/4049/01/2025 [Turn over (c) Determine if STQR is a cyclic quadrilateral. [4]

10 AISS PRELIM/4E/4049/01/2025 7 The diagram shows an isosceles triangle ABC in which ( )3,0A − , 1 ,32B  and ( )5, 2C . M is the foot of perpendicular from B to AC. (a) Find the coordinates of M. [1] (b) Find the equation of the perpendicular bisector of AC. [2] y O M x