CGS 2025 AMATH PRELIM P2 QP

Name: Register No.: Class: CRESCENT GIRLS’ SCHOOL SECONDARY FOUR 2025 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS 4049/02 Paper 2 3 September 2025 Candidates answer on the Question Paper. 2 hours 15 mins No Additional Materials are required. 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. 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. Mathematical Formulae Question 1 2 3 4 5 6 7 8 9 10 Marks Table of Penalties Qn. No. Presentation –1 Significant Figures/ Units –1 Parent’s/Guardian’s Signature For Examiner’s Use This document consists of 16 printed pages. 9

Crescent Girls’ School 202 5 Prelim Sec 4 A Math P2 2 1. ALGEBRA Quadratic Equation For the equation 02 = + +c bx ax, a ac b bx 2 42 − ± −= Binomial expansion 1 22( ) ... ...12 n n n n nr r nnn nab a ab a b a b b r −− −  + = + + ++ ++     , where n is a positive integer and ! ( )...( 1) !( )! ! n n n nr nr r rn r r  − −+= = − 2. TRIGONOMETRY Identities 1cos sin22 = +A A AA 22 tan1sec + = 22cosec 1 cotAA= + B A B A B Asin cos cos sin)sin( ±= ± cos( ) cos cos sin sinAB A B A B±=  tan tantan( ) 1 tan tan ABAB AB ±±=  A A Acos sin2 2sin = AA A A A 2222 sin2 1 1cos2sin cos2cos − = − = − = A AA 2tan1 tan22tan −= Formulae for ∆ABC C c B b A a sin sin sin= = A bc c b acos22 2 2− + = ∆ = C absin2 1

Crescent Girls’ School 202 5 Prelim Sec 4 A Math P2 [Turn over 3 1 (a) Show that the equation 2376xxee  has only one solution and find its value correct to 2 significant figures. [4] (b) Factorise (3x + 1) 3 – 8 completely. [2]

Crescent Girls’ School 202 5 Prelim Sec 4 A Math P2 4 2 (a) Show that x – 1 is a factor of 3x3 + 8x2 – 5x – 6. [1] (b) Hence solve the equation 3x 3 + 8x2– 5x = 6 completely. [4] (c) Hence solve the equation 3(y –2)3 + 8(y – 2)2 – 5y + 4 = 0. [2]

Crescent Girls’ School 202 5 Prelim Sec 4 A Math P2 [Turn over 5 3 (a) Show that xxxdx d 2sec6 2sec6 ) 2(tan 243 −= . [3] (b) Hence, find the exact value of ∫8 0 4 2sec π dx x. [5]

Crescent Girls’ School 202 5 Prelim Sec 4 A Math P2 6 4 The diagram shows two chords AB and CD intersect at X such that angle 90CXB  . Given that AC = 5 cm, BD = 12 cm and angle BDC θ . (a) Prove that 5cos 12sinAB θθ . [3] (b) Express AB in the form )cos( α θ−R where R > 0 and α is an acute angle. [4] 12 cm C D B A X 5 cm θ

Crescent Girls’ School 202 5 Prelim Sec 4 A Math P2 [Turn over 7 (c) Find the maximum length of AB and the corresponding value of θ. H ence state the radius of the circle. [3]

Crescent Girls’ School 202 5 Prelim Sec 4 A Math P2 8 5 (a) Find the range of values of k for which kx2 + 8x + k < 6 for all real values of x. [4] (b) Given the curve y = 8x 2 + mx + m – 6, where m is a constant, find (i) the possible values of m for which the x-axis is a tangent to the curve. [3] (ii) the range of values of m for which the curve has a positive y-intercept. [1] (c) Show that the equation 2x 2 – cx + c2 + 6 = 0 has no real roots for all real values of c. [2]

Crescent Girls’ School 202 5 Prelim Sec 4 A Math P2 [Turn over 9 6 It is given that n x kx      + 2 is a binomial expansion where k and n are positive constants. (a) Write down the first 4 terms in the expansion 5 2      + x kx in terms of k. [2] (b) Hence, find the value of k if the term independent of x in the expansion 5 22 35      +     + x kxxx is 5. [3] (c) By using the formula for the general term, suggest two possible powers of n such that n x kx      + 2 contains a term independent of x . [3]

Crescent Girls’ School 202 5 Prelim Sec 4 A Math P2 10 7 The equation of a circle with centre C is x2 + y2 – 8x – 10y + 16 = 0. A point B (7, 9) lies on the circumference of the circle. P is the lowest point on the circle. (a) Find the coordinates of C and the radius of the circle. [3] (b) Find the equation of the tangent to the circle at B. [3] (c) Explain why P lies on the x-axis. [1]