Beatty AM 3E EOY 2025

[Turn over BEATTY SECONDARY SCHOOL END-OF-YEAR EXAMINATION 2025 SECONDARY THREE EXPRESS / G3 CANDIDATE NAME CLASS REGISTER NUMBER ADDITIONAL MATHEMATICS 4049 7 October 2025 Setter: 2 hours 15 minutes Candidates answer on the Question Paper Additional Materials: Nil READ THESE INSTRUCTIONS FIRST Write your name, class and register number 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. Given 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 of the marks for this paper is 90. For Examiner’s Use This document consists of 19 printed pages and 1 blank pages.

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0, x = Binomial expansion (a + b)n = an + an − 1b + an − 2b2 + ... + an − r br + ... + bn, where n is a positive integer and = = 2. TRIGONOMETRY Identities sin2 A + cos2 A = 1 sec2 A = 1 + tan2 A cosec2 A = 1 + cot2 A Formulae for ABC = = a2 = b2 + c2 − 2bc cos A  = ab sin C a acbb 2 42−−       1 n       2 n       r n       r n )!(! ! rnr n − ! )1)...(1( r rnnn +−− BABABA sincoscossin)sin( = BABABA sinsincoscos)cos( = BA BABA tantan1 tantan)tan(  = AAA cossin22sin = AAAAA 2222 sin211cos2sincos2cos −=−=−= A AA 2tan1 tan22tan −= A a sin B b sin C c sin 2 1

3 [Turn over 1 (a) Express 216 2xx−− in the form of ()2 px q r++ where p, q and r are constants. [2] (b) Find the range of values of the constant k for which the equation 216 2 3xxk x−− =+ does not have two distinct roots. [3]

4 2 (a) Given logmax= and logmby=, express logabm in terms of x and y. [3] (b) Without using a calculator, evaluate 24 1 2 log 3 log 36 log 8−+ . [4]

5 [Turn over 3 (a) The equation of a curve is ()2 ya x b c=+ + where a, b and c are constants. The solutions of ()2 0ax b c++ = on the graph is 2− and 8, and the maximum value of y is 3. Find the values of a, b and c. [3] (b) Using the values of a, b and c found in (a), sketch the graph of ()2 ya x b c=+ + . [2] y x 0

6 4 (a) The diagram shows part of the graph of ()log 5ayx=+ . The vertical asymptote is b and the y-intercept is 1, where a and b are constants. Find the values of a and b. [2] (b) Solve ()3222 1 5 2 x x+ += . [4] 0 y x 1

7 [Turn over 5 The diagram shows a triangle PQR in which PR 12=cm, M is the mid-point of QR, angle PRQ 6 =radians and angle PQR is a right angle. Without using a calculator, find the value of p such that angle 1sin 14 pRPM − = . [5] P Q R M 12cm

8 46 (a) The lengths of the diagonals PR and QS of the rhombus PQRS are ()42 3+cm and 6 23  + cm respectively. Find the value of 2PQ, leaving your answer in the form ab c+. [4] (b) Determine if the curve 2 86yp x x p=+ + − lies completely above or below the x-axis for p > 8. [3]

9 [Turn over 7 Express ()() 2 2 11 9 4 32 1 xx xx ++ ++ in partial fractions. [5]

10 8 It is given that 32f( ) 2 8xxa xb x=+ ++ , where a and b are constants, has a factor of 4x+ and leaves a remainder of 30 when divided by 1x−. (a) Find the values of a and b. [4] (b) Using the values of a and b found in (a), show that the equation f( ) 0x= has only one real root. [3]