AHS Sec 3 EOY

NAME: _______________________________ ( ) CLASS: 3 ( ) 9o ADDITIONAL MATHEMATICS 4049 Candidates answer on the Question Paper and Graph Paper 27 September 2024 2 hours Additional Material: 1 sheet of Graph Paper READ THESE INSTRUCTIONS FIRST Write your name, index number and class on all the work you hand in. Write in dark blue or black pen. You may use a HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. If working is needed for any question, it must be shown with the answer. Omission of essential working will result in loss of marks. The use of an approved scientific calculator is expected, where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For π , use either your calculator value or 3.142, unless the question requires the answer in terms of π. 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 80. --------------------------------------------------------------------------------------------------------------------------- For Examiners’ Use Questions 1 2 3 4 5 6 7 8 Marks Questions 9 10 11 12 13 14 15 Marks Units 80 Clarity/Logic Accuracy/Precision Parent’s Name and Signature: Date: This document consists of 16 printed pages. ANGLICAN HIGH SCHOOL SECONDARY THREE END OF YEAR EXAMINATIONS 2024 S3

Page 2 of 17 Anglican High School End of Year Examination 2024 [Turn over Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0ax bx c+ += , 2 4 2 b b acx a −± −= Binomial expansion ( ) 1 22 ... ...12 n n n n nrr nnn nab a a b a b a b b r −− −  + = + + ++ ++     , where n is a positive integer and ( ) ( ) ( )1 ... 1! !! ! n nn n rn r rnr 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 ±±=  sin 2 2sin cosA AA= 22 2 2cos2 cos sin 2cos 1 1 2sinA AA A A= − = −=− 2 2tantan2 1 tan AA A = − Formulae for ABC∆ C c B b A a sin sin sin= = A bc c b acos22 2 2− + = Area of C absin 2 1= ∆

Page 3 of 17 Anglican High School End of Year Examination 2024 [Turn over 1 (a) Express 22 8 26xx− +− in the form of ( ) 2ax h k−+ . [2] (b) Hence, show that the expression 22 8 26xx− +− is negative for all real values of x, [2]

Page 4 of 17 Anglican High School End of Year Examination 2024 [Turn over 2 The equation of a curve is 11 1xy−= . (i) It is given that the line 1y kx= − , where 0k ≠ , is a tangent to the curve at point Q. Find the value of k. [4] (ii) Using this value of k, find the coordinates of Q. [2] (iii) Deduce the range of values of k given that 1y kx= − does not intersect the curve 11 1xy−= . [2]

Page 5 of 17 Anglican High School End of Year Examination 2024 [Turn over 3 The polynomial ( ) 32p 6 15x x ax bx= + +− has a factor of 23x+ and leaves a remainder of 21 when divided by ( ) 2x+ . (a) Find the value of a and of b. [4] (b) Hence, factorise the expression completely. [3] 4 (i) Simplify the expression 396 11 99 − . [2]

Page 6 of 17 Anglican High School End of Year Examination 2024 [Turn over (ii) Hence, simplify 1 396 11 992 11 ÷−−  . [2] 5 Solve 232 2 43x xx−= − + . [3]

Page 7 of 17 Anglican High School End of Year Examination 2024 [Turn over 6 Express 32 2 9xx− in partial fractions. [4] 7 The diagram shows part of a straight-line graph. (a) Express y in terms of x in the form ax bye += . [3] ln y x O (-1, 2) (2, 8)

Page 8 of 17 Anglican High School End of Year Examination 2024 [Turn over (b) Find the value of x when ln 1.5y= . [1] 8 Solve the following equations. (a) 127 35xx− ×= . [3]

Page 9 of 17 Anglican High School End of Year Examination 2024 [Turn over (b) ( ) ( )71 7 7 1log 1 log 1 2 log 3xx++ −=+ [3]

Page 10 of 17 Anglican High School End of Year Examination 2024 [Turn over 9 Mr Chan makes a cup of tea and leaves it on the table to cool. The temperature of the tea, TC° is given by the equation 27 50 kxTe −= + , where x is in minutes. The time taken for the tea to cool to 42 C° is 11 minutes. Find (a) the initial temperature of the tea, [1] (b) the time taken for the tea to cool to 35 C° . [3] (c) with justification, the temperature of the tea after a long time. [1]