ACSI 2021 Y3 Core Mathematics Paper 2

FINAL EXAMINATION 2021 YEAR 3 INTEGRATED PROGRAMME CORE MATHEMATICS PAPER 2 MONDAY 11th October 2021 1 hour 30 minutes ADDITIONAL MATERIALS: Answer Paper (7 sheets) Graph Paper ( 1 sheet) INSTRUCTIONS TO STUDENTS Do not open this examination paper until instructed to do so. A calculator is required for this paper. Answer all the questions on the answer sheets provided. At the end of the examination, fasten the answer sheets together. Unless otherwise stated in the question, all numerical answers must be given exactly or correct to three significant figures. Answers in degrees are to be given to one decimal place. INFORMATION FOR STUDENTS The maximum mark for this paper is 80. ______________________________________________________________

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2021/FinalExamination 2 This question paper consists of 4 printed pages. [Turn over Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. Where an answer is incorrect, some marks may be given for correct method, provided this is shown by written working. You are therefore advis ed to show all working. Answer all the questions on the answer sheets provided. Begin each question on a new page. 1. [Maximum mark: 12] (a) Evaluate 3 3 ln(3.256) 1.325 (0.25 2.38) e+ − , leaving your answer correct to 3 significant figure. [2] (b) Simplify ( ) 3 2 1 1 2 4 82 4 3 (3 ) 3 x y yx −   , expressing your answer in positive indices. [3] (c) Simplify 13 21 27 2(3 ) 39 yy yy + +− + . [3] (d) Solve for x if 2 651xx+− = . [4] 2. [Maximum mark: 11] (a) Expand and simplify ( )( ) 2 2 2 2 1 n p n m p mm n n m p  − + −+−  − . [3] (b) Find the sum of 35 x x− and 2 6 10 9 25 x x +− − , expressing your answer as a single fraction in its simplest form. [3] (c) Solve the equation ( )( ) 23 5 4 04 2 4 xx x x x + +=− + − , leaving your answers in 2 decimal places. [5]

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2021/FinalExamination 3

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2021/FinalExamination 4 3. [Maximum mark: 9] A circular cylindrical container of base radius xe cm, and height 2xe cm is fully filled with water. (a) Given that the volume of the water in the container is 12900 cm3, find the value of x . [3] The water is poured into a rectangular tank of base area 850 cm2 and height 60 cm. (b) Find the depth of the water in the rectangular tank. [2] (c) A solid metal sphere of radius 14 cm is then put into the rectangular tank. If the sphere is totally immersed in the water, will the water overflow? Explain your answer. [4] 4. [Maximum mark: 7] (a) Solve 33log (5 ) 1 log (1 2 )xx− = − − . [4] (b) Evaluate 2 3 4 5 63log 3 log 4 log 5 log 6 ... log (64)     . [3] 5. [Maximum mark: 15] O, A, B and C are four points on a flat ground. C is on a bearing of 50 o from O. B is due east of A and south of C. OB = OC = 60 m and 90oAOB= . A vertical pole, TC, 25 m high, is located at C. Find (a) the bearing of O from B, [2] (b) the distance OA, [3] (c) the distance BC, [3] (d) the area of OABC, [3] (e) the largest angle of elevation along OB to the top of the pole at C. [4] [Turn over O A B C T 60 m 60 m 25 m

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2021/FinalExamination 5 6. [Maximum mark: 11] (a) Find the set of values of the constant m for which the line 22y x m=− intersects the curve 2 3yx=− at two distinct points. [4] (b) Let 22( ) 2 4 3 5f x x kx k= − + + . (i) Prove that the equation ( ) 0fx = has no real roots. [3] (ii) Find the value(s) of k if the graph of ()fx passes through the point (1, 6). [4] 7. [Maximum mark: 15] Answer the whole of this question on a sheet of graph paper. The variables 𝑥 and 𝑦 are connected by the equation 5 ( 8) 12 xxy x −= + . Some corresponding values of x and y are given in the following table: x 1− 0 1 2 3 4 5 6 y 4.09 a 2.69− 4.29− b 5− 4.41− 3.33− (a) Find the value of a and of b . [2] (b) Taking 2 cm to represent 1 unit on each axis, draw the graph of 5 ( 8) 12 xxy x −= + for 16 x−   . [4] (c) By drawing a tangent, find the gradient of the curve at the point where 3x= . [3] (d) Find the range of values of x for which 4y− . [3] (e) By drawing a suitable straight line on the graph, find the solution of the equation 5 ( 8) ( 2)( 12)x x x x− = − + . [3] ---------------------------------------------------- END OF PAPER 2 --------------------------------------------

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2021/FinalExamination 6 Answers: 1a) -0.565 1b) 9𝑥 𝑦7 1c) 29 1d) 𝑥 = −3 or 𝑥 = 2 2a) 𝑛4 − 𝑚4 2b) 𝑥−2 (3𝑥−5) 2c) 𝑥 = −8.58 or 0.58 3a) 2.08 3b) 15.2 3c) No 4a) 0.188 4b) 6 5a) 310 5b) 50.3 5c) 77.1 5d) 3281.65 5e) 22.9 6a) −2 < 𝑚 < 2 6bi) −8𝑘2 − 40 6bii) 𝑘 = 1 3 or 𝑘 = 1 7a) 𝑎 = 0, 𝑏 = −5 7b) 7c) Gradient = −0.333

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2021/FinalExamination 7 7d) 1.77 ≤ 𝑥 ≤ 5.43 7e) 0.5

FINAL EXAMINATION 2021 YEAR 3 INTEGRATED PROGRAMME CORE MATHEMATICS PAPER 2 XXXXXX xxth October 2021 1 hour 30 minutes ADDITIONAL MATERIALS: Answer Paper (7 sheets) Graph Paper ( 1 sheet) INSTRUCTIONS TO STUDENTS Do not open this examination paper until instructed to do so. A calculator is required for this paper. Answer all the questions on the answer sheets provided. At the end of the examination, fasten the answer sheets together. Unless otherwise stated in the question, all numerical answers must be given exactly or correct to three significant figures. Answers in degrees are to be given to one decimal place. INFORMATION FOR STUDENTS The maximum mark for this paper is 80. _____________________________________________________ This question paper consists of 4 printed pages. [Turn over

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2021/FinalExamination 2 Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. Where an answer is incorrect, some marks may be given for correct method, provided this is shown by written working. You are therefore advised to show all working. Answer all the questions on the answer sheets provided. Begin each question on a new page. 1. [Maximum mark: 12] (a) Evaluate 3 3 ln(3.256) 1.325 (0.25 2.38) e+ − , leaving your answer correct to 3 significant figures. [2] 5.4623 −9.6636 = −0.565 (b) Simplify ( ) 3 2 1 1 2 4 82 4 3 (3 ) 3 x y yx −   , expressing your answer in positive indices. [3] (3𝑥 𝑦2) 3 9𝑦 (1 3 𝑥2) = 27𝑥3 𝑦6 3𝑦𝑥2 = 27𝑥3 3𝑦7𝑥2 = 9𝑥 𝑦7 Most students just use calculator to find the answer, a small group of students still give the wrong answer. Careless in expanding the number 3 for both cube and square. Some students did not leave it in positive indices.

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2021/FinalExamination 3 (c) Simplify 13 21 27 2(3 ) 39 yy yy + +− + . [3] (33)𝑦+1 + 2(33𝑦) 3𝑦+2(32)𝑦−1 = 33𝑦+3 + 2(33𝑦) 3𝑦+232𝑦−2 = 27(33𝑦) + 2(33𝑦) 33𝑦 Let 𝑥 = 33𝑦 = 27𝑥 + 2𝑥 𝑥 = 29𝑥 𝑥 = 29 (d) Solve for x if 2 651xx+− = . [4] 5𝑥2+𝑥−6 = 50 𝑥2 + 𝑥 − 6 = 0 (𝑥 − 2)(𝑥 + 3) = 0 𝑥 = −3 or 𝑥 = 2 Careless