ACSI 2022 Y3 Core Mathematics Paper 2

Anglo - Chinese School (Independent) FINAL EXAMINATION 2022 YEAR 3 INTEGRATED PROGRAMME CORE MATHEMATICS PAPER 2 Thursday 6th October 2022 1 hour 30 minutes ADDITIONAL MATERIALS: Answer Paper (6 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/Y3IPCoreMathPaper2/2022/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 t herefore advised to show all working. Answer all the questions on the answer sheets provided. Begin each question on a new page. 1. [Maximum mark: 6] (a) Simplify 22 22 2a ab b b a ab − + +− − . [3] (b) Subtract 2 4 4x − from 11 22 xx−−+ , expressing your answer as a single fraction in its simplest form. [3] 2. [Maximum mark: 8] Joe bought x number of books, each at the same price, for a total cost of $336. (a) Write down an expression for the cost of each book in terms of x . [1] Joe sold 20 of them for $480, and the rest at a loss of $4 per book. (b) Write down an expression for the total amount, in dollars, he received for all the books. [2] (c) Given that Joe made a profit of $184 altogether, form an equation in x and show that it reduces to 2 94 1680 0xx−+ = . [2] (d) Hence, solve the equation 2 94 1680 0xx−+ = and state the cost price of each book. [3] 3. [Maximum mark: 13] (a) Evaluate 2 3 2 2.15log 0.25 e −  +  , leaving your answer correct to 2 significant figures. [3] (b) Given that ( ) ( ) 31 43 23144 216 2 3 xy zpp p −−÷= , evaluate x , y and z . [4] (c) Simplify 22 1 6 2(3 ) 48 ww w+ + + . [3] (d) Find the range of values of x if 22 5277xx−− < . [3]

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2022/FinalExamination 3 4. [Maximum mark: 8] (a) Solve the equation 134yy+ = . [4] (b) Solve the equation ( ) 2 33 9log log ( 3) log (9 )xx x+ −= . [4] 5. [Maximum mark: 12] In the figure, A, B and C are three points on a horizontal field. A is due west of B, the bearing of B from C is 125o, AB = 430 m and BC = 460 m. (a) Find (i) the distance between A and C, [3] (ii) ACB , [2] (iii) the bearing of C from A, [2] (iv) the area of ABC∆ . [2] At a certain instant, a hot air balloon is at a point which is directly above C. (b) Given that the angle of elevation of the hot air balloon from B is 5.2o, find the angle of elevation of the hot air balloon from A. [3] 6. [Maximum mark: 9] (a) Find the smallest positive integer value of m given that t he straight line 5y mx= − intersects the curve 2 2yx m= − at two distinct points. [5] (b) Prove that 533 25 60 0xxx− += , has only one real root. [4] A B C 125o 460 m 430 m

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2022/FinalExamination 4 7. [Maximum mark: 14] Answer the whole of this question on a sheet of graph paper. The variables x and y are connected by the equation 603 35yx x=+− . Some corresponding values of x and y are given in the following table: (a) Find the value of a and of b . [2] (b) Taking 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y-axis, draw the graph of 603 35yx x=+− for 1.5 7 x≤≤ . [4] Use your graph to find (c) the least value of y , [1] (d) the range of values of x for which 7y≤− , [2] (e) the gradient of the curve at the point 2x= by drawing a suitable straight line, [2] (f) the solutions of the equation 605 40 0x x+−= by drawing a suitable straight line. [3] 8. [Maximum mark: 10] (a) It is given that log 12a x= log 60b x= (i) Express a in terms of b . [3] (ii) If log 6abc x= , prove that log 15c x= . [4] (b) Using the two single digits a and b , explain clearly whether there exists a number, ab , such that the sum of ab and its reverse ba is a prime number. [3] ----------------------------------------------- END OF PAPER 2 ------------------------------------------------ x 1.5 2 2.5 3 4 5 6 7 y 9.5 a 3.5− 6− b 8− 7− 5.4−

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2022/FinalExamination 5 Answers: 1a) ( )1 () ab ab −− + 1b) 2 2x − − 2a) 336 x 2b) ( ) 336480 20 4x x +− −  2d) $4.80 or $14 3a) 0.47− 3b) 7 4 5 x y z = = = 3c) 23 4 w 3d) 1 32 x−<< 4a) 3.82 4b) 6 5ai) 269 5aii) 66.4 5aiii) 011.4 5aiv) 56727 5b) 8.84 6a) 3 7a) 𝑎𝑎 = 1, 𝑏𝑏 = −8

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2022/FinalExamination 6 7b) 7c) 7d) Answer: Answer:

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2022/FinalExamination 7 7e) 7f) 8a) 5ab= Answer: Gradient = -12 Answer: or 6