Green Land International School EcoleInternationale du PréVert Assessment Task Subject: Mathematics Class: MYP/PEI5 Date: 18 -February 2019 Duration: 2 hrs. Student’s Name: --------------------------------------------------- Task: End of Term 2 Exam ( Paper 2 ) Task requirements: Do not open this examination paper until instructed to do so. A Graphic Display Calculator is required for this paper. Section A: answer all of Section A in the spaces provided. Section B: answer all of Section B on the answer sheets provided. - Unless otherwise stated in the question, all numerical answers must be given exactly or to three decimal places. - Your work will be assessed based on criteria A, C and D. Refer to descriptors on the following page. Level: Criterion A (Max. 8) Criterion C (Max. 8) Criterion D (Max. 8) 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 a correct method, provided written working shows this. All students should therefore be advised to show their working. Working may be continued below the lines, if necessary. Comments: ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… Teacher’s signature: Supervisor signature: Page 1 of 13 Criterion A: Knowing and understanding Achievement level Level descriptor The student does not reach a standard described by any of the descriptors 0-1 below. The student is able to: i. select appropriate mathematics when solving simple problems in 1-2 familiar situations ii. apply the selected mathematics successfully when solving these problems iii. generally solve these problems correctly in a variety of contexts. The student solves successfully questions (1,2,3,4 and 5a,and 9a,b). The student is able to: 3-4 i. select appropriate mathematics when solving more complex problems in familiar situations ii. apply the selected mathematics successfully when solving these problems iii. generally solve these problems correctly in a variety of contexts. The student solves successfully questions (5b,6a,bi ,9c,d,e and 10a). The student is able to: 5-6 i. select appropriate mathematics when solving challenging problems in familiar situations ii. apply the selected mathematics successfully when solving these problems iii. generally solve these problems correctly in a variety of contexts. The student solves successfully questions (7, 8 and 10b ) The student is able to: 7-8 i. select appropriate mathematics when solving challenging problems in both familiar and unfamiliar situations ii. apply the selected mathematics successfully when solving these problems iii. generally solve these problems correctly in a variety of contexts. The student solves successfully questions (6bii and 10c,d). Page 2 of 13 Question 1 [2min] State which of the following quadratics its discriminant equals to zero. a) b) ---------------------- c) Question 2 [2 min] The graph of y f x undergoes a translation of 2 units in the positive x direction followed by a reflection on the 𝑥-axis. Select from the table. The equation of the transformed function is -----------------------------------a) y f x 2 b) c) d) y f x 2 y f x 2 y f x 2 Page 3 of 13 Question 3 [2+3=5 min] Find the value of the unknown length 𝑥 a) ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ b) ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ Page 4 of 13 Question 4 [2+3=5min] The following diagram shows a circle of centre O, and radius 15 cm. The arc ACB subtends an angle of 2 radians at the centre O. C A B 15 cm Diagram not to scale 2 rad O AÔB = 2 radians OA = 15 cm Find (a) the length of the arc ACB; (b) the area of the shaded region. ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ - Page 5 of 13 Question 5 [2+3+2=7min] The equation of a curve may be written in the form y = a(x – p)(x – q). The curve intersects the xaxis at A(–2, 0) and B(4, 0). The curve of y = f (x) is shown in the diagram below. y 4 2 –4 A –2 0 2 B 4 6 x –2 –4 –6 (a) (i) Write down the value of p and of q. (ii) Given that the point (6, 8) is on the curve, find the value of a. (iii) Write the equation of the curve in the form y = ax2 + bx + c. ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ Page 6 of 13 Question 6 [5+3+4=12 min] A recreation park has two trains. Train 1 takes visitors from the entrance (E) to the swimming pool (S), to the mini golf (M) and back to the entrance. Train 2 takes visitors from the entrance (E) to the play area (P), to the racing track (R) and back to the entrance. This is shown in the diagram. TRAIN 1 500 m E S 115° TRAIN 2 750 m 400 m ES = 500 m SM = 400 m ER = 750 m ESM = 115° ERP = 50° EPR = 90° [not to scale] P M 50° R (a) Calculate the total distance Train 2 travels in one journey from E to P to R to E. ----------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ Page 7 of 13 (b) (i) Show that EM = 761 m correct to 3 s.f ----------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------- (ii) If the trains travel at 2 ms–1 find the time taken for Train 1 to complete a journey from E to S to M to E. Give your answer to the nearest second. ----------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ Page 8 of 13 Question 7 [3+4+3=10 min] Given that lengths of sides AB= 8 cm , BC = 11cm and DC = 9 cm & Measure of angles ∠𝐀𝐁𝐂 = 75° and ∠𝐀𝐃𝐂 = 120° a) Find the length of AC b) Find the length of AD c) Find the area of the shaded region ABCD ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ Page 9 of 13 Question 8 [4+4=8 min] Given the parabola 𝑦 = 4𝑥 2 − 2𝑘𝑥 + 9 ; 𝑘 ∈ ℝ Find the value(s) of k in each of the following cases: (a) The x-axis is tangent to the parabola. (b) The parabola meets the x-axis in two distinct points. ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------- Page 10 of 13 Question 9 [1+3+4+4+4=16 min] The table below represents the weights, W, in grams, of 80 packets of roasted peanuts. Weight (W) 80 < W 85 85 < W 90 90 < W 95 Number of packets 5 10 15 95 < W 100 100 < W 105 105 < W 110 110 < W 115 26 13 (a) Show that the mid interval of 80 < W ≤ 85 is 82.5 (b) Use the midpoint of each interval to find an estimate for : 7 4 i) The mean of the weights. ii) the standard deviation of the weights. (c) Complete the following cumulative frequency table for the data above. Weight (W) W 85 W 90 Number of packets 5 15 W 95 W 100 W 105 W 110 W 115 80 (d) Draw a cumulative frequency graph with a scale 2 cm for 10 packets on the vertical axis and 2 cm for 5 grams on the horizontal axis for the table question 9c. (e) Use the graph to estimate i) the median. ii) the upper quartile (that is, the third quartile). Give your answers to the nearest gram. Page 11 of 13 Question 10 [3+2+5+4=14 min] The following diagram shows the triangle AOP, where OP = 2 cm, AP = 4 cm and AO = 3 cm. (a) Calculate AÔP , giving your answer in radians. (3) The following diagram shows two circles which intersect at the points A and B. The smaller circle C1 has centre O and radius 3 cm, the larger circle C2 has centre P and radius 4 cm, and OP = 2 cm. The point D lies on the circumference of C1 and E on the circumference of C2.Triangle AOP is the same as triangle AOP in the diagram above. (b) Find AÔB , giving your answer in radians. (2) (c) Given that AP̂B is 1.63 radians, calculate the area of (i) sector PAEB; (ii) sector OADB. (5) (d) The area of the quadrilateral AOBP is 5.81 cm2. (i) Find the area of AOBE. (ii) Hence find the area of the shaded region AEBD. (4) Page 12 of 13 (Total 14 marks) Question 11 [15 min] Criterion D Applying mathematics in real-life contexts Give your answer in the form of a report For a coal mine An engineer designed a parabolic cross-sectioned tunnel (Arch mine) of internal width 3.4 metres and maximum internal height 2.5 meters. 1- Find a suitable quadratic model for the tunnel. A container in the shape of a box is used to transfer the coal outside the mine of width 2 meters and height 2 meters. 2- Justify whether the box will pass through the Arch mine designed by the engineer or not ? 3- Re-design the parabolic Arch mine to allow the container to pass given the following conditions and requirements: - The internal width allowed for the tunnel is 4 meters. - Suggest a minimum clearance between the truck and the tunnel. Justify? - The tunnel cross-section must have a parabolic shape. Hint: The following should be considered in your design: -Write your model giving the parameters to a reasonable degree of accuracy and justify this degree of accuracy. -Explain whether the dimensions make sense in the context of the problem. End of Exam Best wishes Page 13 of 13
