Find two integers x and y such that the greatest

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Unformatted text preview: te for the standard deviation of the weights. (3) (b) Copy and complete the following cumulative frequency table for the above data. Weight (W) Number of packets W ≤ 85 W ≤ 90 5 15 W≤ 95 W≤ 100 W ≤ 105 W ≤110 W ≤ 115 80 (1) 206 (c) A cumulative frequency graph of the distribution is shown below, with a scale 2 cm for 10 packets on the vertical axis and 2 cm for 5 grams on the horizontal axis. 80 70 60 50 Number of packets 40 30 20 10 80 85 90 95 100 Weight (grams) 105 110 115 Use the graph to estimate (i) the median; (ii) the upper quartile (that is, the third quartile). Give your answers to the nearest gram. (4) 207 (d) Let W1, W2, ..., W80 be the individual weights of the packets, and let W be their mean. What is the value of the sum (W1 – W ) + (W2 – W ) + (W3 – W ) + . . . + (W79 – W ) + (W80 – W ) ? (2) (e) One of the 80 packets is selected at random. Given that its weight satisfies 85 < W ≤ 110 , find the probability that its weight is greater than 100 grams. (4) (Total 14 marks) 298. In this question, a unit vector represents a displacement of 1 metre. A miniature car moves in a straight line, starting at the point (2 , 0). After t seconds, its position, (x , y) , is given by the vector equation x 2 0 .7 = +t y 0 1 (a) How far from the point (0 , 0) is the car after 2 seconds? (2) (b) Find the speed of the car. (2) 208 (c) Obtain the equation of the car’s path in the form ax + by = c. (2) Another miniature vehicle, a motorcycle, starts at the point (0 , 2), and travels in a straight line with constant speed. The equation of its path is y = 0.6x + 2, x ≥ 0. Eventually, the two miniature vehicles collide. (d) Find the coordinates of the collision point. (3) (e) If the motorcycle left point (0 , 2) at the same moment the car left point (2 , 0), find the speed of the motorcycle. (5) (Total 14 marks) 299. Note: Radians are used throughout this question. Let f (x) = sin (1 + sin x). (i) Sketch the graph of y = f (x), for 0 ≤ x ≤ 6. (ii) (a) Write down the x-coordinates of all minimum and maximum points of f, for 0 ≤ x ≤ 6 . Give your answers correct to four significant figures. (9) (b) Let S be the region in the first quadrant completely enclosed by the graph of f and both coordinate axes. (i) Shade S on your diagram. (ii) Write down the integral which represents the area of S. (iii) Evaluate the area of S to four significant figures. (5) (c) Give reasons why f (x) ≥ 0 for all values of x. (2) (Total 16 marks) 209 ˆ 300. In the diagram below, the points O(0 , 0) and A(8 , 6) are fixed. The angle OPA varies as the point P(x , 10) moves along the horizontal line y = 10. y P(x, 10) y=10 A(8, 6) O(0, 0) x diagram to scale (i) Show that AP = (ii) (a) x 2 – 16 x + 80. Write down a similar expression for OP in terms of x. (2) (b) Hence, show that ˆ cos OPA = x 2 – 8 x + 40 √ {( x 2 – 16 x + 80) ( x 2 + 100)} , (3) (c) ˆ Find, in degrees, the angle OPA when x = 8. (2) (d) ˆ Find the positive value of x such that OPA = 60° . (4) Let the function f be defined by ˆ...
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