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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 xcoordinates 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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 Fall '13
 Apple
 The Land

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