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Unformatted text preview: d the height of the rock-climber when t = 2.
(1) (b) Sketch a graph of h against t for 0 ≤ t ≤ 5.
(4) (c) Find dh for:
(i) 0≤t≤2 (ii) 2≤t≤5
(2) (d) Find the velocity of the rock-climber when t = 2.
(2) (e) Find the times when the velocity of the rock-climber is zero.
(3) (f) Find the minimum height of the rock-climber for 0 ≤ t ≤ 5.
(Total 15 marks) 132 190. In this question you should note that radians are used throughout.
(i) Sketch the graph of y = x2 cos x, for 0 ≤ x ≤ 2 making clear the approximate
positions of the positive intercept, the maximum point and the end-points. (ii) (a) Write down the approximate coordinates of the positive x-intercept, the maximum
point and the end-points.
(7) (b) Find the exact value of the positive x-intercept for 0 ≤ x ≤ 2.
(2) Let R be the region in the first quadrant enclosed by the graph and the x-axis.
(c) (i) Shade R on your diagram. (ii) Write down an integral which represents the area of R.
(3) (d) Evaluate the integral in part (c)(ii), either by using a graphic display calculator, or by
using the following information. d (x2 sin x + 2x cos x – 2 sin x) = x2 cos x.
(Total 15 marks) 1
191. In this question the vector km represents a displacement due east, and the vector 0 represents a displacement due north. 1 km 0 The point (0, 0) is the position of Shipple Airport. The position vector r1 of an aircraft Air One
is given by
16 12 r1 = + t ,
12 − 5 133 where t is the time in minutes since 12:00.
(a) Show that the Air One aircraft
(i) is 20 km from Shipple Airport at 12:00; (ii) has a speed of 13 km/min.
(4) (b) Show that a cartesian equation of the path of Air One is:
5x + 12y = 224.
(3) The position vector r2 of an aircraft Air Two is given by 23 2.5 r2 = + t , − 5 6 where t is the time in minutes since 12:00. (c) Find the angle between the paths of the two aircraft.
(4) (d) (i) Find a cartesian equation for the path of Air Two. (ii) Hence find the coordinates of the point where the two paths cross.
(5) (e) Given that the two aircraft are flying at the same height, show that they do not collide.
(Total 20 marks) 192. Initially a tank contains 10 000 litres of liquid. At the time t = 0 minutes a tap is opened, and
liquid then flows out of the tank. The volume of liquid, V litres, which remains in the tank after
t minutes is given by
V = 10 000 (0.933t). 134 (a) Find the value of V after 5 minutes.
(1) (b) Find how long, to the nearest second, it takes for half of the initial amount of liquid to
flow out of the tank.
(3) (c) The tank is regarded as effectively empty when 95% of the liquid has flowed out.
Show that it takes almost three-quarters of an hour for this to happen.
(3) (d) (i) Find the value of 10 000 – V when t = 0.001 minutes. (ii) Hence or otherwise, estimate the initial flow rate of the liquid.
Give your answer in litres per minute, correct to two significant figures.
(Total 10 marks) 193. An urban highway has a speed limit of 50 km h–1. It...
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- Fall '13
- The Land