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Unformatted text preview: 08. Three of the coordinates of the parallelogram STUV are S(–2, –2), T(7, 7), U(5, 15).
(a) Find the vector ST and hence the coordinates of V.
(5) (b) Find a vector equation of the line (UV) in the form r = p + λd where λ ∈ .
(2) (c) 1
Show that the point E with position vector is on the line (UV), and find the value of
11 λ for this point.
The point W has position vector , a ∈
17 (d) (i) . If EW = 2 13 , show that one value of a is –3 and find the other possible value
of a. (ii) For a = –3, calculate the angle between EW and ET .
(Total 19 marks) 279 409. A taxi company has 200 taxi cabs. The cumulative frequency curve below shows the fares in
dollars ($) taken by the cabs on a particular morning.
200 180 160 140 Number of cabs 120 100 80 60 40 20 10 20 30 40
Fares ($) 60 70 80 280 (a) Use the curve to estimate
(i) the median fare; (ii) the number of cabs in which the fare taken is $35 or less.
(2) The company charges 55 cents per kilometre for distance travelled. There are no other charges.
Use the curve to answer the following.
(b) On that morning, 40% of the cabs travel less than a km. Find the value of a.
(4) (c) What percentage of the cabs travel more than 90 km on that morning?
(Total 10 marks) 410. Two fair dice are thrown and the number showing on each is noted. The sum of these two
numbers is S. Find the probability that
(a) S is less than 8;
(2) (b) at least one die shows a 3;
(2) (c) at least one die shows a 3, given that S is less than 8.
(Total 7 marks) 411. Consider functions of the form y = e–kx
1 (a) Show that ∫e
0 – kx dx = 1
(1 – e–k).
(3) 281 (b) Let k = 0.5
(i) Sketch the graph of y = e–0.5x, for –1 ≤ x ≤ 3, indicating the coordinates of the
y-intercept. (ii) Shade the region enclosed by this graph, the x-axis, y-axis and the line x = 1. (iii) Find the area of this region.
(5) (c) (i) Find dy
in terms of k, where y = e–kx.
dx The point P(1, 0.8) lies on the graph of the function y = e–kx.
(ii) Find the value of k in this case. (iii) Find the gradient of the tangent to the curve at P.
(Total 13 marks) 412. The mass of packets of a breakfast cereal is normally distributed with a mean of 750 g and
standard deviation of 25 g.
(a) Find the probability that a packet chosen at random has mass
(i) less than 740 g; (ii) at least 780 g; (iii) between 740 g and 780 g.
(5) (b) Two packets are chosen at random. What is the probability that both packets have a mass
which is less than 740 g?
(2) (c) The mass of 70% of the packets is more than x grams. Find the value of x.
(Total 9 marks) 282 413. Let the function f be defined by f(x) =
, x ≠ –1.
1 + x3 (i) Write down the equation of the vertical asymptote of the graph of f. (ii) Write down the equation of the horizontal asymptote of the graph of f. (iii) Sketch the graph of f in the domain –3 ≤ x ≤ 3.
(4) (b) (i) Using the fact that f′(x) =
f ″( x) = (ii) ( ) – 6x 2
, show that the second derivative
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This note was uploaded on 06/24/2013 for the course HISTORY exam taught by Professor Apple during the Fall '13 term at Berlin Brothersvalley Shs.
- Fall '13
- The Land