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Unformatted text preview: the area of the rectangle ABCD.
(2)
(Total 15 marks) 642. (a) Consider the function f(x) = 2 + 1
. The diagram below is a sketch of part of the
x −1 graph of y = f(x).
y
5
4
3
2
1
–5 –4 –3 –2 –1 0
–1 1 2 3 4 5 x –2
–3
–4
–5 Copy and complete the sketch of f(x).
(2) 430 (b) (i) Write down the xintercepts and yintercepts of f(x). (ii) Write down the equations of the asymptotes of f(x).
(4) (i) Find f′(x). (ii) (c) There are no maximum or minimum points on the graph of f(x).
Use your expression for f′ (x) to explain why.
(3) The region enclosed by the graph of f(x), the xaxis and the lines x = 2 and x = 4, is labelled A,
as shown below. y
5
4
3
2 A 1
–5 –4 –3 –2 –1 0
–1 1 2 3 4 5 x –2
–3
–4
–5 (d) ∫ f ( x) dx. (i) Find (ii) Write down an expression that represents the area labelled A. (iii) Find the area of A.
(7)
(Total 15 marks) 431 643. The depth y metres of water in a harbour is given by the equation t
y = 10 + 4sin , 2
where t is the number of hours after midnight.
(a) Calculate the depth of the water
(i) when t = 2; (ii) at 2100.
(3) The sketch below shows the depth y, of water, at time t, during one day (24 hours).
y
15
14
13
12
11
10
9
depth (metres)
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 t
time (hours) (b) (i) Write down the maximum depth of water in the harbour. (ii) Calculate the value of t when the water is first at its maximum depth during the
day.
(3) 432 The harbour gates are closed when the depth of the water is less than seven metres. An alarm
rings when the gates are opened or closed.
(c) (i) How many times does the alarm sound during the day? (ii) Find the value of t when the alarm sounds first. (iii) Use the graph to find the length of time during the day when the harbour gates are
closed. Give your answer in hours, to the nearest hour.
(7)
(Total 13 marks) 644. Dumisani is a student at IB World College.
The probability that he will be woken by his alarm clock is 7
.
8 If he is woken by his alarm clock the probability he will be late for school is 1
.
4 If he is not woken by his alarm clock the probability he will be late for school is 3
.
5 Let W be the event “Dumisani is woken by his alarm clock”.
Let L be the event “Dumisani is late for school”. (a) Copy and complete the tree diagram below.
L W L′
L W′ L′
(4) 433 (b) Calculate the probability that Dumisani will be late for school.
(3) (c) Given that Dumisani is late for school what is the probability that he was woken by his
alarm clock?
(4)
(Total 11 marks) 645. There were 1420 doctors working in a city on 1 January 1994. After n years the number of
doctors, D, working in the city is given by
D = 1420 + 100n.
(a) (i) How many doctors were there working in the city at the start of 2004? (ii) In what year were there first more than 2000 doctors working in the city?
(3) At the beginning of 1994 the city had a population of 1.2 million. After n years, the population,
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