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1) The depth y metres of water in a harbour is given by the equation
t
y = 10 + 4 sin   ,
 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)
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.
2)
The diagram below shows a circle, centre O, with a radius 12 cm. The chord AB subtends
at an angle of 75° at the centre. The tangents to the circle at A and at B meet at P.
A
12 cm
P diagram not to
scale
O 75º
B
(a)
Using the cosine rule, show that the length of AB is 12 21 – cos 75 .
(2)
(b)
Find the length of BP.
(3)
(c)
Hence find
(i)
the area of triangle OBP;
(ii)
the area of triangle ABP.
(4)
(d)
Find the area of sector OAB.
(2)
(e)
Find the area of the shaded region.
(2)
(Total 13 marks)
3)
In a school of 88 boys, 32 study economics (E), 28 study history (H) and 39 do not study
either subject. This information is represented in the following Venn diagram.
U (88)
E (32)
H (28)
a
b
c
39
(a)
Calculate the values a, b, c.
(4)
(b)
A student is selected at random.
(i)
Calculate the probability that he studies both economics and history.
(ii)
Given that he studies economics, calculate the probability that he does
not study history.
(3)
(c)
A group of three students is selected at random from the school.
(i)
Calculate the probability that none of these students studies economics.
(ii)
Calculate the probability that at least one of these students studies
economics.
(5)
(Total 12 marks)