1.	The following graph shows the depth of water, y metres, at a point P, during one day.
The time t is given in hours, from midnight to noon.

(a)	Use the graph to write down an estimate of the value of t when
(i)	the depth of water is minimum;
(ii)	the depth of water is maximum;
(iii)	the depth of the water is increasing most rapidly.
(3)
(b)	The depth of water can be modelled by the function y = A cos (B (t – 1)) + C.
(i)	Show that A = 8.
(ii)	Write down the value of C.
(iii)	Find the value of B.
(6)
(c)	A sailor knows that he cannot sail past P when the depth of the water is less than 12 m. Calculate the values of t between which he cannot sail past P.
(2)
(Total 11 marks)

2.	The following diagram represents a large Ferris wheel, with a diameter of 100 metres.
		
	Let P be a point on the wheel. The wheel starts with P at the lowest point, at ground level. The wheel rotates at a constant rate, in an anticlockwise (counterclockwise) direction. One revolution takes 20 minutes.
 
(a)	Write down the height of P above ground level after
(i)	10 minutes;
(ii)	15 minutes.
(2)
 
	Let h(t) metres be the height of P above ground level after t minutes. Some values of h(t) are given in the table below.

t	h(t)	
0	0.0	
1	2.4	
2	9.5	
3	20.6	
4	34.5	
5	50.0	
 
(b)	(i)	Show that h(8) = 90.5.
(ii)	Find h(21).
(4)

(c)	Sketch the graph of h, for 0 ≤ t ≤ 40.
(3)
 
(d)	Given that h can be expressed in the form h(t) = a cos bt + c, find a, b and c.
(5)
(Total 14 marks)

3.	The graph of y = p cos qx + r, for –5 ≤ x ≤ 14, is shown below.
		
	There is a minimum point at (0, –3) and a maximum point at (4, 7).
(a)	Find the value of
(i)	p;
(ii)	q;
(iii)	r.
(6)
 
(b)	The equation y = k has exactly two solutions. Write down the value of k.
(1)
(Total 7 marks)

4.	Let f(t) = a cos b (t – c) + d, t ≥ 0. Part of the graph of y = f(t) is given below.
		
	When t = 3, there is a maximum value of 29, at M.
When t = 9 , there is a minimum value of 15.
 
(a)	(i)	Find the value of a.
(ii)	Show that b = .
(iii)	Find the value of d.
(iv)	Write down a value for c.
(7)
 
	The transformation P is given by a horizontal stretch of a scale factor of , followed by a translation of .
(b)	Let M′ be the image of M under P. Find the coordinates of M′.
(2)
 
	The graph of g is the image of the graph of f under P.
(c)	Find g(t) in the form g(t) = 7 cos B(t – C) + D.
(4)
 
(d)	Give a full geometric description of the transformation that maps the graph of g to the graph of f.
(3)
(Total 16 marks)

5.	A formula for the depth d metres of water in a harbour at a time t hours after midnight is

	where P and Q are positive constants. In the following graph the point (6, 8.2) is a minimum point and the point (12, 14.6) is a maximum point.

(a)	Find the value of
(i)	Q;
(ii)	P.
(3)
 
(b)	Find the first time in the 24-hour period when the depth of the water is 10 metres.
(3)
 
(c)	(i)	Use the symmetry of the graph to find the next time when the depth of the water is 10 metres.
(ii)	Hence find the time intervals in the 24-hour period during which the water is less than 10 metres deep.
(4)