1.	Solve the equation 2cos x = sin 2x, for 0 ≤ x ≤ 3π.
(Total 7 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 straight line with equation y =  makes an acute angle θ with the x-axis.
(a)	Write down the value of tan θ.
(1)
 
(b)	Find the value of
(i)	sin 2θ;
(ii)	cos 2θ.
(6)
(Total 7 marks)

4.	Let f(x) = cos 2x and g(x) = 2x2 – 1.
(a)	Find .
(2)
 
(b)	Find (g ° f).
(2)
 
(c)	Given that (g ° f)(x) can be written as cos (kx), find the value of k, k  .
(3)
(Total 7 marks)

5.	Let f(x) = 6 + 6sinx. Part of the graph of f is shown below.
		
	The shaded region is enclosed by the curve of f, the x-axis, and the y-axis.
 
(a)	Solve for 0 ≤ x < 2π.
(i)	6 + 6sin x = 6;
(ii)	6 + 6 sin x = 0.
(5)
 
(b)	Write down the exact value of the x-intercept of f, for 0 ≤ x < 2.
(1)
 
(c)	The area of the shaded region is k. Find the value of k, giving your answer in terms of π.
(6)
 
	Let g(x) = 6 + 6sin. The graph of f is transformed to the graph of g.
(d)	Give a full geometric description of this transformation.
(2)
 
(e)	Given that  = k and 0 ≤ p < 2π, write down the two values of p.
(3)
(Total 17 marks)

6.	The diagram shows two concentric circles with centre O.
		
diagram not to scale
	The radius of the smaller circle is 8 cm and the radius of the larger circle is 10 cm.
Points A, B and C are on the circumference of the larger circle such that  is  radians.
 
(a)	Find the length of the arc ACB.
(2)
 
(b)	Find the area of the shaded region.
(4)
(Total 6 marks)

7.	(a)	Show that 4 – cos 2θ + 5 sin θ = 2 sin2 θ + 5 sin θ + 3.
(2)
(b)	Hence, solve the equation 4 – cos 2θ + 5 sin θ = 0 for 0 ≤ θ ≤ 2π.
(5)
(Total 7 marks)

8.	A rectangle is inscribed in a circle of radius 3 cm and centre O, as shown below.
		
	The point P(x, y) is a vertex of the rectangle and also lies on the circle. The angle between (OP) and the x-axis is θ radians, where 0 ≤ θ ≤ .
 
(a)	Write down an expression in terms of θ for
(i)	x;
(ii)	y.
(2)
 
	Let the area of the rectangle be A.
(b)	Show that A = 18 sin 2θ.
(3)
 
(c)	(i)	Find .
(ii)	Hence, find the exact value of θ which maximizes the area of the rectangle.
(iii)	Use the second derivative to justify that this value of θ does give a maximum.
(8)
(Total 13 marks)

9.	The vertices of the triangle PQR are defined by the position vectors
		.
(a)	Find
(i)	;
(ii)	.
(3)
 
(b)	Show that .
(7)
 
(c)	(i)	Find .
(ii)	Hence, find the area of triangle PQR, giving your answer in the form .
(6)
(Total 16 marks)

10.	Let f(x) =  sin x + e2x cos x, for 0 ≤ x ≤ π. Given that , solve the equation
f(x) = 0.
(Total 6 marks)

11.	Solve cos 2x – 3 cos x – 3 – cos2 x = sin2 x, for 0 ≤ x ≤ 2π.
(Total 7 marks)

12.	Let p = sin40° and q = cos110°. Give your answers to the following in terms of p and/or q.
(a)	Write down an expression for
(i)	sin140°;
(ii)	cos70°.
(2)
(b)	Find an expression for cos140°.
(3)
(c)	Find an expression for tan140°.
(1)
(Total 6 marks)

13.	Consider g (x) = 3 sin 2x.
(a)	Write down the period of g.
(1)

(b)	On the diagram below, sketch the curve of g, for 0 £ x £ 2p.

(3)
 
(c)	Write down the number of solutions to the equation g (x) = 2, for 0 £ x £ 2p.
(2)
(Total 6 marks)

14.	(a)	Given that cos A =  and 0 £ A £  find cos 2A.
(3)
(b)	Given that sin B =  and  £ B £ p, find cos B.
(3)
(Total 6 marks)

15.	Let f : x  sin3 x.
(a)	(i)	Write down the range of the function f.
(ii)	Consider f (x) =1, 0 £ x £ 2p. Write down the number of solutions to this equation. Justify your answer.
(5)
(b)	Find f ′ (x), giving your answer in the form a sinp x cosq x where a, p, q Î .
(2)
(c)	Let g (x) =  for 0 £ x £ . Find the volume generated when the curve of g is revolved through 2p about the x-axis.
(7)
(Total 14 marks)

16.	The following diagram shows a semicircle centre O, diameter [AB], with radius 2.
Let P be a point on the circumference, with  = q radians.

(a)	Find the area of the triangle OPB, in terms of q.
(2)
(b)	Explain why the area of triangle OPA is the same as the area triangle OPB.
(3)
Let S be the total area of the two segments shaded in the diagram below.

(c)	Show that S = 2(p − 2 sin q ).
(3)
(d)	Find the value of q when S is a local minimum, justifying that it is a minimum.
(8)
(e)	Find a value of q for which S has its greatest value.
(2)
(Total 18 marks)

17.	The diagram below shows a circle centre O, with radius r. The length of arc ABC is 3p cm and  = 

(a)	Find the value of r.
(2)
(b)	Find the perimeter of sector OABC.
(2)
(c)	Find the area of sector OABC.
(2)
(Total 6 marks)

18.	Let f(x) = sin3 x + cos3 x tan x,  < x < π.
(a)	Show that f(x) = sin x.
(2)
 
(b)	Let sin x = . Show that f(2x) = .
(5)
(Total 7 marks)

19.	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)

20.	The graph of a function of the form y = p cos qx is given in the diagram below.

(a)	Write down the value of p.
(2)
 
(b)	Calculate the value of q.
(4)
(Total 6 marks)

21.	Given that  and that cosθ = , find
(a)	sin θ;
(3)
(b)	cos 2θ;
(3)
(c)	sin (θ + π).
(1)
(Total 7 marks)

22.	(a)	Given that 2 sin2 θ + sinθ – 1 = 0, find the two values for sin θ.
(4)
 
(b)	Given that 0° ≤ θ ≤ 360° and that one solution for θ is 30°, find the other two possible values for θ.
(2)
(Total 6 marks)

23.	A spring is suspended from the ceiling. It is pulled down and released, and then oscillates up and down. Its length, l centimetres, is modelled by the function l = 33 + 5cos((720t)°), where t is time in seconds after release.
(a)	Find the length of the spring after 1 second.
(2)
 
(b)	Find the minimum length of the spring.
(3)
 
(c)	Find the first time at which the length is 33 cm.
(3)

(d)	What is the period of the motion?
(2)
(Total 10 marks)

24.	The diagram below shows part of the graph of y = sin 2x. The shaded region is between x = 0 and x = m.

(a)	Write down the period of this function.
(2)
 
(b)	Hence or otherwise write down the value of m.
(2)
 
(c)	Find the area of the shaded region.
(6)
(Total 10 marks)

25.	The following diagram shows a triangle ABC, where  is 90°, AB = 3, AC = 2 and  is q.

(a)	Show that sin q = .
(b)	Show that sin 2q = .
(c)	Find the exact value of cos 2q.
(Total 6 marks)

26.	The following diagram shows a sector of a circle of radius r cm, and angle q at the centre. The perimeter of the sector is 20 cm.

(a)	Show that q = .
 
(b)	The area of the sector is 25 cm2. Find the value of r.
(Total 6 marks)

27.	The diagram below shows a circle of radius r and centre O. The angle  = q.

The length of the arc AB is 24 cm. The area of the sector OAB is 180 cm2.
Find the value of r and of q.
(Total 6 marks)

28.	The following diagram shows two semi-circles. The larger one has centre O and radius 4 cm. The smaller one has centre P, radius 3 cm, and passes through O. The line (OP) meets the larger semi-circle at S. The semi-circles intersect at Q.

(a)	(i)	Explain why OPQ is an isosceles triangle.
(ii)	Use the cosine rule to show that cos  = .
(iii)	Hence show that sin  = .
(iv)	Find the area of the triangle OPQ.
(7)

(b)	Consider the smaller semi-circle, with centre P.
(i)	Write down the size of 
(ii)	Calculate the area of the sector OPQ.
(3)
 
(c)	Consider the larger semi-circle, with centre O. Calculate the area of the sector QOS.
(3)
 
(d)	Hence calculate the area of the shaded region.
(4)
(Total 17 marks)

29.	The following diagram shows a circle of centre O, and radius r. The shaded sector OACB has an area of 27 cm2. Angle  = θ = 1.5 radians.

(a)	Find the radius.
(b)	Calculate the length of the minor arc ACB.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 6 marks)

30.	The diagram shows a triangular region formed by a hedge [AB], a part of a river bank [AC] and a fence [BC]. The hedge is 17 m long and  is 29°. The end of the fence, point C, can be positioned anywhere along the river bank.
(a)	Given that point C is 15 m from A, find the length of the fence [BC].

(3)

(b)	The farmer has another, longer fence. It is possible for him to enclose two different triangular regions with this fence. He places the fence so that  is 85°.
(i)	Find the distance from A to C.
(ii)	Find the area of the region ABC with the fence in this position.
(5)
 
(c)	To form the second region, he moves the fencing so that point C is closer to point A.
Find the new distance from A to C.
(4)
 
(d)	Find the minimum length of fence [BC] needed to enclose a triangular region ABC.
(2)
(Total 14 marks)

 d31.	The diagram below shows two circles which have the same centre O and radii 16 cm and 10 cm respectively. The two arcs AB and CD have the same sector angle q = 1.5 radians.

	Find the area of the shaded region.
Working:		
	Answer:
…………………………………………..	
(Total 6 marks)

32.	In a triangle ABC, AB = 4 cm, AC = 3 cm and the area of the triangle is 4.5 cm2.
	Find the two possible values of the angle .
Working:		
	Answer:
…………………………………………..	
(Total 6 marks)

33.	The diagram below shows a circle of radius 5 cm with centre O. Points A and B are on the circle, and  is 0.8 radians. The point N is on [OB] such that [AN] is perpendicular to [OB].

	Find the area of the shaded region.
Working:		
	Answer:
…………………………………………........	
(Total 6 marks)

34.	Part of the graph of y = p + q cos x is shown below. The graph passes through the points (0, 3) and (p, –1).

	Find the value of
(a)	p;
(b)	q.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 6 marks)

35.	The diagram below shows a triangle and two arcs of circles.
	The triangle ABC is a right-angled isosceles triangle, with AB = AC = 2. The point P is the midpoint of [BC].
	The arc BDC is part of a circle with centre A.
	The arc BEC is part of a circle with centre P.

(a)	Calculate the area of the segment BDCP.
(b)	Calculate the area of the shaded region BECD.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 6 marks)

36.	The diagram shows a parallelogram OPQR in which  = ,  = 

(a)	Find the vector .
(3)
 
(b)	Use the scalar product of two vectors to show that cos = – 
(4)
 
(c)	(i)	Explain why cos = –cos
(ii)	Hence show that sin = .
(iii)	Calculate the area of the parallelogram OPQR, giving your answer as an integer.
(7)
(Total 14 marks)

37.	The following diagram shows a circle of centre O, and radius 15 cm. The arc ACB subtends an angle of 2 radians at the centre O.

	Find
(a)	the length of the arc ACB;
(b)	the area of the shaded region.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 6 marks)

38.	Consider the following statements
A:	log10 (10x) > 0.
B:	–0.5 £ cos (0.5x) £ 0.5.
C:	–  £ arctan x £ .
(a)	Determine which statements are true for all real numbers x. Write your answers (yes or no) in the table below.
Statement	(a)	Is the statement true for all
	real numbers x? (Yes/No)	(b)   If not true, example	
A			
B			
C			
 
(b)	If a statement is not true for all x, complete the last column by giving an example of one value of x for which the statement is false.
Working:	
(Total 6 marks)

39.	Given that sin x = , where x is an acute angle, find the exact value of
(a)	cos x;
(b)	cos 2x.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 6 marks)

40.	In the following diagram, O is the centre of the circle and (AT) is the tangent to the circle at T.
 Diagram not to scale
	If OA = 12 cm, and the circle has a radius of 6 cm, ﬁnd the area of the shaded region.
Working:		
	Answer:
.......................................................................	
(Total 4 marks)

41.	If A is an obtuse angle in a triangle and sin A = , calculate the exact value of sin 2A.
Working:		
	Answer:
......................................................................	
(Total 4 marks)

42.	Given that sin θ = , cos θ = – and 0° ≤ θ ≤ 360°,
(a)	find the value of θ;

(b)	write down the exact value of tan θ.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

43.	The diagram shows a circle of radius 5 cm.

	Find the perimeter of the shaded region.
Working:		
	Answer:
......................................................................	
(Total 4 marks)

44.	A triangle has sides of length 4, 5, 7 units. Find, to the nearest tenth of a degree, the size of the largest angle.
Working:		
	Answer:
......................................................................	
(Total 4 marks)

45.	O is the centre of the circle which has a radius of 5.4 cm.

	The area of the shaded sector OAB is 21.6 cm2. Find the length of the minor arc AB.
Working:		
	Answer:
......................................................................	
(Total 4 marks)

46.	Solve the equation 3 sin2 x = cos2 x, for 0° £ x £ 180°.
Working:		
	Answer:
......................................................................	
(Total 4 marks)

47.	The diagrams show a circular sector of radius 10 cm and angle θ radians which is formed into a cone of slant height 10 cm. The vertical height h of the cone is equal to the radius r of its base. Find the angle θ radians.

Working:		
	Answer:
......................................................................	
(Total 4 marks)