1.	The following diagram shows triangle ABC.
		
diagram not to scale
	AB = 7 cm, BC = 9 cm and  = 120°.
 
(a)	Find AC.
(3)
(b)	Find .
(3)
(Total 6 marks)

2.	The following diagram shows a waterwheel with a bucket. The wheel rotates at a constant rate in an anticlockwise (counterclockwise) direction.
		
diagram not to scale
	The diameter of the wheel is 8 metres. The centre of the wheel, A, is 2 metres above the water level. After t seconds, the height of the bucket above the water level is given by h = a sin bt + 2.
 
(a)	Show that a = 4.
(2)

	The wheel turns at a rate of one rotation every 30 seconds.
(b)	Show that b = .
(2)
 
	In the first rotation, there are two values of t when the bucket is descending at a rate of
0.5 m s–1.
(c)	Find these values of t.
(6)
 
(d)	Determine whether the bucket is underwater at the second value of t.
(4)
(Total 14 marks)

3.	There is a vertical tower TA of height 36 m at the base A of a hill. A straight path goes up the hill from A to a point U. This information is represented by the following diagram.
		
	The path makes a 4° angle with the horizontal.
The point U on the path is 25 m away from the base of the tower.
The top of the tower is fixed to U by a wire of length x m.
 
(a)	Complete the diagram, showing clearly all the information above.
(3)

(b)	Find x.
(4)
(Total 7 marks)

4.	The diagram below shows a quadrilateral ABCD with obtuse angles  and .
		
diagram not to scale
	AB = 5 cm, BC = 4 cm, CD = 4 cm, AD = 4 cm,  = 30°,  = x°,  = y°.
 
(a)	Use the cosine rule to show that AC = .
(1)
 
(b)	Use the sine rule in triangle ABC to find another expression for AC.
(2)
 
(c)	(i)	Hence, find x, giving your answer to two decimal places.
(ii)	Find AC.
(6)

(d)	(i)	Find y.
(ii)	Hence, or otherwise, find the area of triangle ACD.
(5)
(Total 14 marks)

5.	The diagram below shows a circle with centre O and radius 8 cm.
		
diagram not to scale
	The points A, B, C, D, E and F are on the circle, and [AF] is a diameter. The length of arc ABC is 6 cm.
 
(a)	Find the size of angle AOC.
(2)
 
(b)	Hence find the area of the shaded region.
(6)
 
	The area of sector OCDE is 45 cm2.
(c)	Find the size of angle COE.
(2)

(d)	Find EF.
(5)
(Total 15 marks)

6.	The following diagram shows the triangle ABC.
		
diagram not to scale
	The angle at C is obtuse, AC = 5 cm, BC = 13.6 cm and the area is 20 cm2.
 
(a)	Find .
(4)
 
(b)	Find AB.
(3)
(Total 7 marks)

7.	The circle shown has centre O and radius 3.9 cm.
		
diagram not to scale
	Points A and B lie on the circle and angle AOB is 1.8 radians.
 
(a)	Find AB.
(3)
 
(b)	Find the area of the shaded region.
(4)
(Total 7 marks)

8.	The diagram below shows a triangle ABD with AB = 13 cm and AD = 6.5 cm.
Let C be a point on the line BD such that BC = AC = 7 cm.
		
diagram not to scale

(a)	Find the size of angle ACB.
(3)
 
(b)	Find the size of angle CAD.
(5)
(Total 8 marks)

9.	Let f(x) = 3sinx + 4 cos x, for –2π ≤ x ≤ 2π.
(a)	Sketch the graph of f.
(3)
 
(b)	Write down
(i)	the amplitude;
(ii)	the period;
(iii)	the x-intercept that lies between  and 0.
(3)
 
(c)	Hence write f(x) in the form p sin (qx + r).
(3)
 
(d)	Write down one value of x such that f′(x) = 0.
(2)
 
(e)	Write down the two values of k for which the equation f(x) = k has exactly two solutions.
(2)
 
(f)	Let g(x) = ln(x + 1), for 0 ≤ x ≤ π. There is a value of x, between 0 and 1, for which the gradient of f is equal to the gradient of g. Find this value of x.
(5)
(Total 18 marks)

10.	The following diagram shows a circle with centre O and radius 4 cm.
		
diagram not to scale
	The points A, B and C lie on the circle. The point D is outside the circle, on (OC).
Angle ADC = 0.3 radians and angle AOC = 0.8 radians.
 
(a)	Find AD.
(3)
 
(b)	Find OD.
(4)
 
(c)	Find the area of sector OABC.
(2)
 
(d)	Find the area of region ABCD.
(4)
(Total 13 marks)

11.	Let f(x) =  and g(x) = –0.5x2 + 5x – 8, for 0 ≤ x ≤ 9.
(a)	On the same diagram, sketch the graphs of f and g.
(3)

(b)	Consider the graph of f. Write down
(i)	the x-intercept that lies between x = 0 and x =3;
(ii)	the period;
(iii)	the amplitude.
(4)
 
(c)	Consider the graph of g. Write down
(i)	the two x-intercepts;
(ii)	the equation of the axis of symmetry.
(3)
 
(d)	Let R be the region enclosed by the graphs of f and g. Find the area of R.
(5)
(Total 15 marks)

12.	The diagram below shows triangle PQR. The length of [PQ] is 7 cm, the length of [PR] is 10 cm, and  is 75°.

(a)	Find 
(3)
(b)	Find the area of triangle PQR.
(3)
(Total 6 marks)

13.	Let f (x) = 4 tan2 x – 4 sin x, 
(a)	On the grid below, sketch the graph of y = f (x).

(3)
 
(b)	Solve the equation f (x) = 1.
(3)
(Total 6 marks)

14.	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)

15.	A ship leaves port A on a bearing of 030°. It sails a distance of 25 km to point B.
At B, the ship changes direction to a bearing of 100°. It sails a distance of 40 km to reach point C. This information is shown in the diagram below.
		
diagram not to scale
	A second ship leaves port A and sails directly to C.
 
(a)	Find the distance the second ship will travel.
(4)
 
(b)	Find the bearing of the course taken by the second ship.
(3)
(Total 7 marks)

16.	The diagram below shows a quadrilateral ABCD. AB = 4, AD = 8, CD =12, BD = 25°,  =q.

(a)	Use the cosine rule to show that BD = .
(2)
Let q = 40°.
(b)	(i)	Find the value of sin .
(ii)	Find the two possible values for the size of .
(iii)	Given that  is an acute angle, find the perimeter of ABCD.
(12)
(c)	Find the area of triangle ABD.
(2)
(Total 16 marks)

17.	Let f (x) = 6 sin px , and g (x) = 6e–x – 3 , for 0 £ x £ 2. The graph of f is shown on the diagram below. There is a maximum value at B (0.5, b).

(a)	Write down the value of b.
(b)	On the same diagram, sketch the graph of g.
(c)	Solve f (x) = g (x) , 0.5 £ x £ 1.5.

(Total 6 marks)

18.	Consider the equation 3 cos 2x + sin x = 1
(a)	Write this equation in the form f (x) = 0 , where f (x) = p sin2 x + q sin x + r , and p , q , r Î .
(b)	Factorize f (x).
(c)	Write down the number of solutions of f (x) = 0, for 0 £ x £ 2p.

(Total 6 marks)

19.	Find all solutions of the equation cos 3x = cos (0.5x), for 0 £ x £ p.
Working:		
	Answer:
..................................................................	
(Total 6 marks)

20.	Consider the function f (x) = cos x + sin x.
(a)	(i)	Show that f (–) = 0.
(ii)	Find in terms of p, the smallest positive value of x which satisfies f (x) = 0.
(3)
 
	The diagram shows the graph of y = ex (cos x + sin x), – 2 £ x £ 3. The graph has a maximum turning point at C(a, b) and a point of inflexion at D.

(b)	Find .
(3)
 
(c)	Find the exact value of a and of b.
(4)

(d)	Show that at D, y = .
(5)
 
(e)	Find the area of the shaded region.
(2)
(Total 17 marks)

21.	Let f (x) = sin 2x and g (x) = sin (0.5x).
(a)	Write down
(i)	the minimum value of the function f ;
(ii)	the period of the function g.
(b)	Consider the equation f (x) = g (x).
	Find the number of solutions to this equation, for 0 £ x £ .
Working:		
	Answers:
(a)	(i)	..........................................................
	(ii)	..........................................................
(b)	.................................................................	
(Total 6 marks)

22.	Note: Radians are used throughout this question.
(a)	Draw the graph of y = p + x cos x, 0 £ x £ 5, on millimetre square graph paper, using a scale of 2 cm per unit. Make clear
(i)	the integer values of x and y on each axis;
(ii)	the approximate positions of the x-intercepts and the turning points.
(5)
 
(b)	Without the use of a calculator, show that p is a solution of the equation
p + x cos x = 0.
(3)
 
(c)	Find another solution of the equation p + x cos x = 0 for 0 £ x £ 5, giving your answer to six significant figures.
(2)
 
(d)	Let R be the region enclosed by the graph and the axes for 0 £ x £ p. Shade R on your diagram, and write down an integral which represents the area of R .
(2)
 
(e)	Evaluate the integral in part (d) to an accuracy of six significant figures. (If you consider it necessary, you can make use of the result 
(3)
(Total 15 marks)

23.	In this question you should note that radians are used throughout.
	(a)	(i)	Sketch the graph of y = x2 cos x, for 0 £ x £ 2 making clear the approximate positions of the positive x-intercept, the maximum point and the end-points.
(ii)	Write down the approximate coordinates of the positive x-intercept, the maximum point and the end-points.
(7)
(b)	Find the exact value of the positive x-intercept for 0 £ x £ 2.
(2)
	Let R be the region in the first quadrant enclosed by the graph and the x-axis.
(c)	(i)	Shade R on your diagram.
(ii)	Write down an integral which represents the area of R.
(3)
(d)	Evaluate the integral in part (c)(ii), either by using a graphic display calculator, or by using the following information.
(x2 sin x + 2x cos x – 2 sin x) = x2 cos x.
(3)
(Total 15 marks)

24.	In this part of the question, radians are used throughout.
	The function f is given by
f (x) = (sin x)2 cos x.
	The following diagram shows part of the graph of y = f (x).

	The point A is a maximum point, the point B lies on the x-axis, and the point C is a point of inflexion.
 
(a)	Give the period of f.
(1)
 
(b)	From consideration of the graph of y = f (x), find to an accuracy of one significant figure the range of  f.
(1)

(c)	(i)	Find f¢ (x).
(ii)	Hence show that at the point A, cos x = .
(iii)	Find the exact maximum value.
(9)
 
(d)	Find the exact value of the x-coordinate at the point B.
(1)
 
(e)	(i)	Find dx.
(ii)	Find the area of the shaded region in the diagram.
(4)
 
(f)	Given that f² (x) = 9(cos x)3 – 7 cos x, find the x-coordinate at the point C.
(4)
(Total 20 marks)

25.	The diagram below shows a plan for a window in the shape of a trapezium.
		
	Three sides of the window are 2 m long. The angle between the sloping sides of the window and the base is θ, where 0 < θ < .
(a)	Show that the area of the window is given by y = 4 sin θ + 2 sin 2θ.
(5)
 
(b)	Zoe wants a window to have an area of 5 m2. Find the two possible values of θ.
(4)

(c)	John wants two windows which have the same area A but different values of θ.
	Find all possible values for A.
(7)
(Total 16 marks)

26.	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)

27.	The following diagram represents a large Ferris wheel at an amusement park.
The points P, Q and R represent different positions of a seat on the wheel.
		
	The wheel has a radius of 50 metres and rotates clockwise at a rate of one revolution every 30 minutes.
	A seat starts at the lowest point P, when its height is one metre above the ground.
 
(a)	Find the height of a seat above the ground after 15 minutes.
(2)
 
(b)	After six minutes, the seat is at point Q. Find its height above the ground at Q.
(5)
 
	The height of the seat above ground after t minutes can be modelled by the function
h(t) = 50 sin (b(t – c)) + 51.
(c)	Find the value of b and of c.
(6)
 
(d)	Hence find the value of t the first time the seat is 96 m above the ground.
(3)
(Total 16 marks)

28.	Let f(x) =  +1, g(x) = 4cos – 1. Let h(x) = (g ° f)(x).
(a)	Find an expression for h(x).
(3)
(b)	Write down the period of h.
(1)
(c)	Write down the range of h.
(2)
(Total 6 marks)

29.	A farmer owns a triangular field ABC. One side of the triangle, [AC], is 104 m, a second side, [AB], is 65 m and the angle between these two sides is 60°.
(a)	Use the cosine rule to calculate the length of the third side of the field.
(3)
 
(b)	Given that sin 60°= , find the area of the field in the form  where p is an integer.
(3)

	Let D be a point on [BC] such that [AD] bisects the 60° angle. The farmer divides the field into two parts A1 and A2 by constructing a straight fence [AD] of length x metres, as shown on the diagram below.

(c)	(i)	Show that the area of A1 is given by .
(ii)	Find a similar expression for the area of A2.
(iii)	Hence, find the value of x in the form , where q is an integer.
(7)
 
(d)	(i)	Explain why .
(ii)	Use the result of part (i) and the sine rule to show that .
(5)
(Total 18 marks)

30.	In the triangle PQR, PR = 5 cm, QR = 4 cm and PQ = 6 cm.
Calculate
(a)	the size of ;
(b)	the area of triangle PQR.
(Total 6 marks)

31.	The following diagram shows the triangle AOP, where OP = 2 cm, AP = 4 cm and AO = 3 cm.

(a)	Calculate , giving your answer in radians.
(3)

	The following diagram shows two circles which intersect at the points A and B. The smaller circle C1 has centre O and radius 3 cm, the larger circle C2 has centre P and radius 4 cm, and OP = 2 cm. The point D lies on the circumference of C1 and E on the circumference of C2.Triangle AOP is the same as triangle AOP in the diagram above.

(b)	Find , giving your answer in radians.
(2)
 
(c)	Given that  is 1.63 radians, calculate the area of
(i)	sector PAEB;
(ii)	sector OADB.
(5)
 
(d)	The area of the quadrilateral AOBP is 5.81 cm2.
(i)	Find the area of AOBE.
(ii)	Hence find the area of the shaded region AEBD.
(4)
(Total 14 marks)

32.	The following diagram shows a pentagon ABCDE, with AB = 9.2 cm, BC = 3.2 cm, BD = 7.1 cm,  =110°,  = 52° and  = 60°.

(a)	Find AD.
(4)
 
(b)	Find DE.
(4)
 
(c)	The area of triangle BCD is 5.68 cm2. Find .
(4)
 
(d)	Find AC.
(4)
 
(e)	Find the area of quadrilateral ABCD.
(5)
(Total 21 marks)

33.	The diagram below shows the graph of f (x) = 1 + tan  for −360° £ x £ 360°.

(a)	On the same diagram, draw the asymptotes.
(2)
 
(b)	Write down
(i)	the period of the function;
(ii)	the value of f (90°).
(2)
 
(c)	Solve f (x) = 0 for −360° £ x £ 360°.
(2)
(Total 6 marks)

34.	(a)	Consider the equation 4x2 + kx + 1 = 0. For what values of k does this equation have two equal roots?
(3)

Let f be the function f (q ) = 2 cos 2q + 4 cos q + 3, for −360° £ q £ 360°.
(b)	Show that this function may be written as f (q ) = 4 cos2 q + 4 cos q + 1.
(1)
 
(c)	Consider the equation f (q ) = 0, for −360° £ q £ 360°.
(i)	How many distinct values of cos q satisfy this equation?
(ii)	Find all values of q which satisfy this equation.
(5)
 
(d)	Given that f (q ) = c is satisfied by only three values of q, find the value of c.
(2)
(Total 11 marks)

35.	A Ferris wheel with centre O and a radius of 15 metres is represented in the diagram below. Initially seat A is at ground level. The next seat is B, where  = .

(a)	Find the length of the arc AB.
(2)
 
(b)	Find the area of the sector AOB.
(2)

(c)	The wheel turns clockwise through an angle of . Find the height of A above the ground.
(3)
 
	The height, h metres, of seat C above the ground after t minutes, can be modelled by the function
		h (t) = 15 − 15 cos .
(d)	(i)	Find the height of seat C when t = .
(ii)	Find the initial height of seat C.
(iii)	Find the time at which seat C first reaches its highest point.
(8)
 
(e)	Find h′ (t).
(2)
 
(f)	For 0 £ t £ p,
(i)	sketch the graph of h′;
(ii)	find the time at which the height is changing most rapidly.
(5)
(Total 22 marks)

36.	Let f (x) = a (x − 4)2 + 8.
(a)	Write down the coordinates of the vertex of the curve of f.
(b)	Given that f (7) = −10, find the value of a.
(c)	Hence find the y-intercept of the curve of f.
(Total 6 marks)

37.	The following diagram shows a circle with radius r and centre O. The points A, B and C are on the circle and  =q.

The area of sector OABC is p and the length of arc ABC isp.
Find the value of r and of q.
(Total 6 marks)

38.	Let ƒ (x) = a sin b (x − c). Part of the graph of ƒ is given below.

Given that a, b and c are positive, find the value of a, of b and of c.
(Total 6 marks)

39.	The function f is defined by f : x ® 30 sin 3x cos 3x, 0 £ x £ .
(a)	Write down an expression for f (x) in the form a sin 6x, where a is an integer.
(b)	Solve f (x) = 0, giving your answers in terms of p.
(Total 6 marks)

40.	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.
(b)	Calculate the value of q.
(Total 6 marks)

41.	A farmer owns a triangular field ABC. One side of the triangle, [AC], is 104 m, a second side, [AB], is 65 m and the angle between these two sides is 60°.
(a)	Use the cosine rule to calculate the length of the third side of the field.
(3)
 
(b)	Given that sin 60° = find the area of the field in the form where p is an integer.
(3)
 
	Let D be a point on [BC] such that [AD] bisects the 60° angle. The farmer divides the field into two parts A1 and A2 by constructing a straight fence [AD] of length x metres, as shown on the diagram below.

(c)	(i)	Show that the area of Al is given by .
(ii)	Find a similar expression for the area of A2.
(iii)	Hence, find the value of x in the form , where q is an integer.
(7)
 
(d)	(i)	Explain why .
(ii)	Use the result of part (i) and the sine rule to show that
.
(5)
(Total 18 marks)

42.	Consider y = sin.
(a)	The graph of y intersects the x-axis at point A. Find the x-coordinate of A, where 0 £ x £ π.
(b)	Solve the equation sin= –, for 0 £ x £ 2p.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 6 marks)

43.	Let f (x) = sin 2x + cos x for 0 £ x £ 2p.
(a)	(i)	Find f ¢(x).
	One way of writing f ¢(x) is –2 sin2 x – sin x + 1.
(ii)	Factorize 2 sin2 x + sin x – 1.
(iii)	Hence or otherwise, solve f ¢(x) = 0.
(6)

	The graph of y = f (x) is shown below.

	There is a maximum point at A and a minimum point at B.
 
(b)	Write down the x-coordinate of point A.
(1)
 
(c)	The region bounded by the graph, the x-axis and the lines x = a and x = b is shaded in the diagram above.
(i)	Write down an expression that represents the area of this shaded region.
(ii)	Calculate the area of this shaded region.
(5)
(Total 12 marks)

44.	In triangle PQR, PQ is 10 cm, QR is 8 cm and angle PQR is acute. The area of the triangle is 20 cm2. Find the size of angle 

(Total 6 marks)

45.	Let f (x) = sin (2x + 1), 0 £ x £ π.
(a)	Sketch the curve of y = f (x) on the grid below.

(b)	Find the x-coordinates of the maximum and minimum points of f (x), giving your answers correct to one decimal place.
Working:		
	Answer:
(b)	…………………………………………..	
(Total 6 marks)

46.	Solve the equation 2 cos2 x = sin 2x for 0 £ x £ π, giving your answers in terms of π.
Working:		
	Answer:
…………………………………………..	
(Total 6 marks)

47.	The depth y metres of water in a harbour is given by the equation
y = 10 + 4 sin,
	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).

(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.
(7)
(Total 13 marks)

48.	The following diagram shows a triangle ABC, where BC = 5 cm,  = 60°,  = 40°.

(a)	Calculate AB.
(b)	Find the area of the triangle.
Working:		
	Answers:
(a)	…………………………………………..
(b)	…………………………………………..	
(Total 6 marks)

49.	Let f (x) = 1 + 3 cos (2x) for 0 £ x £ π, and x is in radians.
(a)	(i)	Find f¢ (x).
(ii)	Find the values for x for which f¢ (x) = 0, giving your answers in terms of p.
(6)

	The function g (x) is defined as g (x) = f (2x) – 1, 0 £ x £ .
(b)	(i)	The graph of f may be transformed to the graph of g by a stretch in the x-direction with scale factor  followed by another transformation. Describe fully this other transformation.
(ii)	Find the solution to the equation g (x) = f (x)
(4)
(Total 10 marks)

50.	The points P, Q, R are three markers on level ground, joined by straight paths PQ, QR, PR as shown in the diagram. QR = 9 km,  = 35°,  = 25°.
	Diagram not to scale
(a)	Find the length PR.
(3)
(b)	Tom sets out to walk from Q to P at a steady speed of 8 km h–1. At the same time, Alan sets out to jog from R to P at a steady speed of a km h–1. They reach P at the same time. Calculate the value of a.
(7)
(c)	The point S is on [PQ], such that RS = 2QS, as shown in the diagram.

	Find the length QS.
(6)
(Total 16 marks)

51.	The graph of the function f (x) = 3x – 4 intersects the x-axis at A and the y-axis at B.
(a)	Find the coordinates of
(i)	A;
(ii)	B.
(b)	Let O denote the origin. Find the area of triangle OAB.
Working:		
	Answers:
(a)	(i)	...........................................................
	(ii)	...........................................................
(b)	..................................................................	
(Total 6 marks)

52.	(a)	Factorize the expression 3 sin2 x – 11 sin x + 6.
(b)	Consider the equation 3 sin2 x – 11 sin x + 6 = 0.
(i)	Find the two values of sin x which satisfy this equation,
(ii)	Solve the equation, for 0° £ x £ 180°.
Working:		
	Answers:
(a)	..................................................................
(b)	(i)	...........................................................
	(ii)	...........................................................
	
(Total 6 marks)

53.	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)	Using the cosine rule, show that the length of AB is 12.
(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)

54.	Note: Radians are used throughout this question.
	A mass is suspended from the ceiling on a spring. It is pulled down to point P and then released. It oscillates up and down.

	Its distance, s cm, from the ceiling, is modelled by the function s = 48 + 10 cos 2πt where t is the time in seconds from release.
(a)	(i)	What is the distance of the point P from the ceiling?
(ii)	How long is it until the mass is next at P?
(5)
 
(b)	(i)	Find .
(ii)	Where is the mass when the velocity is zero?
(7)
	A second mass is suspended on another spring. Its distance r cm from the ceiling is modelled by the function r = 60 + 15 cos 4pt. The two masses are released at the same instant.
(c)	Find the value of t when they are first at the same distance below the ceiling.
(2)
 
(d)	In the first three seconds, how many times are the two masses at the same height?
(2)
(Total 16 marks)

55.	Two boats A and B start moving from the same point P. Boat A moves in a straight line at 20 km h–1 and boat B moves in a straight line at 32 km h–1. The angle between their paths is 70°.
	Find the distance between the boats after 2.5 hours.
Working:		
	Answer:
......................................................................	
(Total 6 marks)

56.	The diagram shows a triangle ABC in which AC = 7 , BC = 6,  = 45°.

(a)	Use the fact that sin 45° =  to show that sin  = .
(2)
	The point D is on (AB), between A and B, such that sin  = .
(b)	(i)	Write down the value of  + .
(ii)	Calculate the angle BCD.
(iii)	Find the length of [BD].
(6)
(c)	Show that  = .
(2)
(Total 10 marks)

57.	In triangle ABC, AC = 5, BC = 7,  = 48°, as shown in the diagram.

	Find  giving your answer correct to the nearest degree.
Working:		
	Answer:
......................................................................	
(Total 6 marks)

58.	Consider the trigonometric equation 2 sin2 x = 1 + cos x.
(a)	Write this equation in the form f (x) = 0, where f (x) = a cos2 x + b cos x + c,
and a, b, c Î .
(b)	Factorize f (x).
(c)	Solve f (x) = 0 for 0° £ x £ 360°.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................
(c)	..................................................................	
(Total 6 marks)

59.	The following diagram shows a triangle with sides 5 cm, 7 cm, 8 cm.
 Diagram not to scale
	Find
(a)	the size of the smallest angle, in degrees;
(b)	the area of the triangle.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

60.	(a)	Write the expression 3 sin2 x + 4 cos x in the form a cos2 x + b cos x + c. 
(b)	Hence or otherwise, solve the equation
3 sin2 x + 4 cos x – 4 = 0,	0° £ x £ 90°.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

61.	In the diagram below, the points O(0, 0) and A(8, 6) are ﬁxed. The angle 
varies as the point P(x, 10) moves along the horizontal line y = 10.

Diagram to scale

(a)	(i)	Show that 
(ii)	Write down a similar expression for OP in terms of x.
(2)

(b)	Hence, show that

(3)
 
(c)	Find, in degrees, the angle  when x = 8.
(2)
 
(d)	Find the positive value of x such that .
(4)
	Let the function f be deﬁned by

(e)	Consider the equation f (x) = 1.
(i)	Explain, in terms of the position of the points O, A, and P, why this
equation has a solution.
(ii)	Find the exact solution to the equation.
(5)
(Total 16 marks)

62.	The diagram below shows a sector AOB of a circle of radius 15 cm and centre O. The angle q at the centre of the circle is 2 radians.
Diagram not to scale

(a)	Calculate the area of the sector AOB.
(b)	Calculate the area of the shaded region.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

63.	The diagrams below show two triangles both satisfying the conditions
AB = 20 cm, AC = 17 cm,  = 50°.
Diagrams not
to scale

(a)	Calculate the size of  in Triangle 2.
(b)	Calculate the area of Triangle 1.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

64.	The depth, y metres, of sea water in a bay t hours after midnight may be represented by the function
, where a, b and k are constants.
	The water is at a maximum depth of 14.3 m at midnight and noon, and is at a minimum depth of 10.3 m at 06:00 and at 18:00.
	Write down the value of
(a)	a;
(b)	b;
(c)	k.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................
(c)	..................................................................	
(Total 4 marks)

65.	Town A is 48 km from town B and 32 km from town C as shown in the diagram.

	Given that town B is 56 km from town C, find the size of angle  to the nearest degree.
Working:		
	Answer:
....................................................................	
(Total 4 marks)

66.	(a)	Express 2 cos2 x + sin x in terms of sin x only.
(b)	Solve the equation 2 cos2 x + sin x = 2 for x in the interval 0 £ x £ p, giving your answers exactly.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

67.	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)

68.	Solve the equation 3 cos x = 5 sin x, for x in the interval 0° £ x £ 360°, giving your answers to the nearest degree.
Working:		
	Answer:
......................................................................	
(Total 4 marks)

69.	(a)	Sketch the graph of y = p sin x – x, –3 £ x £ 3, on millimetre square paper, using a scale of 2 cm per unit on each axis.
	Label and number both axes and indicate clearly the approximate positions of the 
x-intercepts and the local maximum and minimum points.
(5)
 
(b)	Find the solution of the equation
	p sin x – x = 0,          x > 0.
(1)
 
(c)	Find the indefinite integral

	and hence, or otherwise, calculate the area of the region enclosed by the graph, the 
x-axis and the line x = 1.
(4)
(Total 10 marks)

70.	The diagram shows a vertical pole PQ, which is supported by two wires fixed to the horizontal ground at A and B.

	 = 40 m 
 = 36°
 = 70° 
 = 30°
 
	Find
(a)	the height of the pole, PQ;
(b)	the distance between A and B.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

71.	f (x) = 4 sin.
	For what values of k will the equation f (x) = k have no solutions?
Working:		
	Answer:
......................................................................	
(Total 4 marks)

72.	The diagram shows the graph of the function f given by
f (x) = A sin + B,
	for 0 £ x £ 5, where A and B are constants, and x is measured in radians.

	The graph includes the points (1, 3) and (5, 3), which are maximum points of the graph.
(a)	Write down the values of f (1) and f (5).
(2)
(b)	Show that the period of f is 4.
(2)
 
	The point (3, –1) is a minimum point of the graph.
(c)	Show that A = 2, and find the value of B.
(5)
(d)	Show that f¢ (x) = p cos .
(4)
 
	The line y = k – px is a tangent line to the graph for 0 £ x £ 5.
(e)	Find
(i)	the point where this tangent meets the curve;
(ii)	the value of k.
(6)
(f)	Solve the equation f (x) = 2 for 0 £ x £ 5.
(5)
(Total 24 marks)