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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

IB Questionbank Maths SL 1

(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)


x43

2πf

2π

(c) Given that (g ° f)(x) can be written as cos (kx), find the value of k, k .(3)

(Total 7 marks)

5. 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)

6. (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)

7. A rectangle is inscribed in a circle of radius 3 cm and centre O, as shown below.


BOA 3π

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)


2π

(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)

8. 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)


ddA

51

6OR and

21

3OQ,

13

4OP

PQ

PR

21QPRcos

QPRsin

3a

9. Let f(x) = sin x + e2x cos x, for 0 ≤ x ≤ π. Given that , solve the equationf(x) = 0.

(Total 6 marks)

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

11. 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)

12. Consider g (x) = 3 sin 2x.

(a) Write down the period of g.(1)


x2e3 31

6πtan

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

(3)

(c) Write down the number of solutions to the equation g (x) = 2, for 0 x 2.(2)

(Total 6 marks)

13. (a) Given that cos A = and 0 A find cos 2A.(3)

(b) Given that sin B = and B , find cos B.(3)

(Total 6 marks)


43210

– 1– 2– 3– 4

π2

32π 2 ππ

y

x

31 ,

2π

32

2π

14. The following diagram shows a semicircle centre O, diameter [AB], with radius 2.

Let P be a point on the circumference, with = radians.

(a) Find the area of the triangle OPB, in terms of .(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( − 2 sin ).(3)

(d) Find the value of when S is a local minimum, justifying that it is a minimum.(8)

(e) Find a value of for which S has its greatest value.(2)

(Total 18 marks)

15. Let f(x) = sin3 x + cos3 x tan x, < x < π.

(a) Show that f(x) = sin x.(2)


BOP

2π

(b) Let sin x = . Show that f(2x) = .(5)

(Total 7 marks)

16. 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)


32

954

6π

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)

17. 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)


21

10

3

(b) Calculate the value of q.(4)

(Total 6 marks)

18. Given that and that cosθ = , find

(a) sin θ;(3)

(b) cos 2θ;(3)

(c) sin (θ + π).(1)

(Total 7 marks)

19. (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)

20. 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)


π2π

1312

(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)

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

(a) Show that sin = .

(b) Show that sin 2 = .

(c) Find the exact value of cos 2.(Total 6 marks)


BCA CAB

35

954

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

(a) Show that = .

(b) The area of the sector is 25 cm2. Find the value of r.(Total 6 marks)

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

(a) On the same diagram, draw the asymptotes.(2)


rr220

2x

(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)

24. 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)

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

the circle and =.

The area of sector OABC is and the length of arc ABC is .

Find the value of r and of .(Total 6 marks)


COA

34

32

26. 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)

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

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 .(Total 6 marks)


BOA

28. 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 .(Total 6 marks)

29. 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)


3π

4 0

3 0

2 0

1 0

– 1 0

– 2 0

– 3 0

– 4 0

/2 x

y

30. 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)


BOA

Or

AC

B

31. 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 2.

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 6 marks)

32. 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)


9x

9x

21

R.QP

Wo rkin g:

A nsw ers:

.. .. ................ .. .. . .. .............. .. .. .. . ........

33. Let f (x) = 6 sin x , 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)


0 1 2

B

x

y

Wo rk ing :

A n sw ers:

(a ) ........... ..... .. .. .. .. .. .. .. .. .. ...............

(b ) .... .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. ... . .. .. ......

34. 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 .

(b) Factorize f (x).

(c) Write down the number of solutions of f (x) = 0, for 0 x 2.

(Total 6 marks)

35. 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 = 1.5 radians.


A n sw ers:

Wo rk in g :

(a ) ....................... .. ..... .. .. .. .. .. .. .. ... .. . ...

(b ) ..... ............................. .. .. .. .. ... .. .. . .. .

(c ) ....... .. .. .. .. .. ... .. .. .. .. .. .. .. .. .. .. ..... .. .. ....

A B

C D

O

Find the area of the shaded region.

Working:

Answer:

…………………………………………..(Total 6 marks)


36. 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)


2

1 .5

1

0 .5

0

– 0 .5

– 1

– 1 .5

– 2

0 .5 1 1 .5 2 2 .5 3 3 .5 x

y

37. 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)

38. Solve the equation 2 cos2 x = sin 2x for 0 x π, giving your answers in terms of π.

Working:

Answer:

…………………………………………..(Total 6 marks)


CAB

39. 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)

40. 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].


B C

6 0 ° 4 0 °

A

B C5 c m

BOA

0 .8

5 c m

N

A

BO

Find the area of the shaded region.

Working:

Answer:

…………………………………………........(Total 6 marks)


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

Find the value of

(a) p;

(b) q.

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 6 marks)


y

x

3

2

1

0

– 1

2

42. Find all solutions of the equation cos 3x = cos (0.5x), for 0 x .

Working:

Answer:

..................................................................

(Total 6 marks)


43. 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)


A

B

C

D

E

P2

2

44. 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)


45. (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)


46. 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)


O

B

C

A

2 radD ia g ra m n o t to sca le

A B = 2 rad ian sO A = 1 5 cm

Ô

47. 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)


48. 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)


2π3

49. 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 allreal 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)


2π

2π

50. 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)


A

A B

C

5 7

4 8 °

d ia g ram n o t to sca le

,B

51. 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)


31

52. 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 .

(b) Factorize f (x).

(c) Solve f (x) = 0 for 0° x 360°.

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(c) ..................................................................(Total 6 marks)


53. 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)


5 7

8

54. (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)


55. In the following diagram, O is the centre of the circle and (AT) is the tangent to the circle at T.


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)

56. The diagram below shows a sector AOB of a circle of radius 15 cm and centre O. The angle at the centre of the circle is 2 radians.



O

T

A

A B

O

(a) Calculate the area of the sector AOB.

(b) Calculate the area of the shaded region.

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 4 marks)

57. The diagrams below show two triangles both satisfying the conditions

AB = 20 cm, AC = 17 cm, = 50°.

Diagrams notto scale


CBA

A

B C

A

B C

Tria n g le 1 T rian g le 2

(a) Calculate the size of in Triangle 2.

(b) Calculate the area of Triangle 1.

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 4 marks)


BCA

58. 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)

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


t

kbay 2cos

AB

C

3 2 k m

4 8 k m

Given that town B is 56 km from town C, find the size of angle to the nearest degree.

Working:

Answer:

....................................................................

(Total 4 marks)

60. (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 , giving your answers exactly.

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 4 marks)


BAC

61. 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)

62. If A is an obtuse angle in a triangle and sin A = , calculate the exact value of sin 2A.

Working:

Answer:

......................................................................

(Total 4 marks)

63. Given that sin θ = , cos θ = – and 0° ≤ θ ≤ 360°,

(a) find the value of θ;


135

21

23

(b) write down the exact value of tan θ.

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 4 marks)

64. 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°


Q

P

A

B3 6 3 0

7 0

BQ

QBP

QAB

QBA

Find

(a) the height of the pole, PQ;

(b) the distance between A and B.

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 4 marks)

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


1 ra d ian

Find the perimeter of the shaded region.

Working:

Answer:

......................................................................

(Total 4 marks)

66. f (x) = 4 sin .

For what values of k will the equation f (x) = k have no solutions?

Working:

Answer:

......................................................................

(Total 4 marks)


23 x

67. 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)

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


O

A B

The area of the shaded sector OAB is 21.6 cm2. Find the length of the minor arc AB.

Working:

Answer:

......................................................................

(Total 4 marks)

69. Solve the equation 3 sin2 x = cos2 x, for 0° x 180°.

Working:

Answer:

......................................................................

(Total 4 marks)


70. 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)


1 0 cm

1 0 cmh

r

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