sin 2

1. (a) Show that = tan θ.

1  cos 2
(2)

π
(b) Hence find the value of cot in the form a + b 2 , where a, b  .
8
(3)
(Total 5 marks)

2. The diagram shows a tangent, (TP), to the circle with centre O and radius r. The size of
PÔA is θ radians.

(a) Find the area of triangle AOP in terms of r and θ.

(1)

(b) Find the area of triangle POT in terms of r and θ.

(2)

(c) Using your results from part (a) and part (b), show that sin θ < θ < tan θ.
(2)
(Total 5 marks)

IB Questionbank Mathematics Higher Level 3rd edition 1

3. Given ΔABC, with lengths shown in the diagram below, find the length of the
line segment [CD].

diagram not to scale

(Total 5 marks)

4. The radius of the circle with centre C is 7 cm and the radius of the circle with centre D is 5 cm.
If the length of the chord [AB] is 9 cm, find the area of the shaded region enclosed by the two
arcs AB.

diagram not to scale

(Total 7 marks)

IB Questionbank Mathematics Higher Level 3rd edition 2

5. The points P and Q lie on a circle, with centre O and radius 8 cm, such that PÔQ = 59°.

diagram not to scale

Find the area of the shaded segment of the circle contained between the arc PQ and
the chord [PQ].
(Total 5 marks)

6. The vertices of an equilateral triangle, with perimeter P and area A, lie on a circle with radius r.
P k
Find an expression for in the form , where k  +.
A r
(Total 6 marks)

1
7. (a) Given that α > 1, use the substitution u = to show that
x

 1 1 1

1 1 x 2
dx   1  u
1 2
du .

(5)

1 π
(b) Hence show that arctan α + arctan  .
 2
(2)
(Total 7 marks)

IB Questionbank Mathematics Higher Level 3rd edition 3

8. (a) Show that sin 2 nx = sin((2n + 1)x) cos x – cos((2n + 1)x) sin x.
(2)

(b) Hence prove, by induction, that

sin 2nx
cos x + cos 3x + cos 5x + ... + cos((2n – 1)x) = ,
2 sin x

for all n  +
, sin x ≠ 0.
(12)

1
(c) Solve the equation cos x + cos 3x = , 0 < x < π.
2
(6)
(Total 20 marks)

 π π
9. If x satisfies the equation sin x    2 sin x sin  , show that 11 tan x = a + b 3 ,
 3 3
where a, b  +
.
(Total 6 marks)

π
10. Throughout this question x satisfies 0 ≤ x < .
2

dy
(a) Solve the differential equation sec2 x   y 2 , where y = 1 when x = 0.
dx

Give your answer in the form y = f(x).

(7)

IB Questionbank Mathematics Higher Level 3rd edition 4

 (b) (i) Prove that 1 ≤ sec x ≤ 1 + tan x.

π
π π 1
(ii) Deduce that
4
 
0
4 sec xdx   ln 2 .
4 2
(8)
(Total 15 marks)

11. The graph below shows y = a cos (bx) + c.

Find the value of a, the value of b and the value of c.

(Total 4 marks)

IB Questionbank Mathematics Higher Level 3rd edition 5

12. In the right circular cone below, O is the centre of the base which has radius 6 cm.
The points B and C are on the circumference of the base of the cone. The height AO of the cone
is 8 cm and the angle BÔC is 60°.

diagram not to scale

Calculate the size of the angle BÂC .

(Total 6 marks)

13. Consider the triangle ABC where BÂC = 70°, AB = 8 cm and AC = 7 cm. The point D on the
BD
side BC is such that = 2.
DC
Determine the length of AD.
(Total 6 marks)

14. The interior of a circle of radius 2 cm is divided into an infinite number of sectors.
The areas of these sectors form a geometric sequence with common ratio k. The angle of the
first sector is θ radians.

(a) Show that θ = 2π(1 – k).

(5)

IB Questionbank Mathematics Higher Level 3rd edition 6

 (b) The perimeter of the third sector is half the perimeter of the first sector.

Find the value of k and of θ.

(7)
(Total 12 marks)

π
15. Consider the function f : x →  arccos x .
4

(a) Find the largest possible domain of f.

(4)

(b) Determine an expression for the inverse function, f–1, and write down its domain.
(4)
(Total 8 marks)

16. Let α be the angle between the unit vectors a and b, where 0 ≤ α ≤ π.

(a) Express │a – b│ and │a + b│ in terms of α.

(3)

(b) Hence determine the value of cos α for which │a + b│ = 3│a – b│.
(2)
(Total 5 marks)

17. (a) A particle P moves in a straight line with displacement relative to origin given by

s = 2 sin (πt) + sin(2πt), t ≥ 0,

where t is the time in seconds and the displacement is measured in centimetres.

IB Questionbank Mathematics Higher Level 3rd edition 7

 (i) Write down the period of the function s.

(ii) Find expressions for the velocity, v, and the acceleration, a, of P.

(iii) Determine all the solutions of the equation v = 0 for 0 ≤ t ≤ 4.

(10)

(b) Consider the function

f(x) = A sin (ax) + B sin (bx), A, a, B, b, x  .

Use mathematical induction to prove that the (2n)th derivative of f is given by

f(2n)(x) = (–1)n (Aa2n sin (ax) + Bb2n sin (bx)), for all n  +.
(8)
(Total 18 marks)

18. Triangle ABC has AB = 5cm, BC = 6 cm and area 10 cm2.

(a) Find sin B̂ .

(2)

(b) Hence, find the two possible values of AC, giving your answers correct to two decimal
places.
(4)
(Total 6 marks)

IB Questionbank Mathematics Higher Level 3rd edition 8

19. The diagram below shows a curve with equation y = 1 + k sin x, defined for 0 ≤ x ≤ 3π.

π 
The point A  ,2  lies on the curve and B(a, b) is the maximum point.
6 

(a) Show that k = –6.

(2)

(b) Hence, find the values of a and b.

(3)
(Total 5 marks)

1 1 π
20. (a) Show that arctan   arctan   .
2  3 4
(2)

(b) Hence, or otherwise, find the value of arctan (2) + arctan (3).
(3)
(Total 5 marks)

IB Questionbank Mathematics Higher Level 3rd edition 9

21. The diagram below shows two straight lines intersecting at O and two circles, each with centre
O. The outer circle has radius R and the inner circle has radius r.

diagram not to scale

Consider the shaded regions with areas A and B. Given that A : B = 2 : 1, find the exact value of
the ratio R : r.
(Total 5 marks)

22. A triangle has sides of length (n2 + n + 1), (2n + 1) and (n2 – 1) where n > 1.

(a) Explain why the side (n2 + n + 1) must be the longest side of the triangle.
(3)

(b) Show that the largest angle, θ, of the triangle is 120°.

(5)
(Total 8 marks)

IB Questionbank Mathematics Higher Level 3rd edition 10

23. Two non-intersecting circles C1, containing points M and S, and C2, containing points N and R,
have centres P and Q where PQ = 50. The line segments [MN] and [SR] are common tangents
to the circles. The size of the reflex angle MPS is α, the size of the obtuse angle NQR is β, and
the size of the angle MPQ is θ. The arc length MS is l1 and the arc length NR is l2. This
information is represented in the diagram below.

diagram not to scale

The radius of C1 is x, where x ≥ 10 and the radius of C2 is 10.

(a) Explain why x < 40.

(1)

x  10
(b) Show that cos θ = .
50
(2)

(c) (i) Find an expression for MN in terms of x.

(ii) Find the value of x that maximises MN.

(2)

(d) Find an expression in terms of x for

(i) α;

(ii) β.
(4)

IB Questionbank Mathematics Higher Level 3rd edition 11

 (e) The length of the perimeter is given by l1 + l2 + MN + SR.

(i) Find an expression, b(x), for the length of the perimeter in terms of x.

(ii) Find the maximum value of the length of the perimeter.

(iii) Find the value of x that gives a perimeter of length 200.

(9)
(Total 18 marks)

5π
24. Consider the graphs y = e–x and y = e–x sin 4x, for 0 ≤ x ≤ .
4

5π
(a) On the same set of axes draw, on graph paper, the graphs, for 0 ≤ x ≤ .
4
π
Use a scale of 1 cm to on your x-axis and 5 cm to 1 unit on your y-axis.
8
(3)

nπ
(b) Show that the x-intercepts of the graph y = e–x sin 4x are , n = 0, 1, 2, 3, 4, 5.
4
(3)

(c) Find the x-coordinates of the points at which the graph of y = e–x sin 4x meets the graph
of y = e–x. Give your answers in terms of π.
(3)

(d) (i) Show that when the graph of y = e–x sin 4x meets the graph of y = e–x, their
gradients are equal.

(ii) Hence explain why these three meeting points are not local maxima of the
graph y = e–x sin 4x.
(6)

IB Questionbank Mathematics Higher Level 3rd edition 12

 (e) (i) Determine the y-coordinates, y1, y2 and y3, where y1 > y2 > y3, of the local maxima
5π
of y = e–x sin 4x for 0 ≤ x ≤ . You do not need to show that they are maximum
4
values, but the values should be simplified.

(ii) Show that y1, y2 and y3 form a geometric sequence and determine the common ratio
r.
(7)
(Total 22 marks)

25. The diagram below shows two concentric circles with centre O and radii 2 cm and 4 cm.
π
The points P and Q lie on the larger circle and PÔQ = x, where 0 < x < .
2

diagram not to scale

(a) Show that the area of the shaded region is 8 sin x – 2x.
(3)

(b) Find the maximum area of the shaded region.

(4)
(Total 7 marks)

IB Questionbank Mathematics Higher Level 3rd edition 13

26. In triangle ABC, AB = 9 cm, AC =12 cm, and B̂ is twice the size of Ĉ .

Find the cosine of Ĉ .

(Total 5 marks)

27. In the diagram below, AD is perpendicular to BC.

CD = 4, BD = 2 and AD = 3. CÂD =  and BÂD = .

Find the exact value of cos ( − ).

(Total 6 marks)

IB Questionbank Mathematics Higher Level 3rd edition 14

28. The diagram below shows the boundary of the cross-section of a water channel.

 x 
The equation that represents this boundary is y = 16 sec   – 32 where x and y are both
 36 
measured in cm.

The top of the channel is level with the ground and has a width of 24 cm. The maximum depth
of the channel is 16 cm.

Find the width of the water surface in the channel when the water depth is 10 cm. Give your
answer in the form a arccos b where a, b .
(Total 6 marks)

29. A system of equations is given by

cos x + cos y = 1.2

sin x + sin y = 1.4.

(a) For each equation express y in terms of x.

(2)

(b) Hence solve the system for 0  x < , 0 < y < .

(4)
(Total 6 marks)

IB Questionbank Mathematics Higher Level 3rd edition 15

30. Find, in its simplest form, the argument of (sin + i (1− cos ))2 where  is an acute angle.
(Total 7 marks)

31. In triangle ABC, BC = a, AC = b, AB = c and [BD] is perpendicular to [AC].

(a) Show that CD = b − c cos A.

(1)

(b) Hence, by using Pythagoras’ Theorem in the triangle BCD, prove the cosine rule for the
triangle ABC.
(4)

1 3
(c) If AB̂C = 60, use the cosine rule to show that c = a  b2  a 2 .
2 4
(7)
(Total 12 marks)

IB Questionbank Mathematics Higher Level 3rd edition 16

32.

The above three dimensional diagram shows the points P and Q which are respectively west and
south-west of the base R of a vertical flagpole RS on horizontal ground. The angles of elevation
of the top S of the flagpole from P and Q are respectively 35 and 40, and PQ = 20 m.

Determine the height of the flagpole.

(Total 8 marks)

33. The depth, h (t) metres, of water at the entrance to a harbour at t hours after midnight on a
particular day is given by

 t 
h (t) = 8 + 4 sin  , 0  t  24.
6

(a) Find the maximum depth and the minimum depth of the water.
(3)

(b) Find the values of t for which h (t)  8.

(3)
(Total 6 marks)

34. Consider triangle ABC with BÂC = 37.8, AB = 8.75 and BC = 6.

Find AC.
(Total 7 marks)

35. (a) Sketch the curve f(x) = sin 2x, 0 ≤ x ≤ π.

(2)

IB Questionbank Mathematics Higher Level 3rd edition 17

 (b) Hence sketch on a separate diagram the graph of g(x) = csc 2x, 0 ≤ x ≤ π, clearly stating
the coordinates of any local maximum or minimum points and the equations of any
asymptotes.
(5)

(c) Show that tan x + cot x ≡ 2 csc 2x.

(3)

(d) Hence or otherwise, find the coordinates of the local maximum and local minimum points
π
on the graph of y = tan 2x + cot 2x, 0 ≤ x ≤ .
2
(5)

π
(e) Find the solution of the equation csc 2x = 1.5 tan x – 0.5, 0 ≤ x ≤ .
2
(6)
(Total 21 marks)

36. In a triangle ABC, Â = 35°, BC = 4 cm and AC = 6.5 cm. Find the possible values of B̂ and
the corresponding values of AB.
(Total 7 marks)

37. Solve sin 2x = 2 cos x, 0 ≤ x ≤ π.

(Total 6 marks)

IB Questionbank Mathematics Higher Level 3rd edition 18

 5
38. The obtuse angle B is such that tan B =  . Find the values of
12

(a) sin B;
(1)

(b) cos B;
(1)

(c) sin 2B;

(2)

(d0 cos 2B.

(2)
(Total 6 marks)

3
39. Given that tan 2θ = , find the possible values of tan θ.
4
(Total 5 marks)

40. Let sin x = s.

(a) Show that the equation 4 cos 2x + 3 sin x cosec3 x + 6 = 0 can be expressed as
8s4 – 10s2 + 3 = 0.
(3)

(b) Hence solve the equation for x, in the interval [0, π].
(6)
(Total 9 marks)

41. (a) If sin (x – α) = k sin (x + α) express tan x in terms of k and α.

(3)

1
(b) Hence find the values of x between 0° and 360° when k = and α = 210°.
2
(6)
(Total 9 marks)

IB Questionbank Mathematics Higher Level 3rd edition 19

42. The angle θ satisfies the equation 2 tan2 θ – 5 sec θ – 10 = 0, where θ is in the second quadrant.
Find the value of sec θ.
(Total 6 marks)

43. The lengths of the sides of a triangle ABC are x – 2, x and x + 2. The largest angle is 120°.

(a) Find the value of x.

(6)

15 3
(b) Show that the area of the triangle is .
4
(3)

p q
(c) Find sin A + sin B + sin C giving your answer in the form where p, q, r  .
r
(4)
(Total 13 marks)

44. A farmer owns a triangular field ABC. The side [AC] is 104 m, the side [AB] is 65 m and the
angle between these two sides is 60°.

(a) Calculate the length of the third side of the field.

(3)

(b) Find the area of the field in the form p 3 , 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 by constructing a straight fence [AD] of length x metres.

65x
(c) (i) Show that the area o the smaller part is given by and find an expression for
4
the area of the larger part.

(ii) Hence, find the value of x in the form q 3 , where q is an integer.

(8)

IB Questionbank Mathematics Higher Level 3rd edition 20

 BD 5
(d) Prove that  .
DC 8
(6)
(Total 20 marks)

IB Questionbank Mathematics Higher Level 3rd edition 21