1. 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 AAHL Higher Level 1

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

diagram not to scale

(Total 5 marks)

3. 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)

IB Questionbank Mathematics AAHL Higher Level 2

4. 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)

5. (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
6. 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)

IB Questionbank Mathematics AAHL Higher Level 3

7. 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)

8. 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)

9. 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 AAHL Higher Level 4

 (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
10. 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)

11. 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 AAHL Higher Level 5

12. 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)

13. 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 AAHL Higher Level 6

14. 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)

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

16. 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)

IB Questionbank Mathematics AAHL Higher Level 7

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

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

Find AC.
(Total 7 marks)

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

(Total 6 marks)

5
19. 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
20. Given that tan 2θ = , find the possible values of tan θ.
4
(Total 5 marks)

IB Questionbank Mathematics AAHL Higher Level 8

21. (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)

22. 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)

23. 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 AAHL Higher Level 9

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

IB Questionbank Mathematics AAHL Higher Level 10