IB Questionbank Mathematics Higher Level 3rd edition 1

1. The diagram below shows a circle with centre O. The points A, B, C lie on the circumference of

the circle and [AC] is a diameter.

Let ba OB and OA .

(a) Write down expressions for CB and AB in terms of the vectors a and b. (2)

(b) Hence prove that angle CBA is a right angle. (3)

(Total 5 marks)

2. In the diagram below, [AB] is a diameter of the circle with centre O. Point C is on the

circumference of the circle. Let cb OC and OB .

IB Questionbank Mathematics Higher Level 3rd edition 2

(a) Find an expression for CB and for AC in terms of b and c. (2)

(b) Hence prove that BCA is a right angle. (3)

(Total 5 marks)

3. Consider the functions f(x) = x3 + 1 and g(x) =

1

13 x

. The graphs of y = f(x) and y = g(x) meet

at the point (0, 1) and one other point, P.

(a) Find the coordinates of P. (1)

(b) Calculate the size of the acute angle between the tangents to the two graphs at the point P. (4)

(Total 5 marks)

4. Consider the vectors baba OC and OB,OA . Show that if a=b then

(a + b)(a b) = 0. Comment on what this tells us about the parallelogram OACB. (Total 4 marks)

5. (a) Consider the vectors a = 6i + 3j + 2k, b = 3j + 4k.

(i) Find the cosine of the angle between vectors a and b.

(ii) Find a b.

(iii) Hence find the Cartesian equation of the plane containing the vectors a and b and passing through the point (1, 1, 1).

(iv) The plane intersects the x-y plane in the line l. Find the area of the finite triangular region enclosed by l, the x-axis and the y-axis.

(11)

IB Questionbank Mathematics Higher Level 3rd edition 3

(b) Given two vectors p and q,

(i) show that p p = p2;

(ii) hence, or otherwise, show that p + q2 = p2 + 2p q + q2;

(iii) deduce that p + qp + q. (8)

(Total 19 marks)

6. Consider the vectors a = sin(2)i cos(2)j + k and b = cos i sin j k, where 0 < < 2.

Let be the angle between the vectors a and b.

(a) Express cos in terms of . (2)

(b) Find the acute angle for which the two vectors are perpendicular. (2)

(c) For = 6

7, determine the vector product of a and b and comment on the geometrical

significance of this result. (4)

(Total 8 marks)

7. Given that a = 2 sin i + (1 sin )j, find the value of the acute angle , so that a is perpendicular to the line x + y = 1.

(Total 5 marks)

8. Two lines are defined by

l1 : r = 4

7

3

4: and

2

2

3

6

4

3

2

yx

l = (z + 3).

IB Questionbank Mathematics Higher Level 3rd edition 4

(a) Find the coordinates of the point A on l1 and the point B on l2 such that AB is

perpendicular to both l1 and l2.

(13)

(b) Find AB. (3)

(c) Find the Cartesian equation of the plane that contains l1 and does not intersect l2. (3)

(Total 19 marks)

9. Given any two non-zero vectors a and b, show that a b2 = a2b2 (a b)2. (Total 6 marks)

10. Three distinct non-zero vectors are given by cba OC and,OB,OA .

If OA is perpendicular to BC and OB is perpendicular to CA , show that OC is

perpendicular to AB . (Total 6 marks)

11. The angle between the vector a = i 2j + 3k and the vector b = 3i 2j + mk is 30.

Find the values of m. (Total 6 marks)

IB Questionbank Mathematics Higher Level 3rd edition 5

12. A triangle has its vertices at A(1, 3, 2), B(3, 6, 1) and C(4, 4, 3).

(a) Show that ACAB = 10. (3)

(b) Find CAB

. (5)

(Total 8 marks)