1.	Let A = .
(a)	Find A^–1.
(2)
 
(b)	Solve the matrix equation AX = .
(4)
(Total 6 marks)

2.	Let A = .
 
(a)	Find the value of x for which A^–1 does not exist.
(3)
 
(b)	Given that A = A^–1, find x.
(5)
(Total 8 marks)

3.	Let W =  and P = .
(a)	Find WP.
(3)

(b)	Given that 2WP + S = , find S.
(3)
(Total 6 marks)

4.	Let A =  and B = .
(a)	Find AB.
(3)
 
(b)	Solve A^–1 X = B.
(2)
(Total 5 marks)

5.	Let A = .
(a)	Find an expression for det A.
(2)
 
(b)	Find the value of x for which A has no inverse. Express your answer in the form a ln b, where a, b  .
(5)
(Total 7 marks)

6.	Let A =  and B = .
(a)	(i)	Find AB.
(ii)	Write down the inverse of A.
(3)
 
	Let X =  and C = .
(b)	Solve the matrix equation AX = C.
(4)
(Total 7 marks)

7.	A matrix M has inverse M^–1 = .
(a)	Find M.
(3)
(b)	Solve the matrix equation MX = B, where B =  and X = .
(3)
(Total 6 marks)

8.	Let A =  and B = .
Find
(a)	A + B;
(2)
(b)	−3A;
(2)
(c)	AB.
(3)
(Total 7 marks)

9.	Let M = .
(a)	Write down the determinant of M.
(1)
 
(b)	Write down M^−1.
(2)
 
(c)	Hence solve M 
(3)
(Total 6 marks)

10.	Let A =  and B = .
(a)	Find AB in terms of p and q.
(2)

(b)	Matrix B is the inverse of matrix A. Find the value of p and of q.
(5)
(Total 7 marks)

11.	Let A = .
(a)	Find A^2.
(2)
 
(b)	Let B = . Solve the matrix equation 3X + A = B.
(3)
(Total 5 marks)

12.	Let M = , and O = . Given that M^2 – 6M + kI = O, find k.
(Total 6 marks)

13.	(a)	Given A = , find A^–1.
(2)
 
(b)	Hence, solve the system of simultaneous equations.
	7x + 8y = 1
2x + 3y = 1
(4)
(Total 6 marks)

14.	(a)	Let + = .
(i)	Write down the value of a.
(ii)	Find the value of b.
(b)	Let 3 – 5= .
Find the value of q.
(Total 6 marks)

15.	Let A =  and B = . Find, in terms of k,
(a)	2A− B;
(b)	det (2A− B).
(Total 6 marks)

16.	Let A =  and B = . Giving your answers in terms of a, b, c, d and e,
(a)	write down A + B;
(b)	find AB.
(Total 6 marks)

17.	Let C =  and D = .
	The 2 × 2 matrix Q is such that 3Q = 2C – D
(a)	Find Q.
(3)
(b)	Find CD.
(4)
(c)	Find D^–1.
(2)
(Total 9 marks)

18.	The matrices A, B, X are given by
	A =  B =  X =  where a, b, c, d ∈ _.
	Given that AX + X = Β, find the exact values of a, b, c and d.
(Total 8 marks)

19.	If A = and det A = 14, find the possible values of p.
Working: |

Answer:
......................................................................

(Total 4 marks)

20.	Let M = , where a ∈ .
(a)	Find M^2 in terms of a.
(4)
(b)	If M^2 is equal to , find the value of a.
(2)
(c)	Using this value of a, find M^–l and hence solve the system of equations:
	–x + 2y = –3
2x – y = 3
(6)
(Total 12 marks)

21.	A and B are 2 × 2 matrices, where A = and BA = .  Find B.
Working: |

Answer:
......................................................................

(Total 4 marks)

22.	Line L_1 passes through points A(1, –1, 4) and B(2, –2, 5).
(a)	Find .
(2)
(b)	Find an equation for L_1 in the form r = a + tb.
(2)

	Line L_2 has equation r = .
(c)	Find the angle between L_1 and L_2.
(7)

(d)	The lines L_1 and L_2 intersect at point C. Find the coordinates of C.
(6)
(Total 17 marks)

23.	In this question, distance is in metres.
	Toy airplanes fly in a straight line at a constant speed. Airplane 1 passes through a point A.
Its position, p seconds after it has passed through A, is given by 

(a)	(i)	Write down the coordinates of A.
(ii)	Find the speed of the airplane in m s^–1.
(4)

(b)	After seven seconds the airplane passes through a point B.
(i)	Find the coordinates of B.
(ii)	Find the distance the airplane has travelled during the seven seconds.
(5)

(c)	Airplane 2 passes through a point C. Its position q seconds after it passes through C is given by , a  .
	The angle between the flight paths of Airplane 1 and Airplane 2 is 40°. Find the two values of a.
(7)
(Total 16 marks)

24.	Let v =  and w = , for k > 0. The angle between v and w is .
Find the value of k.
(Total 7 marks)

25.	A triangle has its vertices at A(–1, 3), B(3, 6) and C(–4, 4).
(a)	Show that  = –9.
(3)

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

26.	Two lines with equations r_1 =  and r_2 =  intersect at the point P. Find the coordinates of P.
(Total 6 marks)

27.	In this question, distance is in kilometres, time is in hours.
	A balloon is moving at a constant height with a speed of 18 km h^–1, in the direction of the vector .
	At time t = 0, the balloon is at point B with coordinates (0, 0, 5).

(a)	Show that the position vector b of the balloon at time t is given by
	b = .
(6)

	At time t = 0, a helicopter goes to deliver a message to the balloon.
The position vector h of the helicopter at time t is given by
	h = .

(b)	(i)	Write down the coordinates of the starting position of the helicopter.
(ii)	Find the speed of the helicopter.
(4)

(c)	The helicopter reaches the balloon at point R.
(i)	Find the time the helicopter takes to reach the balloon.
(ii)	Find the coordinates of R.
(5)
(Total 15 marks)

28.	A triangle has its vertices at A(–1, 3), B(3, 6) and C(–4, 4).
(a)	Show that 
(b)	Show that, to three significant figures, cos
(Total 6 marks)

29.	In this question, distance is in kilometers, time is in hours.
A balloon is moving at a constant height with a speed of l8 km h^–1, in the
direction of the vector 
	At time t = 0, the balloon is at point B with coordinates (0, 0, 5).

(a)	Show that the position vector b of the balloon at time t is given by
b = 
(6)

	At time t = 0, a helicopter goes to deliver a message to the balloon. The position vector h of the helicopter at time t is given by
h = 

(b)	(i)	Write down the coordinates of the starting position of the helicopter.
(ii)	Find the speed of the helicopter.
(4)

(c)	The helicopter reaches the balloon at point R.
(i)	Find the time the helicopter takes to reach the balloon.
(ii)	Find the coordinates of R.
(5)
(Total 15 marks)

30.	Three of the coordinates of the parallelogram STUV are S(–2, –2), T(7, 7), U(5, 15).
(a)	Find the vector  and hence the coordinates of V.
(5)

(b)	Find a vector equation of the line (UV) in the form r = p + λd where λ ∈ .
(2)

(c)	Show that the point E with position vector is on the line (UV), and find the value of λ for this point.
(2)

	The point W has position vector , a ∈ .
(d)	(i)	If  = 2, show that one value of a is –3 and find the other possible value of a.
(ii)	For a = –3, calculate the angle between  and .
(10)
(Total 19 marks)

31.	The following diagram shows the point O with coordinates (0, 0), the point A with position vector a = 12i + 5j, and the point B with position vector b = 6i + 8j. The angle between (OA) and (OB) is θ.
	Diagram not to scale

	Find
(i)	| a |;
(ii)	a unit vector in the direction of b;
(iii)	the exact value of cos θ in the form , where, p, q ∈ .
(Total 6 marks)

32.	In this question, a unit vector represents a displacement of 1 metre.
	A miniature car moves in a straight line, starting at the point (2, 0).
After t seconds, its position, (x, y), is given by the vector equation

(a)	How far from the point (0, 0) is the car after 2 seconds?
(2)

(b)	Find the speed of the car.
(2)

(c)	Obtain the equation of the car’s path in the form ax + by = c.
(2)
	Another miniature vehicle, a motorcycle, starts at the point (0, 2), and travels
in a straight line with constant speed. The equation of its path is
	y = 0.6x + 2,	x ≥ 0.
	Eventually, the two miniature vehicles collide.
(d)	Find the coordinates of the collision point.
(3)
(e)	If the motorcycle left point (0, 2) at the same moment the car left point (2, 0), ﬁnd the speed of the motorcycle.
(5)
(Total 14 marks)

33.	The vectors  and  are perpendicular for two values of x.
(a)	Write down the quadratic equation which the two values of x must satisfy.
(b)	Find the two values of x.
Working: |

Answers:
(a)	..................................................................
(b)	..................................................................

(Total 4 marks)

IB Questionbank Maths SL