TMA 02

Question 1 (Unit 4)
Consider the vectors
a = −i + j + 2k, b = 2i − j.
(a) Find the magnitude of the vector a.
(b) Find a unit vector in the direction of a.
(c) Calculate the scalar product of a and b and hence calculate the angle
between a and b in degrees (to the nearest degree).
(d) Calculate the vector product a × b.

Question 2 (Unit 4) –

Consider a pyramid with a square base with sides of length a and sloping
sides also of length a, as shown below. Choose a co-ordinate system with
origin, O, at one corner, i-direction aligned with −→OA, j-direction aligned with
−−→OC and k-direction perpendicular to this plane such that all the components
of −−→OD are positive.
(a) Write down the vector equation of the line l through the centre of the
square base that is perpendicular to the base. [2]
(b) Use the fact that the vertex D is on the line l together with the fact
that the length of the sloping side is a to calculate the coordinates of
the point D with respect to the origin O

Question 3 (Unit 4) – 7 marks

(a) Calculate the determinant

0 2 1
3 2 2
2 1 3

(b) Hence, or otherwise, state whether the following simultaneous equations
have a unique solution. Briefly justify your answer.
2y + z = 1,
3x + 2y + 2z = 2,
2x + y + 3z = 3

Question 4 (Unit 5)

Consider the system of equations
2x + 3y − z = 6,
4x + 7y + z = 10,
2x + 4y − 3z = 9.

(a) Use the Gaussian elimination method to reduce the system to upper
triangular form. Clearly label the operations that you use.
(b) If the system has no solution, then clearly state this; if it has a unique
solution, then find it; and if it has an infinite number of solutions, then
find the most general solution.

Question 5 (Unit 5)

(a) Find the eigenvalues and eigenvectors of the matrix
A =

1 −2 −2 1
(b) Express the vector v =

2
3

in the form αv1 + βv2, where v1 and v2 are
the eigenvectors that you found in part (a). [4]
(c) Hence, or otherwise, calculate the product A12 v

Question 6 (Unit 6) – 30 marks
(a) Express the following inhomogeneous system of first-order differential
equations for x(t) and y(t) in matrix form:
dx
dt = x + 2y + 2e
−t
dy
dt = −x + 4y + 11e
−t

Write down, also in matrix form, the corresponding homogeneous
system of equations.
(b) Find the eigenvalues of the matrix of coefficients and an eigenvector
corresponding to each eigenvalue.
(c) Hence write down the complementary function for the system of
equations.
(d) Find a particular integral for the original inhomogeneous system.
(e) Hence write down the general solution of the original inhomogeneous
system.
(f) Find the particular solution of the original inhomogeneous system with
x = 4 and y = −1 when t = 0.
(g) What is the long-term behaviour of this particular solution as t becomes
large? Does the ratio y/x tend to a fixed number, and if so what
number?