©2015 Coventry University 101MP /12/Mock
January 2016
Coventry University
Faculty of Engineering, Environment & Computing
101MP
Algebra - Mock Exam Paper
Instructions to candidates:
Time Allowed: 3 hours
This is a Closed Book examination.
This exam is comprised of TWO sections.
There are eight questions in section A, each worth five marks. In section B
there are four questions, each worth twenty marks.
Attempt all of section A and any THREE of section B.
Students who attempt more than three questions in section B will be marked
on their best three attempts.
Start each question on a new page and carefully identify your answers with
the correct question number.
For this examination you will be supplied with:
1 Answer Book
Selected Mathematical Formulae Booklet
You must hand this paper in at the end of the examination.
Page 1 of 5
Section A ©2015 Coventry University 101MP /12/Mock
Attempt all of section A
Each question is worth 5 marks.
A1. Let X and Y be sets defined by
X = {x, y, z},
Y = {1, 2, a}.
(a) Give an example of a bijective function from X to Y .
(b) Write down all the elements of the product X × Y .
(c) Write down all the elements of the disjoint union X q Y . (5 marks)
√
A2. Write both z = 1− 3i and w = 1−i in the form reiθ and hence calculate
z 6
w
giving your answer in the form a + bi. (5 marks)
A3. Find integers m and n such that
17m + 31n = 1
using Euclid’s algorithm. (5 marks)
A4. Compute the matrix C, where
C = AB − BA,
for matrices
1 2 0 5
A= and B= .
2 −1 5 −2
Are each of the matrices A, B and C, symmetric, anti-symmetric or
neither? (5 marks)
Continue
Page 2 of 5
Section A ©2015 Coventry University 101MP /12/Mock
A5. Define power series p(x) and q(x) by
p(x) = 1 − x + x2 − x3 + x4 − x5 + · · ·
q(x) = 1 + 2x + 4x2 + 8x3 + 16x4 + 32x5 + · · ·
Find the first 6 terms (i.e. all terms up to degree 5) of the power series
p(x)q(x). (5 marks)
A6. Find the inverse of the matrix,
3 −1 1
A = 2 −1 1
4 −2 1
explaining your method. (5 marks)
A7. Let
1 0
a = −1 and b = −1 .
0 1
(a) Find the angle θ between the vectors a and b.
(b) What is the area of the parallelogram with vertices
0, a, b and a + b?
(5 marks)
A8. Show that the only eigenvalue of the matrix
1 1 0
M = 0 1 1
0 0 1
is 1 and find all of its eigenvectors. (5 marks)
Continue
Page 3 of 5
Section B ©2015 Coventry University 101MP /12/Mock
Attempt THREE questions of section B
Each question is worth 20 marks.
B1. For each of the following equations, draw the set of complex solutions
in a separate Argand diagram, explaining your reasoning.
(a) zz = 4, (5 marks)
(b) (z − 1)3 = 1, (5 marks)
(c) Im(z) + Re(z) = 1, (5 marks)
2 2
(d) z − z = 4. (5 marks)
B2. Let A and B be symmetric matrices defined by
0 1 22 −6
A= and B =
1 0 −6 17
(a) Write down the corresponding quadratic forms. (2 marks)
(b) Compute the eigenvalues of A and B. (4 marks)
(c) Compute the eigenvectors of A and B. (6 marks)
(d) Hence write B in the form
B = P DP t ,
where P is an orthogonal matrix and D is a diagonal matrix. (4 marks)
(e) Draw the set of solutions to the quadratic form
22x2 − 12xy + 17y 2 = 1.
(4 marks)
Continue
Page 4 of 5
Section A ©2015 Coventry University 101MP /12/Mock
B3. (a) Let (an )n≥1 be the sequence
0.9, 0.99, 0.999, . . .
i.e. an is the real number given in decimal form by 0.9 · · · 9 with n
copies of 9 after the decimal point.
Find the series (bn )n≥1 with bn = |an − 1|. (2 marks)
(b) State the definition of convergence of a sequence to a limit L. (3 marks)
(c) Hence prove that the series (an )n≥1 converges to the limit 1. (7 marks)
(d) Let (cn )n≥1 be the sequence with cn = a2n . Prove from the definition
that this sequence also converges to 1. (8 marks)
B4. Let
g : [0, ∞) → R
be the function defined by g(x) = x3 .
(a) State what it means for a real-valued function f to be strictly in-
creasing and show that g is strictly increasing. (4 marks)
(b) Is g injective? Explain your answer. (3 marks)
(c) Is g surjective? Explain your answer. (3 marks)
Now let h be the function
h : {0, 1, 2, 3, 4, 5, 6} → {0, 1, 2, 3, 4, 5, 6}
defined by h(x) = x3 (mod 7).
(d) Write out the table of values for h. (4 marks)
(e) Is h injective? Explain your answer. (3 marks)
(f) Is h surjective? Explain your answer. (3 marks)
END
Page 5 of 5
