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LANCASTER UNIVERSITY
2008 EXAMINATIONS
PART II
MATHEMATICS & STATISTICS
Math 270 : Minor Course in Mathematics
Paper 2: Matrix Methods and Probability 2 hours
Answer TWO questions in Section A and TWO questions in Section B. Use separate answer booklets
for Section A and Section B.
SECTION A Matrix Methods
A1. (a) Given that z = − 1 − i,
(i) find r and θ such that
z = r(cos θ + i sin θ), r > 0, 0 ≤ θ ≤ 2 π, [4]
(ii) find the cube roots of z in exponential form. [6]
(b) Using De Moivre’s Theorem, show that
cos 4θ = 1 − 8 sin
2 θ + 8 sin
4 θ.
[9]
(c) (i) Show that z 1 z 2 = z 1 z 2 for any complex numbers z 1 , z 2. [4]
(ii) Hence or otherwise prove that |z 1 + z 2 | ≤ |z 1 | + |z 2 | for all z ∈ C. [7]
please turn over
SECTION A continued
A2. (a) Find the values of x for which
x + 1 − 1 x + 3
x
2 x − 2 − 4
= 0. [10]
(b) (i) Using row reduction, find the inverse of the matrix A, where
A =
. [12]
(ii) Multiply your result by A to show that the matrix you have found is A
− 1
. [3]
(iii) Hence solve the set of simultaneous equations
2 x 1 + 2 x 2 − x 3 = − 2 ,
x 1 + 4 x 2 + x 3 = 5 ,
x 1 − 2 x 2 + 4 x 3 = 5.
[5]
please turn over
SECTION B Probability
B1. (a) Two identical 6-sided dice are rolled. Find the probability that their sum is 7.
What is the probability that both dice show the same score? [6]
(b) Which of the following are identically true?
(i) A \ B = B \ A
(ii) A = (A ∩ B) ∪ (A ∩ B
c )
(iii) (A ∪ B) ∩ C = A ∪ (B ∩ C) [3]
(c) (i) State the three axioms of probability. [3]
(ii) Use them to show that if A ⊆ B then P (A) ≤ P (B). [3]
(d) (i) A study into the effect of tobacco on developing lung cancer monitors 600 non-
smokers and 400 smokers. Over the duration of the study one in 75 non-smokers
and one in ten smokers develop lung cancer. Find the probability that a subject
selected at random will develop lung cancer. [3]
(ii) What is the probability of a subject being a smoker given that they develop lung
cancer? [3]
(e) An aircraft has n engines, each of which will, independently, function with probabilty θ.
The aircraft will be able to operate effectively if at least half of its engines function.
(i) What is the probability that exactly 3 engines of a 5-engine aircraft will work? [3]
(ii) Write down the probability that a 5-engine aircraft will be effective. Write down
the same for a 3-engine aircraft. [6]
please turn over
SECTION B continued
B2. (a) A discrete random variable^ R^ has probability mass function (pmf) given by
p(r) =
1 −θ 4
r = − 3
1 − 3 θ 4
r = − 1
1+3θ 4 r^ = 1 1+θ 4
r = 3.
(i) For which values of θ is this a valid pmf? [3]
(ii) Find E(R) and Var(R). [6]
(b) Andrew and Barry take turns kicking a football at a goal until one of them scores at
which point the game stops. Each time Andrew shoots he has a
1 3 chance of scoring and
each time Barry shoots he has a
2 3
chance of scoring. Andrew shoots first.
Find the probability that
(i) Andrew wins on the third kick. [3]
(ii) Andrew wins on the (2n + 1)
th kick. [3]
(iii) Andrew is the eventual winner. [6]
(c) Suppose that X and Y are independent random variables with X ∼ Poisson(μ) and
Y ∼ Poisson(λ). Find P (X + Y = 0), P (X + Y = 1) and P (X + Y = 2). [9]
B3. The lifetime in years of an electronic tube is a random variable X, having a probability
density function (pdf) given by
fX (x) =
xe
−x for x ≥ 0
0 otherwise.
(a) Show that P(X > t) = (1 + t)e
−t for all t ≥ 0. [6]
(b) Find P(X > 5) and P(X > 10 |X > 5). [6]
(c) Find E(X) and Var(X). [12]
(d) A competitor produces a tube whose lifetime in years is a random variable Y , having a
pdf given by
fY (y) =
1 2 e
− 12 y y ≥ 0
0 otherwise.
Which tube is more reliable? [6]
end of exam
