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Linear Algebra Question Paper
Linear Algebra
Course:Bachelor Of Science In Information Technology
Institution: Masinde Muliro University Of Science And Technology question papers
Exam Year:2009
EXAMINATION FOR THE DEGREE IN INFORMATION
TECHNOLOGY
LINEAR ALGEBRA
DATE: NOVEMBER 2009 TIME: 1½HOURS
INSTRUCTIONS: Answer any THREE questions
QUESTION ONE (20 Marks)
a) List the elements of each of the following sets,
(i) {x : x is an integer and -3 < x < 4}
(ii) {x : x is an integer and (3x - 1)(x + 2) = 0} (5 Marks)
b) Use the notation {x : P(x)}, where P(x) is a propositional function, to describe each of the
following sets.
(i) {1, 2, 3, 4, 5}.
(ii) {3, 6, 9, 12, 15, . . . , 27, 30}. (4 Marks)
c) State whether each of the following statements is true or false.
(i) 2 ? {1, 2, 3, 4, 5}
(ii) {2} ? {1, 2, 3, 4, 5}
(iii) 2 ? {1, 2, 3, 4, 5}
(iv) {2} ? {1, 2, 3, 4, 5}
(v) {1, 2, 3, 4, 5} = {5, 4, 3, 2, 1}. (5 Marks)
d) Test the validity of the following argument.
2
“ If you are eligible for admission then you must be under 25 and if you are
not under 25 then you do not qualify for a scholarship. Therefore if you
qualify for a scholarship, you are eligible for admission.” (6 Marks)
QUESTION TWO (20 Marks)
a) Let f (n) = n2 + 3 and g(n) = 5n -11 for n? N . Thus f : N ? N and g(n) : N ? Z .
Calculate
(i) f (3) (1 Mark)
(ii) g(3) (1 Mark)
(iii) g o f (4) (3 Marks)
(iv) f -1 (n) and state its domain (3 Marks)
b) Construct the truth table for [( p ? q) ? r)]?( p ? ¬q) (5 Marks)
c) Use inverse method to solve the following system of linear equations
8
2 3
4 2 4
+ - =
+ - =
- - + =
x y z
x y z
x y z
(7 Marks)
QUESTION THREE (20 Marks)
a) Show that p ? q = ¬( p ? q) (3 Marks)
b) Let U = {1,2,3,4,5,6,7,8,9,10,11,12} , A = {1,3,5,7,9,11}, B = {2,3,5,7,11} , C = {2,3,6,12} and
D = {2,4,8} .
(i) Determine the following sets and draw their venn diagrams
(I) (A? B)nC (3 Marks)
(II) A \ B (2 Marks)
(III) B ? D (2 Marks)
(ii) How many subsets of C are there? (2 Marks)
c) Find the coefficient of x9 in (2 - x)19 . (4 Marks)
3
d) Let P(x) be the statement “x spends more than five hours every weekday in class,” where the
universe of discourse for x is the set of students. Express each of the following quantification in
English.
(i) ?xP(x)
(ii) ?xP(x)
(iii) ?x¬P(x)
(iv) ?x¬P(x) (4 Marks)
QUESTION FOUR (20 Marks)
a) Evaluate the following
(i) P(7,3) (2 Mark)
(ii) C(12,8) (2 Mark)
(iii) C(n,n - 2) (3 Marks)
b) An IT class consists of 35 male students and 43 female students. A group of five students are to
be selected as class representatives. Find:
(i) The number of ways of choosing the representatives (2 Marks)
(ii) The number of ways of choosing the representatives if it is to have 2 males and 3
females. (2 Marks)
(iii) The probability of choosing 2 males and 3 females. (3 Marks)
c) Use Cramers’ rule to solve
2 1
2 3
2
+ + =
+ + =
+ + =
x y z
x y z
x y z
(6 Marks)
QUESTION FIVE (20 Marks)
a) Define the following terms:
(i) Tautology
(ii) Logical implication
(iii) Function
(iv) Cartesian products (4 Marks)
b) Let S = {0,1,2,3,4} andT = {0,2,4}.
(i) How many ordered pairs are in SxT?
(ii) List the elements in the set {(m,n)?S ×T;m < n} (4 Marks)
c) Use mathematical induction to prove that
3 3.5 3.5 ... 3.5 3(5 1)
1
2 -
+ + + + =
n+
n for n=0, 1,
2… (6 Marks)
d) A survey of 400 people in a small town in Kenya provided the following results. Among
these 400 people: 38 drink coffee, 48 drink tea, 39 drink mineral water, 15 drink both
coffee and tea, 13 drink both coffee and mineral water, 9 drink both tea and mineral water,
and 5 drink all three. Using a venn diagram, find how many of the people surveyed drink
(i) Mineral water but neither coffee nor tea
(ii) Tea and coffee but not mineral water
(iii) None of the three. (6 Marks)
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