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Linear Algebra Question Paper
Linear Algebra
Course:Bachelor Of Science In Information Technology
Institution: Kca University question papers
Exam Year:2011
UNIVERSITY EXAMINATIONS: 2010/2011
FIRST YEAR EXAMINATION FOR THE DEGREE OF BACHELOR OF
SCIENCE IN INFORMATION TECHNOLOGY
BIT 1101: LINEAR ALGEBRA
DATE: APRIL 2011 TIME: 2 HOURS
INSTRUCTIONS: Answer question ONE and any other TWO questions
QUESTION ONE (30 Marks)
a) Define the following terms as used in linear algebra
i. A set
ii. Relation
iii. A function
iv. Power set (4 Marks)
b) In a class of 80 students, 50 students know English, 55 know French and 46 know German
language.37 students know English and French, 28 know French and German and 25 students
know English and German.7 students know none of the languages. Find out
i. How many students know all the three languages (4 Marks)
ii. How many students know exactly 2 languages (1 Mark)
iii. How many students know only one language (1 Mark)
c) Given that
? ? ?
?
?
? ? ?
?
?
-
- -
-
=
1 1 1
4 5 3
7 8 5
A . Find A-1 (6 Marks)
d) Determine whether the following is a tautology, a contradiction or fallacy:
[(p ? q)?(p ? r)?(q ? r)]? r (4 Marks)
2
e) Given that A ={1,2,3,4}. Consider the following relation on A:
R ={(1,1),(2,2),(2,3),(3,2),(4,2),(4,4)}
i. Draw its directed graph (1 Mark)
ii. List the elements of RC (3 Marks)
f) Let f (x) = 2x +1 and g(x) = x2 - 2.Find go f and (gof )-1 (6 Marks)
QUESTION TWO (20 MARKS)
a) Let R be the relation on the positive integers defined by the equation x +3y =12
.that is, R ={(x, y): x +3y =12}
i. list the elements of R (2 Marks)
ii. Find the domain of R (2 Marks)
iii. Find the range of R (2 Marks)
iv. Find the elements RoR (3 Marks)
b) Shade the following sets on a Venn-diagram
i. Ac n(C - B) (3 Marks)
ii.
A? (B -C)C (3 Marks)
c) Prove by induction that 1+ 3+ 5 + ................+ (2n -1) = n2 (5 Marks)
QUESTION THREE (20 MARKS)
a) Write the contra positive, converse and inverse of the following statement
“If today is Easter, then tomorrow is Monday”
b) Given the following relations
f ={(1, a),(2, a),(3, c),(4, b)}
g ={(1, c),(2, a),(3, b),(4, c), (2, d )}
h ={(1, c),(2, d),(3, a),(4, b)}
3
w ={(1, d),(2, d),(4, a)}
m ={(1, b),(2, b),(3, b),(4, b)}
In each case
i. Determine whether each relation given below is a function from A ={1,2,3,4}to B = {a, b, c, d}
(5 Marks)
ii. If it is a function, find its domain and range (6 Marks)
iii. Determine if it is one-one or onto or neither. if neither specify the type (3 Marks)
iv. If it is both one-one and onto ,list the elements of the inverse function (1 Mark)
v. State the domain and range of the inverse function (2 Marks)
QUESTION FOUR (20 MARKS)
a) Let A ={1,2,3}, B ={a, b, c}and C = {x, y, z}.Find the consider the following relations R and S from
A to B and from B to C,respectively.
R ={(1, b),(2, a),(2, c)} and S = {(a, y), (b, x), (c, y), (c, z)}
i. List the elements of A× B×C (3 Marks)
ii. Find the composition RoS (3 Marks)
iii. Find the matrices R S M ,M and RoS M (3 Marks)
iv. Find the product R S M M .compare product R S M M to RoS M (3 Marks)
b) Let P(x) be the statement “ x = x 2 ”.if the universe of discourse is the set of integers, what is the
truth values of the following?
i. P(0)
ii. P(1)
iii. P(2)
iv. P(-1)
v. ?xP(x)
vi. ?xP(x) (6 Marks)
c) let A ={0,1,2},find the power set of A (2 Marks)
QUESTION FIVE (20Marks)
4
a) Given the following system of equations
0
4 5 3 3
7 8 5 5
- + =
- + - = -
- + =
x y z
x y z
x y z
Solve the above system using
i. Elimination method
ii. Cramer’s rule
iii. Inverse method (15 Marks)
b) Test the validity of the following argument
“If I study, then I will not fail mathematics. If I do not play basketball, then I will study. But I failed
mathematics. Therefore I must have played basketball.” (5 Marks)
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