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Linear Algebra Question Paper
Linear Algebra
Course:Bachelor Of Science In Information Technology
Institution: Kca University question papers
Exam Year:2009
UNIVERSITY EXAMINATIONS: 2008/2009
FIRST YEAR EXAMINATION FOR THE DEGREE OF BACHELOR OF
SCIENCE IN INFORMATION TECHNOLOGY
BIT 1101: LINEAR ALGEBRA
DATE: AUGUST 2009 TIME: 2 HOURS
INSTRUCTIONS: Answer question ONE and any other TWO questions
QUESTION ONE (30 Marks)
a) Define the following terms
i) A relation
ii) Power set
iii) Functions
iv) Conjunction
v) A set (5Marks)
b) State the converse, inverse and contrapositive of the proposition:
‘If it’s not Sunday then the supermarket is open until midnight’ (4Marks)
c) Prove by induction that 1+ 3 + 5 + ..... + (2n -1) = n2 (4Marks)
d) Each of the 100 students in the first year of Utopia University’s Computer
Science Department studies at least one of the subsidiary subjects: mathematics,
electronics and accounting. Given that 65 study mathematics, 45 study
electronics, 42 study accounting, 20 study mathematics and electronics, 25 study
mathematics and accounting, and 15 study electronics and accounting, find the
number who study:
(i) all three subsidiary subjects;
(ii) mathematics and electronics but not accounting;
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(iii) only electronics as a subsidiary subject. (8Marks)
e) Determine whether the following is a tautology, a contradiction or neither:
[p ? (q ? r )] ? [(p ?q) ? (p ?r )] (5Marks)
f) For each of the following, draw a Venn diagram and shade the
Region corresponding to the indicated set.
(i) A - (B n C)
(ii) (A - B) ? (A - C). (4Marks)
QUESTION TWO (20 MARKS)
a) Test the validity of the following argument: ‘If you are a mathematician then you are clever.
You are clever and rich. Therefore if you are rich then you are a mathematician.’ (6Marks)
b) i. state the principle of duality in set theory
ii Write the dual of the following set
A ? (B n S ) = (A ? F )?B (4Marks)
c) Let A = {a, b, c, d, e}. For each of the following relations R on
A, determine which of the four properties (reflexive, symmetric, antisymmetric,
transitive) are satisfied by the relation. Justify your answers.
(i) R = {(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b),(c, c)}.
(ii) R = {(a, a), (b, b), (c, c), (d, d), (e, e), (a, b), (b, c)}.
(iii) R = {(a, a), (a, d), (b, b), (c, c), (d, d), (d, e), (e, a), (e, e)}.
(iv) R = {(a, b), (b, c), (c, d), (d, e), (e, a)}.
(v) R = {(a, b), (b, a), (b, d), (d, a), (c, e), (e, c), (e, e)}.
(10Marks)
QUESTION THREE (20MARKS)
a. Let A = {1, 2, 3, 4} and define two relations R and S on A by:
R = {(1, 3), (2, 2), (3, 1), (3, 4), (4, 2)}
S = {(1, 2), (2, 3), (3, 4), (4, 1)}.
(i) List the elements of the relations S ? R and R ? S.
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(ii) List the elements of the relations R-1 , S -1 , ( )-1 SoR , and ( )-1 RoS .
(iii) List the elements of the relations R-1oS -1 and S -1oR-1 (9Marks)
b. Let the universal set be integers. let P(x) be the statement x = x2 .Determine the truth value of the
following statements.
i. P(0)
ii P(2)
iii ?xP(x)
iv ?xP(x) (6Marks)
c) Use Cramer’s rule to solve the systems of equations
x + 2y - 4z = 4
x + 3y - 6z = 7
2x + 3y - 5z = 9 (5Marks)
QUESTION FOUR (20 MARKS)
a) Let P ={q, r, s,t},Q ={a,b,c, d}and f = {(q, a),(r,b), (s,c), (t,b)}.Find the domain, co-domain and
range of the given function. (3 Marks)
b) Let f and g be functions from {1,2,3,4}to {a,b, c, d} and from{a,b, c, d}to {1,2,3,4} respectively
such that f(1) = d,f(2) = c,f(3) = a,f (4) = b and
g(a) = 2,g(b) =1,g(c) =3g(d) = 2 .
i.is f one to one? Is g one to one?
ii.is f onto? Is g onto?
iii.Does either f of g have an inverse? If so find this inverse. (8 Marks)
c) Let f (x) = x2 - 8 and g(x) = x +12 .Find f o g and
( f o g)-1 (6 Marks)
d) In how many ways 4 examinations can be scheduled within a six day period so that no two
examinations are scheduled on the same day? (3 Marks)
4
QUESTION FIVE (20Marks)
a) Find the inverse of the matrix
A =
? ? ?
?
?
? ? ?
?
?
- -
1 3 3
1 1 1
2 2 1
Hence solve the system of equations
2x + 2y + z = 4
x - y - z = 1
x + 3y + 3z = 1 (8Marks)
b) Consider the following propositions:
p : Mathematicians are generous.
q : Spiders hate algebra.
Write the compound propositions symbolized by:
(i) p ? q (ii) ¬(q ? p)
(iii) p ?q
(iv) p ? q (6 Marks)
c) Suppose A and B are events with P(A)=0.3, P(B)=p and P(A? B) = 0.5. Find p
If
i.Aand B are mutually exclusive events (3Marks)
ii.A and B are independent (3Marks)
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