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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: 2009/2010
FIRST YEAR EXAMINATION FOR THE DEGREE OF BACHELOR OF
SCIENCE IN INFORMATION TECHNOLOGY
BIT 1101: LINEAR ALGEBRA
DATE: DECEMBER 2009 TIME: 2 HOURS
INSTRUCTIONS: Answer question ONE and any other TWO questions
QUESTION ONE (30Marks)
a) Draw a Venn diagram and shade the region corresponding to the indicated set.
(A - B) n (A - C) (3 Marks)
b) Prove by induction that 4
1 2 3 ..... ( 1)
2 2
3 3 3 3 +
+ + + n = n n
(5 Marks)
c) In a survey of 260 computer science students, the following data were obtained:94 like to work in
USA,64 like to work in UK,58 like to work in Singapore,28 like UK and Singapore,26 USA and
UK,22 USA and Singapore and 14 like all the three places. Find
(i) How many students like none of the three countries?
(ii) How many students like at least one country?
(iii) How many students like only one country (8 Marks)
d) Determine whether the following is a tautology, a contradiction or neither:
{(p ? q)?(q ? r)}?{(p ? q) ? r} (6 Marks)
e) Let A ={1,2,3,4}.Determine whether the following relations are reflexive, symmetric, antisymetric
or transitive.
2
{(1,1),(1,2),(2,1),(2,2),(3,3),(3,4),(4,3), (4,4)} 1 R =
{(1,1),(2,2),(3,3)} 2 R =
{(1,1),(1,3),(3,1),(1,2),(3,3),(4,4)} 1 R =
(8 Marks)
QUESTION TWO (20 MARKS)
a) Given that f (x) = 2x +1 and ( )
3
g x = x .Show that ( )-1 -1 -1 gof = f o g (4 Marks)
b) Let A ={a,b, c, d} and B ={1,2,3}.Determine which of the following relations from A to B
are functions. If it is a function specify its range and type.
i) f ={(a,1),(b,2),(c,1),(d,2)}
ii) f ={(a,1),(b,2),(a,2),(c,1),(d,2)}
iii) f ={(a,3),(b,2),(c,1)}
iv) f ={(a,1),(b,1),(c,1),(d,1)}
v) f ={(a,1),(b,2),(c,3),(d,1)} (10 Marks)
c) Let A ={1,2,3,4}, R ={(1,1),(1,2),(2,3),(2,4),(3,4), (4,1),(4,2)} and
S ={(3,1),(4,4),(2,3),(2,4),(1,1), (1,4)}.Find SoR and RoR. (6 Marks)
QUESTION THREE (20MARKS)
a) Consider the propositions:
p : Mary laughs.
q : Sally cries.
r : Jo shouts.
3
Write in words the following compound propositions:
(i) (r ? q)? p
(ii) (r ? ¬q)?(r ? q)
(iii) p ?(¬q ? ¬r) (6Marks)
b) Test the validity of the following argument:
‘If you insulted Bob then I’ll never speak to you again. You insulted Bob so I’ll never
speak to you again.’ (6Marks)
c) Use inverse method to solve the systems of equations
x + 2y - 4z = 4
x + 3y - 6z = 7
2x + 3y - 5z = 9 (8Marks)
QUESTION FOUR (20 MARKS)
a) Define the following as used in set algebra
i. Universal set
ii. Subset
iii. Venn diagram
iv. Finite set
v. Power set (5 Marks)
b) Given that B ={a,b, c, d}.find the power set of B. (5 Marks)
c) i. State the principle of duality in set theory.
ii. Find the dual of the following set.
( ) ( )B A ? B n S = A ? F ? (5 Marks)
d) If repetitions are not permited, how many four digit numbers less than 5000 can be
formed from digits 1,2,3,7,8 and 5. (5Marks)
4
QUESTION FIVE (20Marks)
a) Solve the following system of equations
x + 2y + z = 3
2x +5y - z = -4
3x -2y -z = 5
Using: i. Cramers rule
ii. Elimination method (10Marks)
b) Let A ={1,2,3,4,5}.Determine the truth value of each of the following statements and justify your
answer in each case
i. (?x ? A)(x + 3 = 10)
ii (?x ? A)(x + 3 < 10)
iii(?x ? A)(x + 3 < 5)
iv(?x ? A)(x + 3 = 7) (10Marks)

Linear Algebra Question Paper

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