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Sma2113:Maths For Science Question Paper

Sma2113:Maths For Science

Course:Bachelor Of Computer Science

Institution: Meru University Of Science And Technology question papers

Exam Year:2014

QUESTION ONE - (30 MARKS)
a) (i) Define the term ring as used in the set of integers. (4 Marks) (ii) Show that the set ?? ???? 2 × 2 matrices over integers form a non-commutative ring with unity under matrix addition and multiplication. (5 Marks)
b) Let ?? be a non-zero integer. Show that if ?? ?? ?????? ?? ??, then ??| ???? + ???? ? ??,?? ? ??. (3 Marks) c) (i) Define the highest common factor (HCF) of two numbers. (2 Marks) (ii) Show that if the HCF of two integers ?? and ?? exists the it must be unique. (3 Marks) d) Compute the phi-function Ø 36 (3 Marks) e) Find the order of 5 modulo 6 (2 Marks) f) Determine the least non-negative residue x such that 6100 = ??(?????? 17). (4 Marks) g) Show that 28??+5 = 25 ?????? 17 where m is any positive integer. (4 Marks)
QUESTION TWO – (20 MARKS)
a) (i) Define a linear Diophantine equation in two variables. (1 Mark)
(ii) Find all the solutions of the Diophantine equation 77?? + 42?? = 35 (7 Marks)
2
b) Suppose that we have postage stamps available in two denominations of £5 ?????? £7. What values can one make using combinations of stamps? (5 Marks)
c) Determine the number of solution of the Congruence ??2 = 8783(?????? 15671) (7 Marks)
QUESTION THREE – (20 MARKS)
a) (i) Define a primitive root of a prime number P. (2 Marks) (ii) Show that 3 in a primitive root of 5 but 4 is not. (6 Marks) b) Determine whether 5 is a primitive root of 11 (4 Marks) c) Find the order of 3 modulo 13. (4 Marks) d) Prove that if ?? = ?? (?????? ??) then ?? and ?? have the same order. (4 Marks)
QUESTION FOUR – (20 MARKS)
a) (i) Define an integral domain. (2 Marks) (ii) Show that the ring ?? = {0,1,2,3,4,5,6,7} under addition and multiplication modulo 8 in commutative but is not an integral domain. (7 Marks)
b) Let R be a system satisfying all the ring axioms with possible exception of ?? + ?? = ?? + ??. If ? an element ?? ? ?? such that ???? = ???? ? ?? = ??, ? ??,?? ? ??. Prove that R is a ring. (6 Marks)
c) Let a be a non-zero integer. Show that: i. If ?? ?? and ?? ??,then ??|?? (3 Marks)
ii. If ?? ?? and ?? ??, then ?? == ±?? (2 Marks)

Sma2113:Maths For Science Question Paper

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