Get premium membership and access revision papers, questions with answers as well as video lessons.

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:2011

QUESTION ONE - (30 MARKS)
a) i) Define a primitive root of a prime number P. (2 Marks) ii) Show that 3 is a primitive root of 5 but 4 is not. (4 Marks) b) Show that 212??+?? = 2?? ???? 13 ,where m and n are any positive integers. (5 Marks) c) Find the order of 2 modulo 17. (3 Marks) d) Show that if the HCF of two integers a and b exists, then it must be unique. (4 Marks) e) Prove that if ?? ?? ,?? ?? and ??,?? = 1 then ????|??. (4 Marks) f) Define the trace of an element as used in number field. (2 Marks) g) i) State the Euclids fomular for generating a pythagoran triple. (3 Marks) ii se the uclid’s forular to generate the ythagorean trile using the air of integers 3 and 4. (3 Marks)
QUESTION TWO (20 MARKS)
a) Compute the Phi-function 540 (4 Marks) b) Determine the order of 2 ???? 13 . (4 Marks) c) Given that the order of ?? ???? ?? is k, prove that ?? = 1 ???? ?? if and only if ?? . (7 Marks) d) Determine the least non-negative residue x such that 668 = ??(???? 17). (5 Marks)
2
QUESTION THREE (20 MARKS)
a) Define the following terms as used in Rings of integers: i. A zero divisor (3 Marks) ii. An integral domain (2 Marks) iii. An invertible element (3 Marks) b) Show that the set ???? 2 × 2 matrices over integers form a non-commutative ring with unity under matrix addition and multiplication. (6 Marks) c) Show that the ring ?? = {0,1,2,3,4,5} under addition and multiplication modulo 6 is commutative but not an integral domain. (6 Marks)
QUESTION FOUR (20 MARKS)
a) i) Define the highest common factor (HCF) of two numbers. (1 Mark) ii) Show that if the HCF of two integers ?? and ?? exists, then it must be unique. (5 Marks)
b) i) Define the term co-prime as used in the set of integers. (1 Mark) ii) Show that two integers ?? and ?? are co-prime if and only if there exist ??,?? such that ???? - ???? = 1 c) Let a, b and c be non-zero integers. Show that: i. If ?? ?? ?????? ?? ?? ,???? ?? ??. (4 Marks) ii. If ?? ?? ?????? ?? ??, ???? ?? = ±??. (3 Marks

Sma2113:Maths For Science Question Paper

More Question Papers

Popular Exams