Number Theory, Complex Variables and 2-D 2019 – BSc Computer Science Part 1

Paper code: 13503
1503
B.Sc. (Computer Science) (Part 1)
Examination, 2019
Paper No. 1.3
NUMBER THEORY, COMPLEX VARIABLES AND 2-D

Time: Three Hours] [Maximum Marks: 50

Note: Attempt all the five questions. All questions carry equal marks. Symbol used are as usual. Attempt any two parts of each question.

1. (a) State and prove Fermat’s theorem.

    (b) Show that 16 ! + 86 is divisible by 323.

    (c) Discuss Euclidean Algorithm with example.

2. (a) Find the equation of the parabola with vertex (2, -3) and (0, 5).

    (b) Given the ellipse . Find the equations and the lengths of the focal radii drawn through the point .

    (c) Find the eccentricity if the hyperbola whose equation is .

3. (a) Prove that :

    (b) If z₁, z₂, z₃, are the vertices of an equilateral triangle having, its circumference at  z₀.

Prove that:

    (c) Define the nth root of unity.
4. (a) Prove that :

    (b) Show that :

, for all

    (c) Simplify :

         (i)

         (ii)

5. (a) State and prove Fundamental Theorem of Arithmetic.

    (b) Discuss relative prime integer with example.

    (c) Find the G. C. D. of 275 and 200 and express it in the form m.275 + n.200.

…………End…………

Thank You !