Number Theory, Complex Variables and 2-D 2018 – BSc Computer Science Part 1

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 fundamental theorem of arithmetic.

(b) State and prove Gauss theorem.

2. (a) Find the equation of the parabola with latus rectum joining the points (3, 5) and (-3, 3).

(b) Find the eccentricity, the co-ordinate of the foci, the length of latus rectum of the ellipse $2x^{2} + 3y^{2} = 1$ .

3. (a) Express $\left ( \frac{1-i}{1+i} \right )^{2}$ in the form A + iB.

(b) Show that the modulus of the product of two complex numbers is the product of their module.

(b) Find $w^{28} + w^{29} + 1 = 0$ .

(c) if $z = \frac{-1 + i\sqrt{3}}{2}$ and $w = \frac{-1 - i\sqrt{3}}{2}$ , represent z and w accurately on complex plane.

5. (a) State and prove fundamental theorem of congruence relation.

(b) If p is a prime numbers and a is any integer, then show that a^p $\cong$ a (modulo p).

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