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

#### Paper code: 135031503B.Sc. (Computer Science) (Part 1)Examination, 2018Paper No. 1.3NUMBER 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 fundamental theorem of arithmetic.

(b) State abd prove Gauss theorem.

(c) Discuss Euclidean algorithm.

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 $2x^{2} + 3y^{2} = 1$.

(c) The foci of a hyperbola are the points $(\pm 5, 0)$. Find the equation of the curve if e = 5/4.

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 moduli.

(c) Find the square root of 2 + 3i.
4. (a) State and prove De-Moiver’s theorem.

(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 prme numbers and a is any integer, then show that ap $\cong$ a (modulo p).

(c) State and prove Wilson’s theorem.

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