Number Theory, Complex Variables and 2-D 2015 – BSc Computer Science Part 1

Note: Attempt all the five questions. All questions carry equal marks. Symbol used are as usual.

1. Solve any two of the following parts:

It the co-ordinates of one extremity of a focal chord of a parabola $y^{2}= 4ax$ are $(at^{2}, 2at)$ , find the co-ordinates of the other extremity.
Find the equation of ellipse whose foci are at the point $s(2,0)$ and ${s}'(-2,0)$ and whose let us rectum is 6.
Find the centre, foci, let us return and equation of directix of the hyperbola:

$x^{2} - y^{2} + 2x + 4y = 4$

2. Attempt any two of the following parts:

$(1+ \cos \theta +i\sin \theta)^{n}+(1+ \cos \theta -i\sin \theta)^{n}=2^{n+1}.\left (\cos \frac {\theta}{2} \right )^{n}.\cos \frac {n\theta}{2}$

3. Attempt any two of the following parts:

$|z_{1}|-|z_{2}|\leq |z_{1}- z_{2}|$ and $|z_{1}|+|z_{2}|\geq |z_{1}+ z_{2}|$
Represent on complex plane the complex number $w_{1}=3+4i$ and $w_{2}=6-3i$ together with $w_{1}+w_{2}$ and $w_{1}-w_{2}$ .
Determine the conjugate and the reciprocal of $\sqrt{-3}-3$ .

4. Attempt any two of the following parts:

$(a^{m}-1)-(a^{n}-1)=(a^{(m,n)}-1)$ , where (m,n) = g.c.d(m,n).

5. Attempt any two of the following parts:

State and prove fundamental theorem of arithmetic.
The liner conquence $ax=\beta(\mod n)$ is solved if $d\mid \beta$ where $d=(a,n)$ if $d\mid \beta$ then it has d in congruent solutions.
Discuss the Fermat’s factorization method with an example.