Saturday, August 1, 2020

NUMBER THEORY, COMPLEX VARIABLE AND 2D | B.SC CS 1ST YEAR | MJPRU | EXAM PAPER | My CS Tutorial

My CS Tutorial August 01, 2020

Number theory,complex variable | My CS Tutorial

Paper Code: 13503
1503
B.Sc. (Computer Science) (Part 1)
Examination, 2017
Paper No. 1.3
NUMBER THEORY, COMPLEX VARIABLES AND 2D

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) Use Fermat’s theorem to determine the remainder, if 8103 is divided by 103.

(b) State and prove Wilson’s theorem.

2. (a) If the co-ordinates of one extremity of a focal chord are $(at_{1}^{2}, 2at_{1})$ , find the co-ordinates of other ectremity of focal chord on parabola $y^{2}=4ax$ .

(b) Find the euqation of the ellipse whose foci are at the points S(2,0) and S'(-2,0) and whose latus rectum is 6.

3. (a) If $x_{r} = \cos\left ( \frac{\pi}{2^{r}} \right )+ i \sin \left ( \frac{\pi}{2^{r}} \right )$ , prove that :

$x_{1}x_{2}x_{3}.....ad inf. = -1$

(b) Find all the values $(8i)^{\frac{1}{3}}$ .

(c) Express $p={\frac{(\sqrt{3}-1)+i(\sqrt{3}+1)}{2\sqrt{2}}}$ in the form $r(\cos\theta + i\sin\theta)$ and derive all the values of $p^\frac{1}{4}$ .
4. (a) Show that the modulus of the product of complex numbers is equal to the product of moduli of those numbers is equal to the sum of those numbers.

(b) Find the moduli and arguments of the following complex numbers :

(i) $\frac{1-i}{1+i}$

(ii) $\frac{1+2i}{1-(1-i)^{2}}$

$\left | z_{1}+z_{2} \right |\leq \left | z_{1} \right |+ \left | z_{2} \right |$

5. (a) Solve the equation $x^{12}-1=0$ and find which of its roots satisfy the equation $x^{4}+x^{2}+1=0$ .

(b) Discuss division algorithm with example.

_______________________________________

Please share this post and blog link with your friends.For more programs use this blog.

If you have any problem, please comment in comment box, subscribe this blog for notifications of new post on your email and follow this blog.If you have any method of this tutorial or program or want to give any suggestion send email on hc78326@gmail.com

Created by-- HARSH CHAUHAN

About My CS Tutorial
SoraTemplates is a blogger resources site is a provider of high quality blogger template with premium looking layout and robust design. The main mission of SoraTemplates is to provide the best quality blogger templates.