Number Theory, Complex Variables and 2-D 2017 – BSc Computer Science Part 1

March 30, 2019 Lokesh Kumar 129 Views 0 Comments 1st Year, BSc, Computer Science, Exam paper

Paper code: 13503
1503
B.Sc. (Computer Science) (Part 1)
Examination, 2017
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) Use Fermat’s theorem to determine the remainder, if 8¹⁰³is 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.