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

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 8¹⁰³ is divided by 103.

    (b) State and prove Wilson’s theorem.

    (c) Discuss basic property of congruence.

2. (a) If the co-ordinates of one extremity of a focal chord are , find the co-ordinates of other ectremity of focal chord on parabola .

    (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.

    (c) Find the eccentricity and co-ordinates of foci of the hyperbola .

3. (a) If , prove that :

    (b) Find all the values .

    (c) Express in the form and derive all the values of .
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)

         (ii)

    (c) Show that :

5. (a) Solve the equation and find which of its roots satisfy the equation .

    (b) Discuss division algorithm with example.

    (c) State and prove Gauss theorem.

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