Number Theory, Complex Variables and 2-D 2022 – BSc Computer Science Part 1

Paper code: 13503
1503
B.Sc. (Computer Science) (Part 1)
Examination, 2022
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) Show that if two integers a and b are relatively prime i.e. if (a,b) = 1 then a/bc ? a/c.

    (b) Show that ‘The relation of divisibility in the set of integers is reflective, transitive but not symmetric.

    (c) If P is a prime and a,b are any integers, then P/ab ? P/a or P/b.

2. (a) If a and b are two integers then a ? b (mod m), if and only if a and b have the same remainder when divided by n.

    (b) Define linear congruence solve 3x ? 4 (mode 5).

    (c) If a ? b (mod m) then for all x ? z

a + x ? b + x (mod m)

ax ? bx (mod m)

3. (a) Find the cube root of unity.

    (b) Solve the equation x³ + 8 = 0 :

    (c) Find the mod z and amp z where :

4. (a) Prove that :

    (b) Show that :

where a and b are complex numbers.

    (c) Express z = -1 -i into polar form :

5. (a) For ellipse :

find its focus, directrix and latus rectum.

    (b) Write equation of two asymptotes of hyperbola :

    (c) For hyperbola x² = 16y find its vertex, focus, latus rectum and length of latus rectum.

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