Number Theory, Complex Variables and 2-D 2023 – BSc Computer Science Part 1

Paper code: 13503
1503
B.Sc. (Computer Science) (Part 1)
Examination, 2023
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) If a = qb + r Then show that g.c.d. of a and b is the same as the g.c.d. of b and r.

    (b) If p is a prime number and p/ab then show that p/a or p/b.

    (c) Show that ‘the linear Diophantine equation ax + by = c has a solution if and only if the greatest common divisor of a and b divides C.

2. (a) List all integers x in the range 1 ≤ x ≤ 100 that satisfy x ≡ 7 (mod 17).

    (b) Solve 7x ≡ 4 (mod 10).

    (c) Discuss Fermat’s theorem or Wilson’s theorem.

3. (a) Express below in standard form x+iy.

    (b) Show that :

    (c) Show that :

4. (a) Simplify :

    (b) Find complex cube root of unity.

    (c) If :

Prove that:

5. (a) Find coordinates of focus, equation of directrix and length of latus rectum for the parabola y² = 5x.

    (b) Find equation of ellipse (referred to its center) whose latus rectum in 5 and eccentricity is 2/3.

    (c) For hyperbola 4x² – 9y² = 36 find eccentricity and length of latus rectum.

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