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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