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 :
![\left | z_{1}+z_{2} \right |^{2}+ \left | z_{1}-z_{2} \right |^{2} = 2\left [ \left | z_{1} \right |^{2}+\left | z_{2} \right |^{2} \right ]](https://s0.wp.com/latex.php?latex=%5Cleft+%7C+z_%7B1%7D%2Bz_%7B2%7D+%5Cright+%7C%5E%7B2%7D%2B+++++++%5Cleft+%7C+z_%7B1%7D-z_%7B2%7D+%5Cright+%7C%5E%7B2%7D+%3D+2%5Cleft+%5B+%5Cleft+%7C+z_%7B1%7D+%5Cright+++++++%7C%5E%7B2%7D%2B%5Cleft+%7C+z_%7B2%7D+%5Cright+%7C%5E%7B2%7D+%5Cright+%5D&bg=ffffff&fg=000&s=0&c=20201002)
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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