Number Theory, Complex Variables and 2-D 2024 – BSc Computer Science Part 1

Paper code: 13503
1503
B.Sc. (Computer Science) (Part 1)
Examination, 2024
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) (i) Express 1 + √(-3)  in the form r(cosθ + isinθ).

        (ii) Express 1 + i in polar form.

    (b) Find real numbers A and B, if A + iB = (3-2i)/(7+4i).

    (c) Prove that the sum and product of two complex numbers are real iff they are conjugate to each other.

2. (a) State and prove Euler’s formula.

    (b) Prove that for two non-zero complex numbers, and arg(z₁z₂) = arg z₁ + arg z₂.

    (c) Show that if the equation z²+αz+β=0 has a pair of conjugate complex roots, then α, β are both real and α² < 4β.

3. (a) State and prove Fermat’s theorem.

    (b) Using Wilson’s theorem, determine 40! mod 43.

    (c) State and prove fundamental theorem of Arithmetic.

4. (a) Find the Vertex, focus, directrix, axis

 and latus rectum of the parabola y² – 4x – 4y = 0.

    (b) The co-ordinates of the vertices of a hyperbola are (9, 2) and (1, 2) and the distance between its two foci is 10. Find its equation and also the length of its latus rectum.

    (c) Determine the equation of the ellipse whose directrices along y = ±9 and foci at (0, ±4). Also find the length of its latus rectum.

5. (a) Find the module and arguments of the following complex numbers :

    (i) 

    (ii)

    (b) State and prove the triangle inequality for complex numbers.

    (c) if z = (cosθ + isinθ), then using De Moivre’s theorem, show that –

and

…………End…………

Thank you: Gourish Rajput (VCB) 🙂