Thursday, August 6, 2020

NUMBER THEORY, COMPLEX VARIABLES AND 2D - 2016 | B.SC CS 1ST YEAR | MJPRU | EXAM PAPER | My CS Tutorial

My CS Tutorial August 06, 2020

Number theory,complex variable | My CS Tutorial

Paper Code: 13503
1503
B.Sc. (Computer Science) (Part 1)
Examination, 2016
Paper No. 1.3
NUMBER THEORY, COMPLEX VARIABLES AND 2D

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) Find the latus rectum, vertex, focus and axis of the parabola:

$y^2-2y+8x-23=0$

(b) Find the equation to the conic section whose focus is (1, -1), eccentricity is and the directrix is the line $x-y=3$ .

(c) Find the points common to the hyperbola $4x^2-4y^2=15$ and the straight line $x-2y-3=0$ . Find also the lengths of the straight line intercepted by the hyperbola.

2. (a) Find four fourth roots of unity.

(b) State and prove De-Moivers theorem.

(c) If $\bar{a}=2\hat{i}-3\hat{j}+5\hat{k}$ and $\bar{a}=3\hat{i}-\hat{j}-2\hat{k}$ . Evaluate $(\bar{a}+\bar{b}).(\bar{a}-\bar{b})$ .

3. (a) Prove that the representative of the complex numbers $1+4i,2+7i,3+10i$ are collinear.

(b) If $\bar{r}=x\hat{i}-y\hat{j}-z\hat{k}$ , then show that:

$\left (\bar{r}\times \hat{i} \right )^{2}=y^{2}+z^{2}$
$\left (\bar{r}\times \hat{i} \right ).\left (\bar{r}\times \hat{i} \right )+xy=0$

4. (a) What do you know about the Euclidean algorithm?

(b) State and prove Gauss theorem.

5. (a) Discuss special divisibility test.

(b) State and prove Farmat’s theorem.

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