# Number Theory, Complex Variables and 2-D 2016 – BSc Computer Science Part 1

#### Paper code: 13503 1503 B.Sc. (Computer Science) (Part 1) Examination, 2016 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) 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:

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

(c) Show that the modulus of the product of two complex numbers is the product of their moduli.

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

(b) State and prove Gauss theorem.

(c) Find the set of relatively prime integers out of first twenty natural numbers.

5. (a) Discuss special divisibility test.

(b) State and prove Farmat’s theorem.

(c) What are the basic properties of congruence?

…………End…………

#### Lokesh Kumar

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