LOYOLA COLLEGE (Autonomous), chennai – 600 034

# B.Sc. degree examination – computer science

third semester -april 2003

## cs 3100/ Csc 100 applicable mathematics

## 07.04.2003 Max.: 100 Marks

## 9.00 – 12.00

**PART ****–**** A** (10 ´ 2 = 20 Marks)

*Answer ALL the questions*.

01.Show that .

- Find the rank of the matrix .
- Form a rational cubic equation, whose roots are 1, 3 -, 3 +.
- If sin (A+iB) = x + iy, prove that .
- State Euler’s theorem for a homogenous function f(x,y,z) of degree ‘n’.
- Examine the function f(x,y) = 1+ x
^{2}-y^{2}for maxima and minima. - Evaluate .
- Find .
- Solve q = 2yp
^{2}. - Find the solution of (D
^{2 }+ 2D +1) y = 0

**PART ****–**** B** (5 ´ 8 = 40 Marks)

*Answer ALL the questions*.

# 11. Find the sum to infinity of the series

(OR)

Verify Cayley -Hamilton theorem for the matrix .

- Find by Horner’s method the root of the equation x
^{3}-3x +1 = 0 which lies between 1 and 2, up to two decimal places.

(OR)

Expand sin^{3}q cos^{5}q inseries of sines of multiples of ‘q’.

- Find the radius of curvature at to the curve x
^{3 }+ y^{3 }=3axy.

(OR)

Using Lagrange’s multiplier method, find the minimum of the function

u = xyz subject to xy + yz +zx = a (x >0, y>0, z >0).

- By changing order of integration, evaluate.

(OR)

Integrate with respect to ‘x’.

- Solve .

(OR)

Find the solution of (D^{2}-3D +2)y = sin 3x.

**PART ****–**** C ** (2 ´ 20 = 40 Marks)

*Answer any two questions*

- (a) Find the eigen values and eigen vectors of the matrix.

- If tan log (x + iy) = a + ib, where a
^{2 }+ b^{2 }¹1, show that

tan log (x^{2 }+ y^{2}) =.

- (a) Solve the reciprocal equation 6x
^{6}-35x^{5 }+ 56x^{4}-56x^{2}35x-6 = 0

- Investigate the maximum and minimum values of

4x^{2 }+ 6xy + 9y^{2 }– 8x -24y + 4.

- (a) Solve p tan x + q tan y = tan z.

- Evaluateover the positive quadrant of the circle

x^{2 }+ y^{2} = a^{2} .

**Latest Govt Job & Exam Updates:**