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+ x2-y2 for maxima and minima.
- Evaluate .
- Find .
- Solve q = 2yp2.
- Find the solution of (D2 + 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 x3-3x +1 = 0 which lies between 1 and 2, up to two decimal places.
(OR)
Expand sin3q cos5q inseries of sines of multiples of ‘q’.
- Find the radius of curvature at to the curve x3 + y3 =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 (D2-3D +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 a2 + b2 ¹1, show that
tan log (x2 + y2) =.
- (a) Solve the reciprocal equation 6x6-35x5 + 56x4-56x2 35x-6 = 0
- Investigate the maximum and minimum values of
4x2 + 6xy + 9y2 – 8x -24y + 4.
- (a) Solve p tan x + q tan y = tan z.
- Evaluateover the positive quadrant of the circle
x2 + y2 = a2 .
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