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.
1
1
1
 1


 ......  log 1   .
3
n  1 2(n  1) 3(n  1)
 n
01.Show that
1 4 5


02. Find the rank of the matrix  2 5 7  .
3 6 9


03. Form a rational cubic equation, whose roots are 1, 3   2 , 3 +  2 .
04. If sin (A+iB) = x + iy, prove that
x2
y2

 1.
sin 2 A
cos 2 A
05. State Euler’s theorem for a homogenous function f(x,y,z) of degree ‘n’.
06. Examine the function f(x,y) = 1+ x2y2 for maxima and minima.
07. Evaluate
08. Find
x
x e
3
x
2
dx
.
 2x  5
dx .
09. Solve q = 2yp2.
10. 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
7 7.9
7.9.11


 ....
9 9.12 9.12.15
(OR)
1 0 3 


Verify Cayley Hamilton theorem for the matrix  2 1  1 .
1 1 1 


12. 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 sin3 cos5 inseries of sines of multiples of ‘’.
 3a 3a 
13. Find the radius of curvature at  ,  to the curve x3 + y3 =3axy.
 2 2 
(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).
1
14. By changing order of integration, evaluate
1 x
  (x
2
 y 2 )dx dy .
x 0 y 0
(OR)
Integrate
15. Solve
5x  1
with respect to ‘x’.
x  2 x  48
2
p  q  1.
(OR)
Find the solution of (D23D +2)y = sin 3x.
PART  C
(2  20 = 40 Marks)
Answer any TWO questions
 3 10 5 


16. (a) Find the eigen values and eigen vectors of the matrix   2  3  4  .
 3
5
7 

(b) If tan log (x + iy) = a + ib, where a2 + b2 1, show that
tan log (x2 + y2) =
17.
2a
.
1 a2  b2
(a) Solve the reciprocal equation 6x635x5 + 56x456x2 35x6 = 0
(b) Investigate the maximum and minimum values of
4x2 + 6xy + 9y2  8x 24y + 4.
18. (a) Solve p tan x + q tan y = tan z.
(b) Evaluate  xy dx dy taken over the positive quadrant of the circle
x2 + y2 = a2 .
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