LOYOLA COLLEGE (AUTONOMOUS), CHENNAI  600 034
B.C.A. DEGREE EXAMINATION
THIRD SEMESTER APRIL 2003
CA 3100/CAP 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


 ......  log 2.
01. Show that
2
1.2
2.2
3.2 3
02. Define characteristic equation and eigen values.
03. Write down the expansion of cos 8 using Demoive’s theorem.
04. If , , one the roots of the equation x3 7x + 6 = 0 find the value of 3 + 3 + 3.
u
xy
u
, show that x
y
 u.
05. If u 
x y
x
y
06. Write the Cartesian formula for radius of curvature.
dx
07. Find  2
.
x  2x  3
08. Evaluate
x
3
cos 2 x dx .
09. Solve pq = x.
10. Find the solution of (D2 6D +13) y = 0.
PART  B
(5  8 = 40 Marks)
Answer ALL the questions
11. Find the inverse of the matrix using Cayley  Hamilton theorem
 1 1 2 


3 
 2 1
 3
2  3 

(OR)
Find the sum to infinity of the series
2.5
2.5.8
2.5.8.11


 ...
6.12
6.12.18 6.12.18.24
12. If the sum of two roots of the equation x4 + px3 + qx2 + rx + s = 0 equals the
sum of the other two prove that p3+8r = 4pq.
(OR)
sin 6
Express
interms of cos .
sin 
1
13. Find the radius of curvature at the point
a 4 , a 4 to the curve to the curve
x y a .
(OR)
Find the maximum and minimum value of the function f (x,y) = x2 y2 x2 y2 .
14. Evaluate  xydx dy taken over the positive quadrant of the circle x2 + y2 = a2
(OR)
5x  1
Integrate 2
with respect to x.
x  2 x  35
15. Solve (y + z)p + (z +x)q = x + y.
(OR)
d 2 y dy
Find the solution of 3 2   14 y  13e 2 x
dx
dx
PART  C
(2  20 = 40 Marks)
Answer any TWO questions
16. a) Find the eigen values and eigen vectors of the matrix
 1 1 3


1 5 1
 3 1 1


b) Sum the series to infinity
1  3 1  3  3 2 1  3  3 2  33


 ......
2!
3!
4!
1
17. a) Show that cos5  sin7    11 [sin 12 2sin 10 4sin 8 +10 sin 6
2
+ 5 sin 4 20sin 2].
1
b) Diminish by 1, the roots of x4  4x3  7x2 + 22x + 24 = 0 and hence solve it.

17. a) By changing the order of integration, evaluate
0 x
e y
dx dy
y
b) Solve (D2 + 4D + 5) y = ex + x3+cos 2x.
* * * * *
2