Roll No.
B.E. / B.Tech. (Part Time) D E G R E E END S E M E S T E R EXAMINATIONS - APRIL / MAY 2014
(Common to All Branches)
FIRST S E M E S T E R
PTMA130 / PTMA171 Mathematics - I
PTMA9111 A P P L I E D MATHEMATICS
(REGULATION 2002 / 2005 / 2009)
Max. Marks: 100
Time: 3 Hours
Answer A L L Questions
Part-A
(10 x 2 = 20 Marks)
2
0
0^
1
2
0
0
1
2
f
1) Find the Eigen values of 2A given that A
2) Give the quadratic f o r m corresponding to the symmetric matrix
-x
a
and z = sin x then find — .
dx
A
4) If * = ~
2
-\v
X
then find
= y/2
^
5(w,v)
2
5) Find the image of | z | = 2 under the mapping w = 5z
6) Show that an analytic function with constant imaginary part is constant.
7) Find the Taylor's series expansion of f ( z ) - e
z
about
z-a.
8) Determine the residue of sinz at the pole.
9) Find L
1} Find L~
3/2
Part-B
(5 x 16 = 80 Marks)
2
2
2
11)rJ.) Reduce the quadratic form 8x] + 7 x
+3x
- 1 2 x ^ 2 - 8 x X 3 + 4 x x j i n t o the
canonical form through an orthogonal transformation. Write down the orthogonal
transformation, which you use.
2
3
2
f j j) Verify Cayley - Hamilton theorem for the matrix A =
1
2
3
4
3
(10)
(6)
12) a i) Discuss the maxima and minima and obtain the extreme values of the function
x + 3xy
3
- 1 5 x - 1 5 y + 72x
2
2
2
(10)
2
2
i*
•
u
du = 3 tan u
n) If
sin
w = — =y — ithen
prove that x — + y—
x
•N
:
I
U
dx
x+y
(6)
4
dy
(OR)
d
2
b i) If u = x
2
d
2
t a n ~ 0 ; / x ) - j / t a n " " ' ( x / ^ ) then prove that — - = — ]
2
(8)
7
ii) If v = log r where r
7
=x
+y
dv
8v
dx
dy
2
7
then find the value of
2
2
13) a) i) Find the analytic function f ( z ) - u + iv, given that u = x
(8)
2
- 3jcy + 3 x - 3y
+1
(8)
ii) Find the bilinear transformation which maps the points z = 1, i, -1 into the points
w = i, 0,- i respectively.
(8)
3
2
2
2
(OR)
b) i) Find the image of the circle | z - 3i | = 3 under the map w =
ii) If f(z)is
analytic, show that
+ dy f \
14) a) i) USlffd/^ontour integration on unit circle evaluate
=\f
2n
J
0
1
(8)
-i2
(8)
d8
2 + cosi9
ii) Find the Laurent's expansion of
/(*)
=
Cg-j
about z=1
z(l-z)
(OR)
b) i) Using Cauchy's residue theorem, evaluate
dz where c:| z | = 4
ii) Determine F ( 2 ) , F ( 4 ) , F ( - 3 z ) if
r5z
2
F(<x) =
-4z +3
z—
J
c
dz where
OC
C is the ellipse 1 6 x + 9 ^
2
15 a) i) Solve y"-4y'+3y
2
=144.
(8)
= e ' ;y(0) = ^ ^ ' ( O ) = Oby Laplace transform method.
ii) Using Convoluion theorem, find the inverse Laplace transform of
(8)
1
(8)
(OR)
b) (i) Find the Laplace transform of the function
f ( t ) = {t -3t + 2) sin 3/
(6)
2
(ii) Find the inverse Laplace transform of
s
2
(s
2
+2s + 3
+2s + 2 ) 0
2
(10)
+ 2 ^ + 5)
&&&&&