B.Sc. Mathematics: Analysis II Annamalai University Question Paper december 2014

Annamalai University question paper december 2014
B.Sc. DEGREE EXAMINATION December 2014
(MATHEMATICS)
(SECOND YEAR)
(PART - III)
GROUP – A : Main
640: ANALYSIS-II
Time: Three hours Maximum: 100 marks
 Answer any FIVE questions (5× 20=100)
1. Evaluate the following integrals
a) ∫
−
dx
e
e
x
x
1
2
(5)
b) ∫
+ −
2
1 x x
dx (5)
c) ∫
( + )1
3
x x
dx (5)
d) 2
2
1
x − x
(5)
2.
a) Show that ∫
=
+
π
π
θ
θ
0
4 5 3cos
d
(10)
b) Show that ( ) ∫
= +
+
2
0
log 2 1
sin cos 2 2
π
π
x x
xdx (10)
3.
a) Establish a reduction formula for ∫
x ax dx n
sin and hence find ∫
2
0
3
sin
π
x x dx and
hence find ∫
2
0
3
sin
π
x x dx
(10)
b) If ∫
=
2
0
cos cos ( , )
π
x nx dx f m n
m
prove that ( ,1 )1
1
( , ) − −
+
= f m n
m
m
f m n hence prove that
1
2
( , )
+
= n
f n n
π
(10)
2
5272
4. Discuss the convergence of the following
a) ∫
∞
−
0
e dx x
(5)
b) ∫
∞
1
x
dx (5)
c) ∫
−∞
−
0
2
1( 3x)
dx (5)
d) ∫
−
1
1 3
1
x
dx (5)
5. a) Find the area bounded by the parabolas y2=4ax and x2=4by. (10)
b) Find the volume bounded by the cylinder x2+y2=4, the planes y+z=4 and z=0. (10)
6. a) Evaluate ( )
∫∫ x + y dxdy over the region in the positive quadrant bounded by the
ellipse 1 2
2
2
2
+ =
b
y
a
x
(10)
b) Evaluate ( )
∫∫∫ + +
v
xy yz zx dxdydz , where v is the region of space bounded by x=0,
x=1, y=0, y=2, z=0 and z=3
(10)
7. a) Solve (1+y2)dx+(x–tan–1y)dy=0 (10)
b) Solve y=2px+y2p3 (10)
8. a) Solve y″+4y′+13y=2e–x given y(0)=0 and y′(0)= –1 (10)
b) Solve (D2–2D+2)y= ex sin x (10)
9. a) Solve x2y″–xy′+4y=cos (log x) + x sin (log x) (10)
b) Solve xy″–(2x+1)y′+(x+1)y=x2ex (10)
10. a) Solve p + q =1
(5)
b) Solve z=px+qy+p2q2 (5)
c) Solve p − q + 3x = 0
(10)
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