MS – 280
*MS280*
VI Semester B.A./B.Sc. Examination, May/June 2014 (Semester Scheme) (2013-14 and Onwards) (N.S.) MATHEMATICS (Paper – VII) Time : 3 Hours
Max. Marks : 100
Instruction : Answer all questions. I. Answer any fifteen questions :
(15×2=30)
1) Find the locus of the point z satisfying z − 1 ≤ 4 . ⎛ ⎞ z2 ⎟. 2) Evaluate lim ⎜ 4 2 iπ⎜ z + z + 1⎟⎠ 4⎝ z→e
3) Show that f(z) = cos z is an analytic function. 4) Prove that u = x3 – 3xy2 is a harmonic function. 5) Define bilinear transformation. 6) Evaluate
2 ∫ (z ) dz around the circle | z | = 1.
C
7) Evaluate
π z 3 3 dz where C: z = . z −1 2
cos
∫
C
8) State Liouville’s theorem. 9) Evaluate
[ (x
∫
C
2
)
) ]
(
− y dx + y 2 + x dy where C is the curve given by
x = t, y = t2 + 1, 0 ≤ t ≤ 1. 10) Show that
1
∫ ∫ 0
1
11) Evaluate ∫
0
0
π
∫
2
3
dx dy
(1 + x )(1 + y ) 2
2
=
π2 . 12
r 3 sin2 θ dθ dr .
0
P.T.O.
MS – 280 ∞ ∞
12) Evaluate
*MS280*
-2-
∫ ∫
e − (x
2 + y2
) dx dy by changing into polar coordinates.
0 0
1 2 2
13) Evaluate ∫
∫∫
x 2 yz dx dy dz .
0 0 1
14) State Green’s theorem in the plane.
(
)
15) Show that ∫∫∫ div x ˆi + yˆj + zkˆ dv = 3v . V
16) Using Stokes theorem prove that div curl F = 0 .
(
)
17) Define interior point on topology. 18) State Bolzano-Weierstrass theorem on R. 19) Write all possible topologies for X = {3, 4}. 20) Define sub base for a topology. II. Answer any four questions.
(4×5=20)
⎛ z − 1⎞ π ⎟⎟ = is a circle. Find 1) Show that the locus of a point z satisfying amp ⎜⎜ ⎝z + 2⎠ 3 its centre and radius.
2) If f(z) = u + iv be an analytic function in the domain D of a complex plane then u = c1 and v = c2. Where c1 and c2 are constants represents orthogonal family of curves. 3) Find the analytic function whose imaginary part is tan −1
y and hence find its x
real part. 4) Find the analytic function f(z) = u + iv given that u – v = ex(cosy – siny). 5) Discuss the transformation w = z2. 6) Find the bilinear transformation which maps the points 1, –i, –1 on to the points 0, i, ∞ .
*MS280*
MS – 280
-3-
III. Answer any two questions. 1) Evaluate
(2×5=10)
sin π z 2 + cos π z 2 dz where C is the circle z = 4 . (z − 1) (z − 2)
∫
C
e2 z
2) Show that Ο ∫
(z − 2)
3
C
dz = 4 πi e 4 where C is the circle z = 3 .
3) State and prove the fundamental theorem of algebra in complex variables. IV. Answer any four questions : 1) Evaluate
(4×5=20)
xy (x + y ) dx dy over the domain D between y2 = x and y = x.
∫∫ R
1 1
2) Evaluate
∫∫ 0 y
(x
2a
3) Show that
∫ 0
2
)
+ y 2 dx dy by changing the order of integration. 2 ax − x 2
∫
0
(x
2
)
+ y 2 dy dx =
3πa 4 by changing into polar 4
coordinates. a
4) Evaluate ∫
0
∫
a2 − x2
0
∫
a2 − x2 − y2
dx dy dz 2
2
2
a −x −y −z
0
2
.
5) Find the surface area of the sphere x2 + y2 + z2 = a2 by using double integration. 6) Evaluate
∫∫∫
xyz dx dy dz where R is the positive octant of the sphere
R
x2
+
y2
+ z2 = a2 by transforming in to cylindrical polar coordinates.
V. Answer any two questions.
(2×5=10)
1) Using Green’s theorem evaluate
∫
C
[e
−x
]
sin y dx + e − x cos y dy where C is the rectangle with vertices (0, 0),
⎛ π⎞ ⎛ π⎞ ⎜ 0, ⎟ , ⎜ π, ⎟ , (π, 0 ) . ⎝ 2⎠ ⎝ 2⎠
MS – 280
-4-
*MS280*
2) State and prove the Gauss divergence theorem. 3) Verify Stoke’s theorem for the function F = y 2 ˆi + xyˆj − xzkˆ , where S is the hemisphere x2 + y2 + z2 = a2, z ≥ 0 . VI. Answer any two questions.
(2×5=10)
1) Prove that the union of an arbitrary collection of open sets is open. 2) Define topological space. Let X = {m, n} and τ = { X, φ, {m}, {n} } then show that τ is a topology on X. 3) If (X, τ) be a topological space and A, B ⊂ X then prove that i) A ⊂ B ⇒ A ⊂ B ii) (A ∩ B ) = A ∩ B . 4) Show that every convergent sequence is a Cauchy sequence. _______________________
MS – 280
-5-
*MS280*
VI Semester B.A./B.Sc. Examination, May/June 2014 (Semester Scheme) (Prior to 2013-14) (OS) MATHEMATICS (Paper – VII) Time : 3 Hours
Max. Marks : 90
Instruction : Answer all questions. I. Answer any fifteen questions :
(15×2=30) 1) In any vector space V over a field F, prove that c, 0 = 0, ∀ c ∈ F ; where 0 be the zero vector of V. 2) Prove that the subset w = {(x, y, z)|x = y = z} is a subspace of V3(R).
3) Verify whether (3, 2, 1), (1, 0, 2), (2, 2, 1) are linearly independent. 4) Prove that any two basis of a vector space have same number of elements. 5) Show that T : V3 (R ) → V2 (R ) defined by T(x, y, z) = (x, y) is a linear transformation. 6) If T : V3 (R ) → V2 (R ) is defined by T(x, y, z) = (y + x, y – 2z). Find matrix of T. 7) Define Eigen values and Eigen vectors of a linear transformation. 8) Evaluate
∫
5xy dx + y 2 dy , where C is the curve y = 2x2 from (0, 0) to (1, 2).
C 2 2
9) Evaluate ∫
∫
0 1 π
10) Evaluate ∫
0
(x
)
2
+ y 2 dx dy .
a sin θ
∫
rdr dθ .
0
11) If A is the region representing the projection of a surface S on the ZX plane. Write the formula for the surface area of S. 2 2 2
12) Evaluate ∫
∫∫
x 2 yz 2 dx dy dz .
0 0 1
13) The acceleration of a particle at any time ‘t’ is given by 4 cos 2 t ˆi − 8 sin 2tˆj + 16 tkˆ . Find the displacement.
MS – 280 14) Find
*MS280*
-6-
∫
C
F . d r where F = x 2 ˆi + y 2 ˆj and C is the curve y = 2x2 from (0, 0) to
(1, 1). 15) State Green’s theorem. 16) Evaluate ∫∫ r . nˆ ds using Gauss divergence theorem, where S is the surface S
of the cube − 1 ≤ x ≤ 1, − 1 ≤ y ≤ 1, − 1 ≤ z ≤ 1. 17) Using Stokes theorem, show that
∫
yzdx + zxdy + xydz = 0 .
C
18) Write Euler’s equation when the function is independent of y ′ . ⎛ dy ⎞ d 19) Prove that δ ⎜ ⎟ = (δy ) . ⎝ dx ⎠ dx
20) Define isoperimetric problem. II. Answer any four questions.
(4×5=20)
1) Prove that the necessary and sufficient condition for a non empty subset W of a vector space V(F) to be a subspace of V(F) is, ∀a, b ∈ F and α, β ∈ W ⇒ aα + bβ ∈ W .
{
}
2) Prove that the set V = a + b 2 | a, b ∈ Q , Q is the field of rationals forms a vector space w.r.t. addition and multiplication of rational numbers. 3) Express one of the vectors (2, 4, 2), (1, –1, 0), (0, 3, 1), (1, 2, 1) as a linear combination of vectors. ⎡ − 1 2 1⎤ 4) Find the linear transformation of the matrix ⎢ ⎥ with respect to the ⎣ 1 0 3⎦ basis
B1 = {(1, 2, 0), (0, –1, 0), (1, –1, 1)} B2 = {(1, 0), (2, –1)}. 5) State and prove Rank-Nullity theorem. 6) Let T : R 2 → R 2 defined by T(1, 0) = (1, 2), T(0, 1) = (4, 3). Find all eigen values of T and the corresponding eigen vectors.
*MS280*
MS – 280
-7-
III. Answer any three questions. 1) Evaluate
∫
(3×5=15)
3 x 2 dx + (2 xz − y ) dy + zdz , where C is the line joining the
C
points (0, 0, 0) and (2, 1, 3). 2) Evaluate ∫∫ xy (x + y ) dx dy over the region R bounded between the parabola R
y=
x2
and the line y = x. 1
3) Evaluate
1
∫∫ 0 x
4) Evaluate
(x
2
)
+ y 2 dy dx by changing the order of integration.
∫∫∫ (x + y + z) dx dy dz , where R is the region bounded by the R
planes x = 0, y = 0, z = 0 and x + y + z = 1. 5) Find the volume of the sphere x2 + y2 + z2 = a2 using triple integrals. IV. Answer any three questions. (3×5=15) 1) Evaluate ∫∫ F . nˆ dS where S denote the part of the plane 2x + y + 2z = 6, S
which lies in the positive octant and F = 4 x ˆi + yˆj + zkˆ .
2) Verify Green’s theorem for
∫
e − x sin y dx + e − x cos y dy , where C is the
C
rectangle with vertices (0, 0), (π, 0 ) , (π, π 2 ) and (0, π 2 ) . 3) State and prove Gauss divergence theorem. 4) Using divergence theorem, evaluate ∫∫ F . nˆ dS , where F = 4x ˆi − 2y 2 ˆj + z 2kˆ S
and S is the surface enclosing the region for which x 2 + y 2 ≤ 4 and 0 ≤ z ≤ 3 . 5) Using Stoke’s theorem, evaluate
∫ (x + y ) dx + (2x − z ) dy + (y + z ) dz ,
C
where C is the boundary of the triangle with vertices at A(1, 0, 0), B (0, 2, 0) and C (0, 0, 3).
MS – 280
*MS280*
-8-
V. Answer any two questions.
(2×5=10)
1) Prove that the necessary condition for the integral I =
x2
∫ f (x, y, y′) dx , where
x1
y(x1) = y1 and y(x2) = y2 to be an extremum is that π
2) Find the extremal of the functional I =
∫
[y
2
0
2
∂f d ⎛ ∂f ⎞ ⎜ − ⎟ = 0. ∂y dx ⎝⎜ ∂y ′ ⎠⎟
]
− (y ′) − 2 y sin x dx with 2
( 2 ) = 0.
y(0) = 0, y π
3) Find the geodesics on a right circular cylinder. 1
4) Find the extremal of the functional ∫
0
[ (y′)
2
]
+ x 2 dx subject to the constraint
1
∫
ydx = 2 and having conditions y(0) = 0, y(1) = 1.
0 _______________________
