Bangalore University 2007-1st Sem B.Sc Mathematics Vester B.A./,2008 - Question Paper

15)    Evaluate using Greens theorem <1>_C 2xy dx + x² dy where C is the circle x² + y² = a².

16)    Evaluate JJ- ds using Gauss divergence theorem where S is the surface of

o

the cube - 1 < x < 1,-1 < y < 1,-1 < z < 1.

17)    Using Stoke's theorem evaluate 0_C yzdx + 3xdy + xydz where C is the circle x² + y² = 1, z = 2.

18)    Write Euler's equation when the function is independent of y.

^x2

2 /,a2 , o_ftx

19) Show that Euler's equation for the extremum of J ⁺²y^e

^xi

reduces to y" - y = e^x.

20) Define :

i)    Isoperimetric problem

ii)    Geodesics on a surface

II. Answer any four questions    (4x5=20)

1) Prove that the set V = {a + b , b e Q} is a vector space with respect to't' and 7 over Q.

linn i n i ii ii

VI Semester B.A./B.Sc. Examination, June 2008 (Semester Scheme) MATHEMATICS (Paper - VII)

Max. Marks : 90

Instructions: 1) Answer all questions.

2) Answer should be written completely either in Kannada or in English.

I. Answer any fifteen questions.

1)    If V (F) is a vector space, prove that a. ( a - j3) = a. a - a. p V a e F and a, p e V.

2)    Prove that W = { (x, y, z) / x, y, z e Q} is not a subspace of V₃ (R).

3)    Prove that S - { (1, 1), (3, 1) } is a basis of V₂ (R).

4)    Show that (1, 0, 1), (1, 1, 0) and (-1, 0, -1) are linearly independent in V₃ (R).

. 5) Show that T : V₂ (R) > V₃ (R) defined by T (x, y) = (y, x, x + y) is a linear transformation.

6)    Find the matrix of the linear transformation T : V (R) -> V₃ (R) defined by T (x, y, z) = (x, 2y, 3z) with respect to standard basis.

7)    If T = V₄ (R) > V₂ (R) is a linear transformation and nullity is 1 find the rank of T.

8)    Evaluate J* dy - y dx where c is the curve x = a cos t, y = a sin t, o <t < y₂.

c

(2,3)

9) Evaluate J 2xydx + x²dy.

(i.i)

10) The acceleration of a particle at any time t is given by 4 Cos 2 ti- 8 sin2 t ] + 16t K find the displacement.

P.T.O.

SM - 224

-3-

2)    Express one of the vectors (2, 4, 2), (1, -1, 0) (0, 3, 1) (1, 2, 1) as a linear combination of others.

3)    State and prove Rank-nullity theorem.

4)    Find the linear transformation of the matrix A =

bases = {(1, 2), (-2, 1)}

B₂= {(1,-1, 1), (1, 2, 3), (-1,0, 2)}

5)    Find the linear transformation T = V₃ (R) V³ (R) such that T(l, 0, 0) = (4, 5, 8), T(l, -1, 0) = (8, 10, 8) T (0, 1, 1) = (-3, - 4, -7).

6)    Find all the eigen values and a basis'for eigen space of the linear transformation T = v (R) > V (R) defined by T (x, y, z) = (x + y + z, 2y + z, 2y + 3z).

with respect to the

3 v-'/ ' 3

III. Answer any three questions

1) Evaluate J 3x²dx + (2xz - y) dy + zdz where C is the line joining (0, 0, 0)

c

and (2, 1, 3).

l ^x

2) By changing into polar coordinates evaluate I I \/^x + y ^dy ^dx.

o o

a x

f f C^s y

3)    Evaluate changing the order of integration J J n-77-r dydx.

0 0 vv^a-xMa-yj

4)    If IR is the region bounded byx = 0, y = 0, z = 0andx + y + z = a show that IJ J (^x + y + ^z) dxdydz =

R

5) Find the volume common to the sphere x² + y² + z² = a² and the cylinder

x² + y² = ax.

IV. Answer any three questions.    ^x5

1)    Evaluate J Jf, fids _where F-4xi + yj + zK and S is the part of the surface 2x + y + 2z = 6 in the first octant.

2)    State and prove Gauss divergence theorem.

3)    Verify Green's theorem for (j)_c (_x² + y) dx - xy²dy taken around the square whose vertices are (0, 0), (1, 0), (1, 1) and (0, 1).

4)    Evaluate JJf, nds _wh_ere f = zi + xj - 3y²zK and S is the surface of the

cylinder x² + y² = 16 bounded by z = 0 and z = 5 in I^st octant (use divergence theorem).

5)    Verify Stokes theorem for jF.dr for p = y² [ + xy j - x zKand C is the

c

boundary of the hemisphere x² + y² + z² = a².

V. Answer any two questions.    (2x5-]0)

1) Show that the external of I = J jy (1 + (y')²) dx is a parabola.

X

2)    Find the geodesics on a right circular cylinder.

3)    If a cable hangs freely under gravity from two fixed points then show that the shape of the curve is a catenary.

2 --- 2

4)    Find the extremum of the functional ⁺    under jy    given

-2 -2

that y (2) = y (-2) = 0.