Nalanda Open University 2009 B.Sc Mathematics Bachelor of Science ( Hors), Part-III Term End , -VII - Question Paper
Bachelor of Science (Mathematics Honours), Part-III Term End Examination, 2009
Paper-VII (Mathematics)
Niiliind;! Open University
Bachelor of Science (Mathematics Honours), Part-HI Term End Examination* 200?
Paper-VII (Mathematics) iTiine: 3.00 Hi s. Full Marks: 75
Answer any Six; Questions, selecting at least one question from:each.|up.
Group-A
1. (a) Define a convex set. Prove that the intersection of two convex sets is also a convex set.
(b) Solve the foil owning L.P. problem graphically Max z 5x.j + 3x2 subject to the fojlo.b constraints 3xj + 5x2 1 1'5'
5xj + 2x2 110
x, X2 2 0
Use the simplex method to solve the follo'wilig:iirie'ar1 programming problem: Maximize z = 7xj + 5x2 subject to Xj + 2x2 1 6
4k j + 3x2 112
Xj , x2 2 0
Find the feasible solution to the following transportation problem.
|
Warehouse > |
wt |
w2 |
w3 |
w4 |
Factory capacity |
|
Factory 4- Fj |
19 |
30 |
50 |
10 |
7 |
|
70 |
30 |
40 |
60 |
9 | |
|
40 |
8 |
70 |
20 |
18 | |
|
Ware house requirement |
5 |
8 |
7 |
14 |
34 |
Group-B
4. (a) State and prove the necessary condition for the total differential equation to be mtegrable; (b) Solve the differential equation:
(y2 + yz) dx+(z2+xz) dy + (y2-xy) dz = 0.
5. (a) Apply Charpit's method to solve z2(p2z2+q2)=l (b) Discuss Char pit! s. method of solving the partial differ ehtM ifequati'dri of flfst: >rde.r given fey
f(x, y, z, p, q) = 0.
5. (a) Explain Monge's method to solve-.RrTHSs+Tt=\/:, symbols having.usual meaning,
(b) Solve..by:-Monge's method 2x2r-5xys+2yt+2(px+qy)=0.
3xdy 3y2
(b) Find the integral ofq2r - 2pqs+pt0 m the:form y+xf(z)=F(z) and:show that this represents a surface generattedby straight lines fha't-.are parallel to a fixed plane.
7. (a) Solve x
3x2
+ 2xy
O'
+ r
Group- C
(a). Findthe att faction of at hin uniform rod upon an external point.
'(b) Aright circular cylinder is of .infinite lengjlv iii one directionand is homogeneous, the'firiite exte remity b eing'perpendicular ta gerw'rafor: E rdve-'tf ,tte at tract io n at t-he.je ntie'of this end is 2MT/a ? where M is the mass per unit length and a the radius.
>L. (a) Findthe potentialofa uniform solid sphere at an internal point.
(b) Show that a family of right circular cones with a common axis and vert ex is a possible family .of equipotential surfaces and findthe potential function.,..
Group-D
L0:(a) Prove that pressure of a heavy homogeneous fluid at all points in the same horizontal plane' is given by p=pgz where symbols have usual meaning:
(b) Findthe depth of the centre of pressure of a circular area of radius a immersed vertically in a ho mogene ous liquid with it s centre at a dept h hb elo w the free s urfac e. flJeO'Atone floats with its axis horizontal in a liquid of density double its own. Findthe pressure on its base and prove that if 6 be the inclination to the vertical o f t he resultant t hrust on t he 'umyed surface, andipo- th?: semi-VeticaT ingle-iif t'he.:.c one, t he n tan 0 = tan a,.
(b) Show-that & right circular-cone of density p and s &mj vertical sngle go can float vertex .dp wnwards in .liquid of density cr wit hone ge n e rat :<?r ve r t ic a 1 a rid t he. b as e j:us.t: cle' sli.: ij f t lie' liquid if p= cr(Cos2oo)3-Lj2.(a) Show that dp = .p QCdx+Ydy+Zdz) where the terms have their usual meanings
(b) A liquid of given volume V is at rest under the forces
aJ bf
Show that the s urface s of equip res sure are s imilar ellips oids.
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| Earning: Approval pending. |
