BABSc, B.Com & BCA Questions _III - Nalanda Open University

Nalanda Open University
Annual Exam-2010,
Bachelor of Science (Mathematics) Hons, Part-III
Paper-VII
Time: 3Hrs Full Marks: 75
Answer Six questions, selecting at least one question from each group.
Group-A
1. (a) Prove that the set of all convex combinations of a finite number of linearly independent
vectors V 1 , V 2 , V 3 , ....., V m is a convex set.
(b) Solve the following L.P. problem graphically.
Max Z = 5x+7y
subject to the following constraints
x + y ≤ 4
2. Use the simplex method to solve
Max Z = 3x 1 +9x 2
subject to
x + 4x
≤ 8
x
1 2
+ 2x
≤ 4
1 2
3x
+ 8y
≤ 24
10x
+ 7y
≤ 35
x,
y ≥ 0
x1,
x2
≥ 0
3. Obtain an initial basic feasible solution to the following transportation problem
using the north-west corner method
D E F G Available
A 11 13 17 14 250
B 16 18 14 10 300
C 21 24 13 10 400
Requirement 200 225 275 250
Group-B
4. (a) Solve (y 2 +yz+z 2 )dx+(z 2 +zx+x 2 )dy+(x 2 +xy+y 2 )dz=0
(b) Solve
dx dy dz
= =
2 2 2
x − yz y − zx z − xy
5. (a) Apply Charpit's method to find complete integral p 2 +q 2 -2px-2qy+1=0.
(b) Solve the following Lagrange's equation (y 3 x-2x 4 )p+(2y 4 -x 3 y)q=9z(x 3 -y 3 ).
6. (a) Solve r = b 2 t.
(b) Find the solution of Lagrange's differential equation pq+Qq=R.
7. (a) Find the orthogonal projection on the xz-plane of the curves which lie on the
paraboloid 3z=x 2 +y 2 and satisfy the equation 2dz = (x+z) dx+ydy.
dx
x y t dy
2
+ 2 − 3 = , − 3x + 2y = e t
(b) Solve the simiultaneous equations dt
dt
.
Group-C
8. (a) Find the attraction of a thin uniform spherical shell at an internal point.
(b) A uniform solid sphere of mass M is cut in 2 parts by a diametral plane. Prove
2
3
γ M 2
that the resultant attraction between the halves is 16 a , where a is the radius
of the sphere and γ the constant of gravitation.