Nalanda Open University 2009 B.Sc Mathematics Bachelor of Science Hons., Part-II Final , -IV (Differential Equation, Vector, Calculus, Statics and Dynamic) - Question Paper
Bachelor of Science (Mathematics) Hons., Part-II Final Examination, 2009
Paper-IV (Differential Equation, Vector, Calculus, Statics and Dynamic)
Bachelor of Science (Mathematics) Hons., Part-H Final Examination, 200?
Paper-IV (Differential Equation, Vector, Calculus, Statics anil Dynamic)
[Tiine: 3.00 Hi's. Full Marks: 75
Answer any Six Questions, selecting at least one question.from ac-la.grup.
Group-A
1- (a) Solve any two of the following:
(i) y = px + (ii) x2fy-pxQyp2 (iii) xyp2 - Cf -y2 \p - xy = 0 where P = ~.
(b) Find the orthogonal trajectories of the family of parabolas. y=4ax;_
(a) Solved-3 + 2yJl
$$ 4*.
d2y dy *
(b) Solve 2 +4y = e Cosx dx dx
2 d y dy 2
3. (a) Solve by the method of variation of parameters x Y+x-y=x s
dx dx
'a <Py rl , - *
X
(b) Solve x2 y- Gf +2x|- + I3+2Q= x1 e
s-f-r .-i1
dx dx
Group-B
4.. (a) Give a geometrical interpretation of scalar triple product [a t c (b) Prove that a x(|xc j= \c .
P. Ca)ProvE*a.di = g.-+.-
iit iM -at
(b) Find the moment about the point i + 2j +3%; of a force represented by i +j + acting
.... through the' point -2i + 3j + k .
5. (a) State and prove Green's theorem.
(b) Prove that div rtexsg where a and are constant vectors.
Group-C
7. (a) Obtain the equation of line of action of the resultant of a system of coplanar forces acting upon a rigid body.
(b) A square of side 2 a is placed with-its'.plane vertical betw.een two smooth pegs which are in the same horizontal line at a distance C apart. Show that it will be. in equilibrium when the
1 -ife _c2 |
inclination of one ofits edges with the horizontal is either /4 or 2" r"fc2 V
B. (a) Discuss the nature of forces which can be omitted infomiing the equati-tili of virtual W5rk'; (b) A string of length a forms the shorter diagonal of a rhombus of four uniform rods each of length b and weight W which are hinged together. If one of the rods be supported in a
horizont.posftion, prove that the tension in the string is
fja) Stats'. and prove, energy test of stability.
(b) A heay right circular cone rests with its base on a fixed rough of radius a. Find the greatest height of the cone if it is in stable equilibrium.
Group-D
10. (a) State and prove Hooke's law.
(b) Prove that the work done against the- tensi oil' iii str etching a light elastic string is equal to the product of its extension and the- A.M; of-initial and final tensions.
11..(a) Obtain expressions for radial and transverse accelerations.
(b) The velocity of a.pjarticle along and perpendicular to the radius from a fixed origin are and f4&-. Find' tfie'.path.
12. (a) Explain simple pendulam and obtain its periodic time for small oscillation.
(b) A particle describes the curve
rx = a*Cosn8
ytider a force towards the pole. Find-the law of force.
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| Earning: Approval pending. |
