2013

(November)

MATHEMATICS

(General)

Course: 501

((A) Analysis – II, (B) Mechanics)

Full Marks: 80

Pass Marks: 32

Time: 3 hours

The figures in the margin indicate full marks for the questions

(A) Analysis – II (Complex Analysis)

(Marks: 35)

1. (a) Define an analytic function. 1

(b) Show that the functionis harmonic. 3

(c) State and prove the necessary conditions of Cauchy-Riemann equations. 6

Or

Construct the analytic function, when.

2. (a) Define line integral and closed contour. 1+1

(b) Answer any two: 5x2=10

- State and prove Cauchy’s integral formula.
- Ifis an analytic function ofandis continuous at each point within and on a closed contour, then prove that
- Show that, ifis analytic in the domainand its derivative at any pointis again analytic.

3. (a) Write in short (any two): 1+1=2

Isolated singularity, Poles, Removable singularity, Essential singularity

(b) State Taylor’s theorem. Expandin a Taylor series about. 2+3=5

Or

State Laurent’s theorem and expand with the help of this theorem.

(c) Find the residue of

Or

Find the value of:

, if

(B) Mechanics

(Marks: 45)

(a) Statics

4. (a) What do you mean by screw? 1

(b) Write down the equation of central axis for a finite number of forces acting on a rigid body. 2

(c) Answer any one: 7

- Prove that when a rigid body under the action of the three forces is in equilibrium, the forces are either parallel or concurrent.
- Find the principal pitch of any wrench.

5. Answer any two: 5x2=10

- Explain the forces which can be omitted while forming the equation of virtual work.
- Deduce the Cartesian equation of common catenary.
- A kite flying at a height withy a length of wire paid out, and with the vertex of the catenary on the ground, show that at the kite the inclination of the wire to the ground is , and its tensions there and at the ground are and, whereis the weight of the wire per unit length.

(b) Dynamics

6. (a) Define simple harmonic motion and mention its nature. 1+1=2

(b) Answer any one: 6

- The distances of a particle performing SHM from the middle point of its path at three consecutive times observed to be. Show that the time of the complete oscillation is
- A particle is moving along a curve. Find the acceleration among the tangent and normal to the path of the particle.

7. (a) Find the differential equation of a central orbit in the form 7

(b) A particle under a central accelerationis projected with velocityat a distance, show that the path is a rectangular hyperbola if the angle of projection is

8. Answer any two: 5x2=10

- Show that the moment of inertia of a rectangle whose sides are a and b, and mass M about a diagonal is

- Determine the moment of inertia of a thin uniform rod of mass M and length 2a about one end.
- Find the moment of inertia of a thin uniform rod about a line through its centre and perpendicular to its length.

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