2015

(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

1. (a) Write when the limit of a complex function exists. 1

(b) Prove that is continuous at . 3

Or

Show that is non-analytic 3

(c) State and prove the necessary condition for a function to be analytic.

Or

If , then find , such that is analytic.

2. (a) Derivative of an analytic function is not analytic. State true or false.

(b) Prove that.

(c) State and prove Cauchy’s integral formula.

Or

Evaluate along the curve given by the line from to and the line from to.

(d) Define complex line integral. 3

3. (a) Write the condition for the convergence of the Taylor’s series. 1

(b) Find the singularity of the function. 3

(c) Find the residue of any one of the following: 3

- at
- at

(d) Evaluate any one of the following by using contour integration: 6

(B) Mechanics

(Marks: 45)

(a) Statics

4. (a) Define screw.

(b) Write the name used for denoting a single reduced force and a couple of a system of forces.

(c) Write the conditions that the system of coplanar forces may be in equilibrium.

(d) Find the equation of a system of coplanar forces acting at different points of a rigid body.

Or

Show that any system of forces acting on a rigid body can be reduced to a single force together with a couple whose axis is along the direction of the force.

5. (a) Define virtual displacement. 1

(b) Define span of a common catenary. 1

(c) Write two forces which can be omitted while writing the equation of virtual work. 2

(d) State and prove the principle of virtual work for a system of coplanar forces. 6

(b) Dynamics

6. (a) Define angular velocity of a point. 1

(b) Simple harmonic motion of a particle is described by. Find the time period.

(c) Find the tangential and normal components of velocity of a moving point along a plane curve. 5

Or

A particle describes the curve with a constant velocity. Find the components of velocity along the radius vector and perpendicular to it.

7. (a) Define central orbit. 1

(b) A particle describes a curve under a force to the pole. Find the law of force. 6

Or

A particle is projected upwards under gravity in a resisting medium, whose resistance varies as the square of the velocity. Find the velocity of the particle at any position. 6

8. (a) State the theorem of parallel axes of moment. 2

(b) Define radius of gyration. 2

(c) Find the moment of inertia of a uniform rod of length and mass about an axis through the middle point and perpendicular to it. 6

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