## Thursday, October 05, 2017

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 function is 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
1. State and prove Cauchy’s integral formula.
2. If is an analytic function of and is continuous at each point within and on a closed contour , then prove that 3. Show that , if is analytic in the domain and its derivative at any point is again analytic.
3. (a) Write in short (any two): 1+1=2
Isolated singularity, Poles, Removable singularity, Essential singularity
(b) State Taylor’s theorem. Expand in 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
1. Prove that when a rigid body under the action of the three forces is in equilibrium, the forces are either parallel or concurrent.
2. Find the principal pitch of any wrench.
1. Explain the forces which can be omitted while forming the equation of virtual work.
2. Deduce the Cartesian equation of common catenary.
3. 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 , where is the weight of the wire per unit length.

(b) Dynamics
6. (a) Define simple harmonic motion and mention its nature. 1+1=2
1. 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  (b) A particle under a central acceleration is projected with velocity at a distance , show that the path is a rectangular hyperbola if the angle of projection is  