JNTU

JNTU B.Tech II MATHEMATICS-IV Previous Question Paper (September - 2021 - R16)

B.Tech | II | II-I | R16 | September - 2021

Answer PaperPaper Code: 133BDMaximum Marks: 75Time: 3 hours

Answer any five questions

All questions carry equal marks

Main

115 marks

Find the analytic function whose real part is

e^x(x cos y - y sin y)

8 marks

Answer

Given the real part u(x, y) = e^x(x cos y - y sin y). To find the analytic function f(z) = u + iv, we use the Milne-Thomson method. First, calculate the partial derivatives:

u_x = e^x(x cos y - y sin y) + e^x(cos y) = e^x((x+1)cos y - y sin y)

u_y = e^x(-x sin y - sin y - y cos y) = -e^x((x+1)sin y + y cos y)

Let f'(z) = u_x + iv_x. By Cauchy-Riemann equations, v_x = -u_y, so f'(z) = u_x - i u_y. Substituting z for x and 0 for y:

f'(z) = e^z((z+1)cos 0 - 0 sin 0) - i(-e^z((z+1)sin 0 + 0 cos 0))

f'(z) = e^z(z+1)

Integrating with respect to z:

f(z) = ∫(z e^z + e^z) dz = z e^z - e^z + e^z + C = z e^z + C

Let u and v denote the real and imaginary components of the function f defined by means of the equations

f(z) = z²/z (z ≠ 0), 0 (z = 0)

. Verify whether the Cauchy–Riemann equations are satisfied or not at the origin

(0,0)

7 marks

Answer

Given f(z) = z²/z for z ≠ 0 and f(0) = 0. This simplifies to f(z) = z for all z. Thus, u + iv = x + iy, implying u = x and v = y. At the origin (0,0):

u_x = ∂/∂x(x) = 1, u_y = ∂/∂y(x) = 0

v_x = ∂/∂x(y) = 0, v_y = ∂/∂y(y) = 1

The Cauchy-Riemann equations are u_x = v_y and u_y = -v_x. Substituting the values:

1 = 1 (Satisfied)

0 = -0 (Satisfied)

Therefore, the Cauchy-Riemann equations are satisfied at the origin.

215 marks

Prove that if

w = f(z) = u + iv

is analytic in a region R, then

dw/dz = ∂w/∂x = -i(∂w/∂y)

7 marks

Answer

Evaluate

∫_C (x + 2y) dx + (y - 2x) dy

along:

the parabola
y = 2x²
from (1, 2) to (2, 8)
the straight lines from (1, 1) to (1, 8) and then from (1, 8) to (2, 8)
the straight line from (1, 1) to (2, 8).

8 marks

Answer

315 marks

Evaluate

∫_C (e^z / (z-1)²) dz

using Cauchy’s integral formula, where C is

|z| = 2

7 marks

Answer

Find the poles and the corresponding residues of each pole of

f(z) = (z-1) / ((z-2)(z-3)²)

8 marks

Answer

415 marks

Find the Maclaurin series and its radius of convergence

(z+2)/(1-z²)

7 marks

Answer

Find the Laurent series that converges for

0 < |z - z₀| < R

and determine the precise region of convergence. Show details

(z²-3i) / (z-3)²

z₀ = 3

8 marks

Answer

515 marks

Find all the singularities in the finite plane and the corresponding residues of

cos z / z⁴

8 marks

Answer

Evaluate (counterclockwise)

∫_C (z-23) / (z²-4z-5) dz

, where

C: |z - 2 - i| = 3.2

7 marks

Answer

Evaluate the integral

∫_-∞^∞ x / (8-x³) dx

15 marks

Answer

715 marks

Obtain the Fourier series for

f(x) = x² in [-π, π]

7 marks

Answer

Find the cosine transform

f_c(w)

f(x) = {x, 0 < x < 2; 0, x > 2}

8 marks

Answer

Find the deflection

u(x,t)

for the string of length L and

c² = 1

when the initial velocity is zero and the initial deflection with

(say, 0.01) is

k sin(3πx)

15 marks

JNTU B.Tech II MATHEMATICS-IV Previous Question Paper (September - 2021 - R16)

Main

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

About this paper

Other papers in MATHEMATICS-IV