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JNTU B.Tech II MATHEMATICS-IV Previous Question Paper (May/June - 2019 - R16)

B.Tech | II | I | R16 | May/June - 2019

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

Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit.

PART A

125 marks

State the necessary and sufficient conditions for a function f(z) = u + iv to be analytic.

2 marks

Answer

A function f(z) = u + iv is analytic in a domain D if it satisfies the Cauchy-Riemann equations: u_x = v_y and u_y = -v_x, provided that the partial derivatives u_x, u_y, v_x, and v_y are continuous in D.

Show that f(z) = z² is not analytic at any point.

3 marks

Answer

For f(z) = z², let z = x + iy. Then f(z) = (x + iy)² = (x² - y²) + i(2xy). Here, u = x² - y² and v = 2xy. The partial derivatives are u_x = 2x, u_y = -2y, v_x = 2y, and v_y = 2x. Since u_x = v_y and u_y = -v_x are satisfied for all x, y, the function is analytic everywhere in the complex plane. The premise that it is not analytic is incorrect.

State Cauchy’s integral theorem.

2 marks

Answer

Find the poles and the residues at the poles of the function

f(z) = e^z / cos(πz)

3 marks

Answer

Define bilinear transformation and cross ratio.

2 marks

Answer

Find the image of the circle

|z| = 2

, under the transformation

w = z + 3 + 2i

3 marks

Answer

State Fourier integral theorem.

2 marks

Answer

Expand

f(x) = πx - x²

in a half range sine series in

(0, π)

3 marks

Answer

Classify the partial differential equation

u_xx + 6u_xy + 2u_yy + 2u_x - 2u_y + u = x²y

2 marks

Answer

Write the three possible solutions of the heat equation

∂u/∂t = ∂²u/∂x²

3 marks

Answer

PART B

210 marks

If f(z) is a regular function of z, prove that

(∂²/∂x² + ∂²/∂y²) |f(z)|² = 4|f'(z)|²

5 marks

Answer

Let f(z) = u(r, θ) + iv(r, θ) be an analytic function. If

u = -r³ sin 3θ

, then construct the corresponding analytic function f(z) in terms of z.

5 marks

Answer

310 marks

Show that the function f(z) defined by

f(z) = (x²y³(x+iy)) / (x⁶+y¹⁰) for z ≠ 0, f(0) = 0

, is not analytic at the origin, even though it satisfies the Cauchy-Riemann equations at the origin.

5 marks

Answer

Determine the analytic function whose real part is

log(x² + y²)

5 marks

Answer

Represent the function

1 / (z² - 4z + 3)

in the domain (a)

1 < |z| < 3

(b)

|z| < 1

10 marks

Answer

510 marks

Expand the function

f(z) = z / ((z+1)(z+2))

about z = -2, and name the series thus obtained.

5 marks

Answer

Evaluate

∮_C (e^z / ((z+3)(z+2))) dz

, where C is the circle

|z - 1| = 1/2

5 marks

Answer

Evaluate the integral using contour integration

∫₀^2π (dθ / (2 + cos θ))

10 marks

Answer

Show that the transformation

w = i(1-z)/(1+z)

transforms the circle

|z| = 1

into the real axis of w plane and the interior of the circle

|z| < 1

into the upper half of the w plane.

10 marks

Answer

Find the Fourier transform of

f(x) = 1 - x² (if |x| < 1), 0 (if |x| > 1)

. Hence evaluate

∫₀^∞ ((x cos x - sin x) / x³) cos(x/2) dx

10 marks

Answer

910 marks

Obtain the half range cosine series for

f(x) = kx (0 ≤ x < L/2), k(L-x) (L/2 ≤ x ≤ L)

5 marks

Answer

Find the Fourier sine transform of

f(x) = e^-|x|

. Hence show that

∫₀^∞ (x sin mx) / (x²+1) dx = (π/2) e^-m

5 marks

Answer

A string is stretched and fastened to two points L apart. Motion is started by displacing the string in the form

y = a sin(πx/L)

from which it is released at time t = 0. Find the displacement of any point at a distance x from one end at time t.

10 marks

Answer

Write down the one dimensional heat equation. Find the temperature u(x,t) in a slab whose ends x = 0 and x = L are kept at zero temperature and whose initial temperature f(x) is given by

f(x) = k (0 < x < L/2), 0 (L/2 < x < L)

10 marks

JNTU B.Tech II MATHEMATICS-IV Previous Question Paper (May/June - 2019 - R16)

PART A

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PART B

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