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JNTU B.Tech II MATHEMATICS-IV Previous Question Paper (February - 2024 - R16)

B.Tech | II | II-I | R16 | February - 2024

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

Note: i) Question paper consists of Part A, Part B. ii) Part A is compulsory, which carries 25 marks. In Part A, answer all questions. iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.

Part A

1.a

Define Analytic function.

2 marks

Answer

A complex-valued function f(z) is said to be analytic at a point z₀ if it is single-valued and possesses a unique derivative f'(z) at every point in some neighborhood of z₀. A function that is analytic at every point in a domain D is called an analytic function in that domain.

1.b

w = log z

, find

dw/dz

and determine where w is non-analytic.

3 marks

Answer

Given the complex function

w = log z

, where

z = re^iθ

, the derivative is found using the standard derivative rule for the complex logarithm:

dw/dz = 1/z

The function

w = log z

is defined as

log z = ln|z| + i arg(z)

. This function is non-analytic where it fails to satisfy the Cauchy-Riemann equations or where it is multi-valued and discontinuous. Specifically, the function is non-analytic at

z = 0

, where the derivative is undefined, and along the branch cut, typically taken as the negative real axis (

z = x + i0, x ≤ 0

), where the function is discontinuous.

1.c

Find the zeros of the function

sin(1/z)

2 marks

Answer

1.d

Show that the function

e^z

has an essential singularity at

z = ∞

3 marks

Answer

1.e

Define conformal mapping.

2 marks

Answer

1.f

Find the fixed points of the transformation

w = (z - 1 + i)/(z + 2)

3 marks

Answer

1.g

State dirichlet conditions.

2 marks

Answer

1.h

State Fourier integral in complex form.

3 marks

Answer

1.i

In heat flow, what is meant by transient state.

2 marks

Answer

1.j

Classify the second order partial differential equations.

3 marks

Answer

Part B

210 marks

Show that

f(z) = xy + iy

is everywhere continuous but it is not analytic.

5 marks

Answer

f(z) = u + iv

is an analytic function, show that

u(x,y) = C

and

v(x,y) = K

represent two families of orthogonal curves.

5 marks

Answer

310 marks

Find the conjugate harmonic function of the harmonic function

u = a(1 + cosθ)

5 marks

Answer

Show that

xy^2

cannot be real part of an analytic function.

5 marks

Answer

410 marks

Evaluate

∫(y - x - 3x^2i)dz

where c consists of the line segments from

z = 0

z = i

and the other from

z = i

z = 1 + i

5 marks

Answer

Expand the function

f(z) = z / ((z - 1)(z - 3))

as Taylor’s series about

0 < |z - 1| < 2

5 marks

Answer

Show that using residue theorem,

∫[0 to 2π] (1 / (a + b sin θ)) dθ = (2π) / √(a^2 - b^2)

a > b > 0

10 marks

Answer

Evaluate

∫[0 to 2π] (sin^2θ / (a + b cos θ)) dθ

; (

a > b > 0

10 marks

Answer

Find the bilinear transform which maps the points

z = 0, -i, -1

into the points

w = i, 1, 0

. Find the image of the line

y = mx

under this transformation.

10 marks

Answer

Find Fourier sine transform of

f(x) = 1 / (x(x^2 + a^2))

and hence deduce cosine transform of

1 / (x^2 + a^2)

10 marks

Answer

f(x) = x

for

0 < x ≤ π/2

and

f(x) = π - x

for

π/2 < x < π

, then Show that:

f(x) = 4/π [sin x - (sin 3x)/3^2 + (sin 5x)/5^2 - ...]

f(x) = π/4 - 2/π [cos 2x/1^2 + cos 6x/3^2 + cos 10x/5^2 + ...]

10 marks

Answer

The ends A and B of a rod 20 cm long have the temperatures at 30°C and 80°C until steady-state prevails. The temperatures of the ends are changed to 40°C and 60°C respectively. Find the temperature distribution in the rod at time t.

10 marks

Answer

A tightly stretched string with fixed end points

x = 0

and

x = l

is initially in a position given by

y = y0 sin^2(πx/l)

. If it is released from rest from this position, find the displacement

y(x,t)

10 marks

JNTU B.Tech II MATHEMATICS-IV Previous Question Paper (February - 2024 - R16)

Part A

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Part B

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