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JNTU B.Tech II MATHEMATICS-II Previous Question Paper (May/June - 2019 - R15)

B.Tech | II | II-I | R15 | May/June - 2019

Answer PaperPaper Code: 123AAMaximum 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

(25 Marks)

Find curl f where

f = xyi + yzj + zxk

2 marks

Answer

The curl of a vector field

f = F₁i + F₂j + F₃k

is defined as the cross product of the del operator and the vector field, given by

∇ × f = (∂F₃/∂y - ∂F₂/∂z)i + (∂F₁/∂z - ∂F₃/∂x)j + (∂F₂/∂x - ∂F₁/∂y)k

Given

f = xyi + yzj + zxk

, we have

F₁ = xy

F₂ = yz

, and

F₃ = zx

. Calculating the partial derivatives:

∂F₃/∂y = 0, ∂F₂/∂z = y

∂F₁/∂z = 0, ∂F₃/∂x = z

∂F₂/∂x = 0, ∂F₁/∂y = x

Substituting these into the formula, we get

curl f = (0 - y)i + (0 - z)j + (0 - x)k = -yi - zj - xk

Find the circulation of an irrotational vector function over the rectangle

x = ±a, y = ±b

3 marks

Answer

The circulation of a vector field F around a closed curve C is defined by the line integral ∮_C F · dr. According to Stokes' Theorem, this line integral is equal to the surface integral of the curl of the vector field over the surface S enclosed by the curve, given by ∬_S (∇ × F) · dS.

Since the vector function is stated to be irrotational, by definition, its curl is zero everywhere in the region, i.e., ∇ × F = 0. Therefore, the surface integral of the curl over the rectangle defined by x = ±a and y = ±b is ∬_S 0 · dS = 0. Consequently, the circulation of an irrotational vector function over any closed path, including the specified rectangle, is 0.

Find the leading term in the Fourier series expansion of

f(x) = e^x

(−π, π)

2 marks

Answer

State Fourier integral theorem.

3 marks

Answer

Write the normal equations for a quadratic curve.

2 marks

Answer

Prove that

Δ^2 x^(m) = m(m-1)x^(m-2)

, m is a positive integer and h = 1.

3 marks

Answer

What is Interpolation?

2 marks

Answer

Under what conditions Gauss-Seidal method is applicable.

3 marks

Answer

Write the iteration formulae of modified Euler’s method.

2 marks

Answer

dy/dx = x + y

and y(0), h = 0.1 then find K1 and K2 in Runge Kutta fourth order formula.

3 marks

Answer

Part B

(50 Marks)

f = 3x^2yz i + x^2z^2 j + 2x^3yz k

. Show that

∫_C f · dr

is independent of the path of integration. Hence evaluate the integral when C is any path joining (0, 0, 0) to (1, 2, 3).

10 marks

Answer

Prove that if

is the position vector of any point in space then

r^n r

is irrotational and is solenoidal if n = -3.

10 marks

Answer

410 marks

State and prove shifting property of Fourier transform.

5 marks

Answer

Find the inverse Fourier sine transform (x) of

Fs{p} = e^{-ap}/p

and hence deduce

Fs^{-1}{1/p}

5 marks

Answer

Find the Fourier series expansion of the function

f(x) = { 0, -π ≤ x ≤ 0; sin x, 0 ≤ x ≤ π }

and deduce

1/(1·3) - 1/(3·5) + 1/(5·7) + ⋯ = (π-2)/4

10 marks

Answer

Fit a curve of the form

y = a/x + bx

to the data given below:

x	1.0	1.2	1.4	1.6	1.8
y	0	0.1280	0.5440	1.2960	2.4320

10 marks

Answer

710 marks

Given

u0 + u8 = 1.924; u1 + u7 = 1.959; u2 + u6 = 1.982 and u3 + u5 = 1.996

find

, stating the assumption made.

5 marks

Answer

Given

sin 45° = 0.7071, sin 50° = 0.7660, sin 55° = 0.8192 and sin 60° = 0.8660

. Find

sin 52°

using Newton’s interpolation formula.

5 marks

Answer

By using Regula Falsi method, find a positive root of the equation

tan x + tanh x = 0

10 marks

Answer

Solve the system of equations by Gauss seidal method:

4x + 2y + z = 10, x + 2y + 7z = 12, 5x + 8y - 3z = -1

10 marks

Answer

Find the successive approximate solution of the differential equation

y' = y, y(0) = 1

by Picard’s method and compare it with exact solution.

10 marks

Answer

Using the shooting method, solve the first boundary value problem:

u'' = u + 1, 0 < x < 1; u(0) = 0; u(1) = e - 1

10 marks

JNTU B.Tech II MATHEMATICS-II Previous Question Paper (May/June - 2019 - R15)

Part A

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Part B

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