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JNTU B.Tech II MATHEMATICS-II Previous Question Paper (December - 2018 - R13)

B.Tech | II | II-II | R13 | December - 2018

Answer PaperPaper Code: 114DDMaximum 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. Each question carries 10 marks and may have a, b, c as sub questions.

PART- A

(25 Marks)

125 marks

State Divergence theorem.

2 marks

Answer

The Divergence Theorem states that the surface integral of a vector field F over a closed surface S is equal to the volume integral of the divergence of F over the volume V enclosed by S:

∬_S F · n dS = ∭_V (∇ · F) dV

In what direction from the point (−1, 2, 2) is the direction derivative of 𝜙 = 𝑥𝑦²𝑧³ a maximum. Find the magnitude of this maximum.

3 marks

Answer

The direction of maximum increase is given by the gradient ∇φ. Given φ = xy²z³, the gradient is ∇φ = (y²z³)i + (2xyz³)j + (3xy²z²)k. At (-1, 2, 2):

∇φ = (4·8)i + (2·-1·2·8)j + (3·-1·4·4)k = 32i - 32j - 48k

The magnitude is |∇φ| = √(32² + (-32)² + (-48)²) = √(1024 + 1024 + 2304) = √4352 = 8√68 = 16√17.

State the Dirichlet’s conditions of 𝑓(𝑥).

2 marks

Answer

Write the statement of Fourier integral theorem.

3 marks

Answer

With usual notations, show that 𝐸^(1/2) = 𝜇 + (1/2)𝛿

2 marks

Answer

What is least square curve fitting? Write down the normal equations for fitting a straight line for the set of points (𝑥𝑖, 𝑦𝑖), 𝑖 = 1, 2, … 𝑛.

3 marks

Answer

Explain bisection method.

2 marks

Answer

What is the condition for convergence of Gauss-seidel iteration method?

3 marks

Answer

Write the formula to evaluate

∫ y dx

from 𝑎 to 𝑏 by i) simpson’s 1/3rd rule ii) Simpson’s 3/8th rule.

2 marks

Answer

Write the merits and demerits of modified Euler’s method.

3 marks

Answer

PART-B

(50 Marks)

210 marks

If 𝐹 = (4𝑥𝑦 − 3𝑥²𝑧²)𝑖 + 2𝑥²𝑗 − 2𝑥³𝑧𝑘, prove that

∫ F ⋅ dr

along 𝐶 is independent of the curve joining two points.

5 marks

Answer

Evaluate

∫ F ⋅ dr

along 𝐶 by Stokes theorem if 𝐹 = 𝑦²𝑖 + 𝑥²𝑗 + (𝑥 + 𝑧)𝑘 and 𝐶 is the boundary of the region with vertices (0, 0, 0), (1, 0, 0) and (1, 1, 0).

5 marks

Answer

Verify Gauss Divergence theorem for 𝐹 = 2𝑥²𝑦𝑖 − 𝑦²𝑗 + 4𝑥𝑧²𝑘 taken over the region of the first octant of the cylinder 𝑦² + 𝑧² = 9 and 𝑥 = 2.

10 marks

Answer

410 marks

Obtain the Fourier series for the function 𝑓(𝑥) = (𝜋 − 𝑥)² , 0 < 𝑥 < 2𝜋. Hence deduce that 1/1² + 1/2² + 1/3² + ⋯ = 𝜋²/6.

5 marks

Answer

Find Fourier sine transform of 𝑥𝑒^(-𝑎𝑥).

5 marks

Answer

510 marks

If 𝑓(𝑥) = {𝑘𝑥, 0 ≤ 𝑥 ≤ 𝜋/2; 𝑘(𝜋 − 𝑥), 𝜋/2 ≤ 𝑥 ≤ 𝜋}, obtain the half-range sine series.

5 marks

Answer

Find the Fourier transform of 𝑓(𝑥) = {1, |𝑥| ≤ 𝑎; 0, |𝑥| > 𝑎}. Hence prove that

∫ (sin²(𝑎𝑥)/𝑥²) dx = 𝜋𝑎/2

from 0 to ∞.

5 marks

Answer

610 marks

Construct the difference table and evaluate 𝑌(2.5) from the following table: 𝑥: 1, 2, 3, 4, 5; 𝑌(𝑥): 4, 8, 32, 76, 102.

5 marks

Answer

Using Lagrange interpolation, fit a polynomial 𝑦(𝑥) for the following data and hence find 𝑓(1.5): 𝑥: -1, 1, 4, 5; 𝑓(𝑥): -2, 5, 12, 20.

5 marks

Answer

710 marks

Find the cubic polynomial that takes the following values and hence obtain 𝑦(4): 𝑦(0) = 1, 𝑦(1) = 0, 𝑦(2) = 1, and 𝑦(3) = 10.

5 marks

Answer

Using least square method, fit an exponential curve of the form 𝑦 = 𝑎𝑒^(𝑏𝑥) to the following data and hence evaluate Y(3.0). 𝑥: 1, 1.1, 1.2, 1.3, 1.4, 1.5; 𝑌(𝑥): 0.23, 0.67, 1.56, 3.56, 5.66, 10.23.

5 marks

Answer

Solve the following system of equations using Jacobi iteration method: 𝑥 + 10𝑦 + 𝑧 = 6; 10𝑥 + 𝑦 + 𝑧 = 6; 𝑥 + 𝑦 + 10𝑧 = 6.

10 marks

Answer

910 marks

Find a real root of the equation 𝑓(𝑥) = 𝑥 + log(𝑥) − 2 = 0 using Newton-Raphson method.

5 marks

Answer

Solve the system of equations 𝑥 + 𝑦 + 𝑧 = 1; 3𝑥 + 𝑦 − 3𝑧 = 5; 𝑥 − 2𝑦 − 5𝑧 = 10, using decomposition method.

5 marks

Answer

1010 marks

Evaluate

∫ sin(𝑥) dx

from 0 to 𝜋/2 using Simpson’s 1/3 rule with 6 sub intervals.

5 marks

Answer

Evaluate

∫ (1 + 𝑒^(2𝑥) + 𝑥²) dx

from -1 to 1 using Gauss-Legendre 1-point, 2-point and 3-point formula.

5 marks

Answer

Use Taylor’s method to solve the differential equation 𝑑𝑦/𝑑𝑥 = 𝑥 + 𝑦, 𝑦(0) = 1 and evaluate 𝑦(0.2) correct to 4 decimal places. Taking h = 0.1.

10 marks

JNTU B.Tech II MATHEMATICS-II Previous Question Paper (December - 2018 - R13)

PART- A

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PART-B

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