JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I MATHEMATICAL METHODS Previous Question Paper (December - 2018 - R13)

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

Answer PaperPaper Code: 111ALMaximum Marks: 75Time: 3 hours

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.

Part A

125 marks
a

Prove that

E = ehD
where D is a differential operator.

2 marks

Answer

By definition, the shift operator E acts on a function f(x) as E f(x) = f(x+h). Using Taylor's series expansion for f(x+h):

f(x+h) = f(x) + hf'(x) + (h2/2!)f''(x) + ...

Since the differential operator D is defined as D f(x) = f'(x), we have Dn f(x) = f(n)(x). Substituting this:

f(x+h) = (1 + hD + (hD)2/2! + ...) f(x) = ehD f(x)

Thus, E = ehD.

b

Write the Normal equations for fitting a curve

y = a + b/x
from the given data points
(xi, yi)
.

3 marks

Answer

For the curve y = a + b/x, let u = 1/x, so y = a + bu. The sum of squares of residuals is S = Σ(yi - a - bui)2. Minimizing S with respect to a and b gives the normal equations:

Σyi = na + bΣ(1/xi)
Σ(yi/xi) = aΣ(1/xi) + bΣ(1/xi2)
c

Explain the solution of the equation

x + 1 = 0
graphically.

2 marks

Answer

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d

Evaluate

∫01 1/(1+x) dx
using Trapezoidal Rule.

3 marks

Answer

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e

Write the finite Fourier cosine transform of f(x) in [0,π].

2 marks

Answer

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f

Find the half range sine series of f(x) = x in [0,1].

3 marks

Answer

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g

Form PDE by eliminating constants from

z = ax + by + ab
.

2 marks

Answer

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h

Solve the PDE

px + qy = 1
.

3 marks

Answer

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i

State stokes theorem.

2 marks

Answer

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j

Find the angle between the surface

x2+y2+z2=9
and
z = x2+y2-3
at the point (2,-1,2).

3 marks

Answer

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Part B

210 marks
a

Find f(2.5) from the following table using Gauss Forward interpolation formula.

5 marks

Answer

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b

Fit the curve

y = a + bx + cx2
from the following table.

5 marks

Answer

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OR
310 marks
a

Find f(0.5) from the following table using Newton forward interpolation formula.

5 marks

Answer

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b

Find polynomial which is passing through the following points (1, 21), (3, 15), (4, 18), (6, 25).

5 marks

Answer

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410 marks
a

Find a positive root of

x3 - x - 2 = 0
using Bisection Method.

5 marks

Answer

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b

Solve the system of equation using LU decomposition method (Crout’s): x+5y+z = 14, 2x+y+3z = 13, 3x+y+4z = 17.

5 marks

Answer

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OR
510 marks
a

Evaluate y(1.2) and y(1.4) by RK method of fourth order if

y' = xy + 1, y(1) = 1
.

5 marks

Answer

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b

Solve the system of equation using Gauss –seidel method: 8x-3y+2z = 20, 4x+11y-z = 33, 6x+3y+12z = 35.

5 marks

Answer

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610 marks
a

Find the Fourier series of f(x).

5 marks

Answer

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b

Find the Fourier sine transform of f(x) defined by piecewise function.

5 marks

Answer

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OR
710 marks
a

Using Fourier integral, Show that

∫0∞ (sin λπ - λcos λπ)/(1 - λ2) * sin λx dλ
identity.

5 marks

Answer

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b

Find the Half range cosine series for f(x) = x+x2 in [0, 2].

5 marks

Answer

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810 marks
a

Solve the PDE

(x2-y2)z = px - qy
.

5 marks

Answer

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b

Solve

(ux) - 2(ut) = u, u(x,0) = 6e-3x
by method of separation of variables.

5 marks

Answer

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OR
910 marks
a

Solve the PDE

p(1-x2) + q(x) = y
.

5 marks

Answer

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b

Solve the Laplace equation

uxx + uyy = 0
with the given boundary conditions.

5 marks

Answer

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1010 marks
a

If f and g are any two vector functions, then show the identity involving curl, div and gradients.

5 marks

Answer

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b

Evaluate the line integral along the closed curve formed by y = 0, x = 1 and y = x by Green’s theorem.

5 marks

Answer

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OR
1110 marks
a

Prove that div curl f = 0 where f is a vector point function.

5 marks

Answer

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b

Evaluate the surface integral of A over cylinder surface.

5 marks

Answer

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This page contains the question paper of MATHEMATICAL METHODS from JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD for B.Tech IT I I-I, December - 2018, R13.

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