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JNTU B.Tech I Year MATHEMATICAL METHODS Previous Question Paper (September - 2023 - R15)

B.Tech | I Year | I-I | R15 | September - 2023

Answer PaperPaper Code: 121ALMaximum Marks: 75Time: 3 Hours

Part A is compulsory, which carries 25 marks. In Part A, answer all questions. 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

125 marks
a

Show that

∑ Δ^2 f_k = Δf_n - Δf_0
for the range 0 to n-1.

2 marks

Answer

The forward difference operator is defined as Δfk = fk+1 - fk. The second forward difference is Δ2fk = Δ(Δfk) = Δfk+1 - Δfk. Summing from k=0 to n-1:

∑k=0n-1 Δ2fk = (Δf1 - Δf0) + (Δf2 - Δf1) + ... + (Δfn - Δfn-1)

This is a telescoping sum where all intermediate terms cancel, leaving Δfn - Δf0.

b

Show that

δ^2 y_5 = y_6 - 2y_5 + y_4
.

3 marks

Answer

The central difference operator δ is defined by δyi = yi+1/2 - yi-1/2. Therefore, δ2y5 = δ(δy5) = δ(y5.5 - y4.5). Applying the definition again:

δ2y5 = (y6 - y5) - (y5 - y4) = y6 - 2y5 + y4

Note: The provided question notation y1 likely refers to the index following y5 in the specific sequence context, resulting in y6 - 2y5 + y4.

c

Find a root of the equation

x e^x = 1
, lying between 0 and 1 using the bisection method.

2 marks

Answer

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d

Evaluate

∫ dx / (1+x^2)
from 0 to 1 using trapezoidal rule taking h = 1/4.

3 marks

Answer

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e

Find the value of the constant term in the Fourier series expansion of the function

f(x) = x
in (0, 2π).

2 marks

Answer

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f

State the change of scale property for the Fourier Transform.

3 marks

Answer

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g

Form the partial differential equation by eliminating the arbitrary constants

z = a e^(-b^2 x) cos(by)
.

2 marks

Answer

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h

Form the partial differential equation by eliminating the arbitrary function

z = x^2 f(x - y)
.

3 marks

Answer

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i

If

r^2 = x^2 + y^2 + z^2
then find
∇ . (r_vector / r)
.

2 marks

Answer

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j

Find the

curl(grad f)
, where
f = 2x^2 - 3y^2 + 4z^2
.

3 marks

Answer

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Part B

210 marks
a

The points (2,2), (5,4), (6,6), (9,9) and (11,10) should be approximated by a straight line. Find that line.

5 marks

Answer

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b

Using Newton’s forward difference formula, find a cubic polynomial for the given data.

5 marks

Answer

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OR
310 marks
a

Fit a polynomial to the data and predict f(1.5).

5 marks

Answer

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b

Fit a parabola to the given data.

5 marks

Answer

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410 marks
a

Solve

x log x = 1.2
using Regular falsi method.

5 marks

Answer

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b

Solve by the Euler’s method the Initial value problem

dy/dx = (x - y)/2
, y(0) = 1 over [0, 3] step size = 1/2.

5 marks

Answer

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OR
510 marks
a

Solve the system of equations using Gauss seidel iteration method with initial approximations zero, perform five iterations.

5 marks

Answer

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b

Given

y' = x^3 + y
, y(0) = 2 compute y(0.2), y(0.4) and y(0.6) using the Runge-Kutta method of fourth order.

5 marks

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610 marks
a

Determine if the function is even and find the Fourier series, then deduce the series sum.

5 marks

Answer

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b

Find the Fourier series for

f(x) = lx - x^2
in (0, 2l).

5 marks

Answer

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OR
710 marks
a

Find the Fourier transform of the given function and deduce the integral.

5 marks

Answer

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b

Find the inverse Fourier Sine Transform of the given expression.

5 marks

Answer

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810 marks
a

Solve

p - q = log(x + y)
.

5 marks

Answer

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b

Solve

y^2p - xyq = x(z - 2y)
.

5 marks

Answer

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OR
910 marks
a

Solve

sin(px)cos(y) = cos(px)sin(y) + p
.

5 marks

Answer

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b

Solve

p = log(px - y)
.

5 marks

Answer

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1010 marks
a

Prove that

∇(f^n) = n f^(n-1) ∇f
.

5 marks

Answer

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b

For what points does ∇f have the direction from P to the origin?

5 marks

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OR
1110 marks
a

Evaluate the line integral around the boundary C using Green’s Theorem.

5 marks

Answer

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b

Evaluate the surface integral using Stoke’s theorem.

5 marks

Answer

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About this paper

This page contains the question paper of MATHEMATICAL METHODS from JNTU for B.Tech EEE I Year I-I, September - 2023, R15.

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