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JNTU B.Tech I Year MATHEMATICAL METHODS Previous Question Paper (December - 2017 - R13)

B.Tech | I Year | I-I | R13 | December - 2017

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 and may have a, b, c as sub questions.

PART - A

125 marks

Prove that

h log 1 log 1 sin .           

2 marks

Answer

The operators are defined as follows: the forward difference operator

Δf(x) = f(x+h) - f(x)

, the backward difference operator

∇f(x) = f(x) - f(x-h)

, and the central difference operator

δf(x) = f(x+h/2) - f(x-h/2)

. Using the shift operator

E = e^(hD)

, we have

Δ = E - 1

∇ = 1 - E^(-1)

, and

δ = E^(1/2) - E^(-1/2)

. Thus,

μδ = (1/2)(E^(1/2) + E^(-1/2))(E^(1/2) - E^(-1/2)) = (1/2)(E - E^(-1)) = (1/2)(Δ + ∇)

y = a1 + a2x, Σy = 110, Σx = 30, Σx² = 15, Σxy = 55

then find a1.

3 marks

Answer

Using the normal equations for the line of best fit

y = a_1 + a_2x

, we have

Σy = na_1 + a_2Σx

and

Σxy = a_1Σx + a_2Σx^2

. Given

n=5

(assuming 5 data points based on provided sums),

110 = 5a_1 + 30a_2

and

55 = 30a_1 + 15a_2

. Solving the system,

55 = 30a_1 + 15((110-5a_1)/30)

leads to

110 = 60a_1 + 110 - 5a_1

, hence

55a_1 = 0

, so

a_1 = 0

Find square root of a number N by Newton-Raphson method.

2 marks

Answer

Find first approximation of

y' = x + y

y(0) = 1

by picards method.

3 marks

Answer

Express f(x) = x as a Fourier Series in

(-π, π).

2 marks

Answer

Prove that

F{f(ax)} = 1/|a| F(p/a)

where F(p) = Fourier transform of f(x).

3 marks

Answer

Eliminate arbitrary function from

z = f(x² + y² + z²).

2 marks

Answer

Find the general solution of

p + q = 1.

3 marks

Answer

φ = 3x²y - y³z²

find grad φ at (1, -2, -1).

2 marks

Answer

Find the curl of the vector

xyz i + x²y j - 3xz²y k.

3 marks

Answer

PART - B

Find f(2.5) using Newton’s forward formula and Lagranges interpolation formula.

10 marks

Table for Newton Forward: x(0,1,2,3,4,5,6), y(0,1,16,81,256,625,1296)

5 marks

Answer

Table for Lagrange: x(5,6,9,11), y(12,13,14,16)

5 marks

Answer

310 marks

Fit a second degree polynomial to data: x(0,1,2,3,4), y(1,1.8,1.3,2.5,6.3)

5 marks

Answer

Find parabola y = ax²+bx+c through (-1,2), (0,1), (1,4)

5 marks

Answer

410 marks

Find root of x-cos(x)=0 by bisection method.

5 marks

Answer

Find f'(6) from: x(0,2,3,4,7,9), f(x)(4,26,58,112,466,922)

5 marks

Answer

510 marks

Solve y'=x²-y, y(0)=1 using R-K 4th order.

5 marks

Answer

Use Milne’s method for y'=y+x².

5 marks

Answer

Find Fourier series of f(x) = sin(x) in (-π, π) and deduce sum.

10 marks

Answer

710 marks

Show Fourier integral result.

5 marks

Answer

Find inverse Fourier transform.

5 marks

Answer

810 marks

Solve u_x = u_t + u.

5 marks

Answer

String displacement problem.

5 marks

Answer

Temperature of an infinitely long plane plate.

10 marks

Answer

Show Gauss divergence theorem for F = z i + x j - 3y²z k over cube.

10 marks

Answer

Verify Stoke’s theorem for hemisphere.

10 marks

JNTU B.Tech I Year MATHEMATICAL METHODS Previous Question Paper (December - 2017 - R13)

PART - A

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PART - B

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