If a = a_1 i + a_2 j + a_3 k and r = xi + yj + zk , then find ∇(r.a) .

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JNTU B.Tech II MATHEMATICS - II Previous Question Paper (April - 2018 - R15)

B.Tech | II | II | R15 | April - 2018

Answer PaperPaper Code: 124DDMaximum 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

1.a

a = a_1 i + a_2 j + a_3 k

and

r = xi + yj + zk

, then find

∇(r.a)

2 marks

Answer

Given the vectors

a = a_1 i + a_2 j + a_3 k

and

r = xi + yj + zk

, their dot product is

r ⋅ a = a_1 x + a_2 y + a_3 z

. The gradient operator is defined as

∇ = i(∂/∂x) + j(∂/∂y) + k(∂/∂z)

. Applying the gradient to the scalar product, we get

∇(r ⋅ a) = i(∂/∂x)(a_1 x + a_2 y + a_3 z) + j(∂/∂y)(a_1 x + a_2 y + a_3 z) + k(∂/∂z)(a_1 x + a_2 y + a_3 z)

. Calculating the partial derivatives yields

∇(r ⋅ a) = a_1 i + a_2 j + a_3 k = a

1.b

State Divergence theorem.

3 marks

Answer

The Divergence Theorem, also known as Gauss's Theorem, states that the outward flux of a vector field F across a closed surface S is equal to the triple integral of the divergence of F over the volume V enclosed by that surface.

∬_S F · n dS = ∭_V (∇ · F) dV

In this expression, F is a continuously differentiable vector field, S is a closed surface enclosing a volume V, and n is the outward-pointing unit normal vector to the surface S.

1.c

If f is defined as

f(x) = -x in (-π, 0) and x in (0, π)

, find a₀ in Fourier series.

2 marks

Answer

1.d

Write the Euler’s formulae for Fourier series in the interval (-λ, λ).

3 marks

Answer

1.e

Prove that

δ = E^{1/2} - E^{-1/2}

2 marks

Answer

1.f

If h = 1, evaluate

Δ(1/x!)

3 marks

Answer

1.g

Find the two values between which the root of

x^3 - cos x - 1 = 0

lies.

2 marks

Answer

1.h

A = [[3, 4], [6, 16]]

, find LU decomposition.

3 marks

Answer

1.i

dy/dx = x - y

and y(0) = 1, then find first approximation up to 2nd degree terms y(x) using Picard's method.

2 marks

Answer

1.j

y'' = x - y

, with h = 0.25, find the recurrence formula relating

y_{i-1}, y_i, y_{i+1}

3 marks

Answer

Part B

210 marks

Find the directional derivative of f(x,y,z)=xy²+yz³ at the point (2,-1,1) in the direction of the vector i + 2j + 2k.

5 marks

Answer

Evaluate

∫(yzdx + xzdy + xydz)

over a helix x = a cos t, y = a sin t, z = kt as t varies from 0 to 2π.

5 marks

Answer

Use Stoke’s Theorem to evaluate

∫((x + y) dx + (2x - z) dy + (y + z) dz)

where C is the boundary of the triangle with vertices (2,0,0), (0,3,0) and (0,0,6).

10 marks

Answer

410 marks

Obtain Fourier series for

f(x) = x^2 + x

-π < x < π

5 marks

Answer

Find the Fourier Transform of f(x) = cos x for 0 < x < a and 0 otherwise.

5 marks

Answer

Find the fourier sine transform of

1/(x^2 + a^2)

10 marks

Answer

610 marks

Find y(42) from the data: x=20, 25, 30, 35, 40, 45; y=352, 332, 291, 260, 231, 204 using Newton’s forward interpolation formula.

5 marks

Answer

Apply Lagrange’s formula to find f(5) given that f(1) = 2, f(2) = 4, f(3) = 8, f(4) = 16 and f(7) = 128.

5 marks

Answer

Fit an equation of the form y = ab^x to the following data: x = 2, 3, 4, 5, 6; y = 144, 172.8, 207.4, 248.8, 298.5.

10 marks

Answer

810 marks

Find a root of the equation sin x = 1 - x using Newton Raphson method.

5 marks

Answer

Find a real root of

x log_{10} x - 1.2 = 0

correct to four decimal places using Regula falsi method.

5 marks

Answer

Solve the system of equations using Gauss Seidal method: x + 5y + 2z = 7; 7x - y + z = 2; x + 2y + 5z = 9.

10 marks

Answer

Find y(0.1) and y(0.2) using Runge Kutta method given that

y' = x^2 - y, y(0) = 1

10 marks

Answer

Solve the boundary value problem

y'' - y = 2

with h = 0.5 given that y(0) = 0, y(2) = 2 by finite difference method.

10 marks

JNTU B.Tech II MATHEMATICS - II Previous Question Paper (April - 2018 - R15)

Part A

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Part B

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