A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 - 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) is

Laplace transform of the function f(t) is given by $${\text{F}}\left( {\text{s}} \right) = {\text{L}}\left\{ {{\text{f}}\left( {\text{t}} \right)} \right\} = \int_0^\infty {{\text{f}}\left( {\text{t}} \right){{\text{e}}^{ - {\text{st}}}}{\text{dt}}{\text{.}}} $$ Laplace transform of the function shown below is given by

A

$$\frac{{1 - {{\text{e}}^{ - 2{\text{s}}}}}}{{\text{s}}}$$

B

$$\frac{{1 - {{\text{e}}^{ - {\text{s}}}}}}{{2{\text{s}}}}$$

C

$$\frac{{2 - 2{{\text{e}}^{ - {\text{s}}}}}}{{\text{s}}}$$

D

$$\frac{{1 - 2{{\text{e}}^{ - {\text{s}}}}}}{{\text{s}}}$$

A complex function f(z) = u(x, y) + iv(x, y) and its complex conjugate, f'(z) = u(x, y) - iv(x, y) are both analytic in the entire complex plane, where z = x + iy and $${\text{i}} = \sqrt { - 1} .$$ The function f is then given by

A

f(z) = x + iy

B

f(z) = x2 - y2 + i2xy

C

f(z) = constant

D

f(z) = x2 + y2

The given equation satisfies the Laplace equation. V = x2 + y2 – z2. State True/False.

A

True

B

False

Consider the following assertions.
1. Synthetic apirori judgment is the standard of knowledge.
2. Synthetic a posteriori judgement is the standard of knowledge.
3. Analytic apriori judgement is the standard of knowledge.
4. Analytic a posteriori judgement is the standard of knowledge.
Select the correct answer:

A

Only 1 is correct

B

Only 2 is correct

C

1 and 4 are correct

D

Only 3 is correct

Consider the following assertions
1. Synthetic apriori judgement is the standard of knowledge
2. Synthetic a posteriori judgement is the standard of knowledge.
3. Analytic apriori judgement is the standard of knowledge.
4. Analytic a posteriori judgement is the standard of knowledge

A

Only 1 is correct

B

Only 1 and 4 are correct

C

Only 2 is correct

D

Only 3 is correct

When a potential satisfies Laplace equation, then it is said to be

A

Solenoidal

B

Divergent

C

Lamellar

D

Harmonic

If Laplace equation satisfies, then which of the following statements will be true?

A

Potential will be zero

B

Current will be infinite

C

Resistance will be infinite

D

Voltage will be same

The Laplace transform of a function f(t) u(t), where f(t) is periodic with period T, is A(s) times the Laplace transform of its first period. Then

A

A(s) = s

B

$$A\left( s \right) = {1 \over {\left( {1 - \exp \left( { - {T_s}} \right)} \right)}}$$

C

$$A\left( s \right) = {1 \over {\left( {1 + \exp \left( { - {T_s}} \right)} \right)}}$$

D

A(s) = exp(Ts)

The function f(t) satisfies the differential equation $$\frac{{{{\text{d}}^2}{\text{f}}}}{{{\text{d}}{{\text{t}}^2}}} + {\text{f}} = 0$$ and the auxiliary conditions, f(0) = 0, $$\frac{{{\text{df}}}}{{{\text{dt}}}}\left( 0 \right) = 4.$$ The Laplace transform of f(t) is given by

A

$$\frac{2}{{{\text{s}} + 1}}$$

B

$$\frac{4}{{{\text{s}} + 1}}$$

C

$$\frac{4}{{{{\text{s}}^2} + 1}}$$

D

$$\frac{2}{{{{\text{s}}^2} + 1}}$$

Given the potential V = 25 sin θ, in free space, determine whether V satisfies Laplace’s equation.

A

Yes

B

No

C

Data sufficient

D

Potential is not defined