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

4xy - 2x2 + 2y2 + constant

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
Transform Theory mcq question image

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

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

Consider the following remarks pertaining to the irrotational flow: 1. The Laplace equation of stream function $\frac{{{\delta ^2}{\rm{\Psi }}}}{{\delta {x^2}}} + \frac{{{\delta ^2}{\rm{\Psi }}}}{{\delta {y^2}}} = 0$ must be satisfied for the flow to be potential. 2. The Laplace equation for the velocity potential $\frac{{{\delta ^2}{\rm{\varphi }}}}{{\delta {x^2}}} + \frac{{{\delta ^2}{\rm{\varphi }}}}{{\delta {y^2}}} = 0$ must be satisfied to fulfil the orate . of mass conservation i.e continuity equation.. Which of the above statements is/are correct?

A

1 only

B

Both 1 and 2

C

2 only

D

Neither 1 or 2

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

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

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

A

True

B

False

The value of y which satisfies the equation 5(y + 3) + 7(3y + 2) = (30y + 17) Also satisfies the equation

A

5(3y - 5) = 4y + 1

B

(y - 8) = 2(y - 3) + 5

C

2(y + 5) = 5(3y - 5) - 4

D

4(y - 3) = y + 3

The value of x which satisfies the equation 10 (x + 6) + 8(x - 3) = 5(5x - 4) Also satisfies the equation

A

3(3x - 5) = 2x + 1

B

2(x + 3) = 5(x - 5) + 4

C

5(x - 5) = 2(x - 3) + 5

D

5(x - 3) = x + 5

Consider the following statements in respect of an arbitrary complex number Z: 1. The difference of Z and its conjugate is an imaginary number. 2. The sum of Z and its conjugate is a real number. Which of the above statements is/are correct?

A

1 only

B

2 only

C

Both 1 and 2

D

Neither 1 nor 2

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(T_s)

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

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