F(z) is a function of the complex variable z = x + iy given by F(z) = iz + k Re(z) + i$$I$$m(z) For what value of k will F(z) satisfy the Cauchy-Riemann equations

A point z has been plotted in the complex plane, as shown in figure below.
Complex Variable mcq question image

The plot for point $$\frac{1}{{\text{z}}}$$ is

A

B

C

D

Contour C in the adjoining figure is described by x² + y² = 16. The value $$\oint\limits_{\text{c}} {\frac{{{{\text{z}}^2} + 8}}{{0.5{\text{z}} - 1.5{\text{j}}}}{\text{dz}}} $$ is (Note: $${\text{j}} = \sqrt { - 1} $$ )
Complex Variable mcq question image

A

-2πj

B

2πj

C

4πj

D

-4πj

Consider the line integral $$I = \int_{\text{c}} {\left( {{{\text{x}}^2} + {\text{i}}{{\text{y}}^2}} \right){\text{dz,}}} $$ where z = x + iy. The line c is shown in the figure below
Complex Variable mcq question image

The value of $$I$$ is

A

$$\frac{1}{2}{\text{i}}$$

B

$$\frac{2}{3}{\text{i}}$$

C

$$\frac{3}{4}{\text{i}}$$

D

$$\frac{4}{5}{\text{i}}$$

If the semi-circular contour D of radius 2 is as shown in the figure, then the value of the integral $$\oint\limits_{\text{D}} {\frac{1}{{\left( {{{\text{s}}^2} - 1} \right)}}{\text{ds}}} $$ is
Complex Variable mcq question image

A

jπ

B

-jπ

C

-π

D

π

যদি z=x+iy,z1=x1+iy1, z2=x2+iy2 তিনটি জটিল সংখ্যা হয়, তবে – i. Rez≤zii. argz1z2≤argz1+argz2iii.z1-z2≥z1-z2নিচের কোনটি সঠিক?

A

i ও ii

B

i ও iii

C

ii ও iii

D

i, ii ও iii

The figure shows the plot of y as a function of x
Differential Equations mcq question image

The function shown is the solution of the differential equation (assuming all initial conditions to be zero) is

A

$$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} = 1$$

B

$$\frac{{{\text{dy}}}}{{{\text{dx}}}} = {\text{x}}$$

C

$$\frac{{{\text{dy}}}}{{{\text{dx}}}} = - {\text{x}}$$

D

$$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \left| {\text{x}} \right|$$

Statement I The product of coefficient of correlation, standard deviation of variable 'X' and Standard deviation of variable 'Y' gives the measure of covariance between X and Y variables.
Statement II The product of coefficient of correlation between X and Y variables and the ratio between standard deviation of X variable to standard deviation of Y variable measures the slope of the regression line of X on Y variable.

A

Both statements are correct

B

Both statements are incorrect

C

Statement I is correct while Statement II is incorrect

D

Statement I is incorrect while Statement II is correct

In the following question, two statements are given each followed by two conclusions I and II. You have to consider the statements to be true even if theyseem to be at variance from commonly known facts. You have to decide which of the given conclusions, if any, follows from the given statements. Statements:
(I) Some polynomials are linear equations.
(II) Some linear equations are quadratic. Conclusion:
(I) Polynomials are quadratic.
(II) Linear equations are quadratic.

A

Conclusion I follows

B

Conclusion II follows

C

Neither I nor II follows

D

Both I and II follows

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

An analytic function f(z) of complex variable z = x + iy may be written as f(z) = u(x, y) + iv(x, y). Then, u(x, y) and v(x, y) must satisfy,

A

$$\frac{{\partial {\text{u}}}}{{\partial {\text{x}}}} = \frac{{ - \partial {\text{v}}}}{{\partial {\text{y}}}}{\text{ and }}\frac{{\partial {\text{u}}}}{{\partial {\text{y}}}} = \frac{{\partial {\text{v}}}}{{\partial {\text{x}}}}$$

B

$$\frac{{\partial {\text{u}}}}{{\partial {\text{x}}}} = \frac{{ - \partial {\text{v}}}}{{\partial {\text{y}}}}{\text{ and }}\frac{{\partial {\text{u}}}}{{\partial {\text{y}}}} = \frac{{ - \partial {\text{v}}}}{{\partial {\text{x}}}}$$

C

$$\frac{{\partial {\text{u}}}}{{\partial {\text{x}}}} = \frac{{\partial {\text{v}}}}{{\partial {\text{y}}}}{\text{ and }}\frac{{\partial {\text{u}}}}{{\partial {\text{y}}}} = \frac{{ - \partial {\text{v}}}}{{\partial {\text{x}}}}$$

D

$$\frac{{\partial {\text{u}}}}{{\partial {\text{x}}}} = \frac{{\partial {\text{v}}}}{{\partial {\text{y}}}}{\text{ and }}\frac{{\partial {\text{u}}}}{{\partial {\text{y}}}} = \frac{{\partial {\text{v}}}}{{\partial {\text{x}}}}$$

F(z) is a function of the complex variable z = x + iy given by F(z) = iz + k Re(z) + i$$I$$m(z) For what value of k will F(z) satisfy the Cauchy-Riemann equations

Related Questions

F(z) is a function of the complex variable z = x + iy given by F(z) = iz + k Re(z) + i$$I$$m(z)
For what value of k will F(z) satisfy the Cauchy-Riemann equations