COMEDK · Maths · 3. Complex Number

If \(z=r e^{i \theta}\), then \(\left|e^{i z}\right|=\)

A 1
B \(e^{2 r \sin \theta}\)
C \(e^{r \sin \theta}\)
D \(e^{-r \sin \theta}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(e^{-r \sin \theta}\)

Step-by-step Solution

Detailed explanation

We have,…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

The statement \(P(n): ' 1 \times 1 !+2 \times 2 !+3 \times 3\) !
\(+\ldots+n \times n !=(n+1) !-1\) is

COMEDK 2017 Medium
The altitude of a cone is 20 cm and its semi vertical angle is \(30^\circ\). If the semi vertical angle is increasing at the rate of 2 radians per second, then the radius of the base is increasing at the rate of

COMEDK 2023 Medium
The general solution of \(|\sin x|=\cos x\) is (when \(n \in Z\) ) given by

COMEDK 2019 Medium
The solution of the differential equation \(\left(1+y^{2}\right)+\left(x-e^{\tan ^{-1} y}\right) \dfrac{d y}{d x}=0\) is

COMEDK 2021 Hard
Let \((3,4,-1)\) and \((-1,2,3)\) be the end points of a diameter of a sphere. Then, the radius of the sphere is equal to

COMEDK 2017 Easy
The conjugate of the complex number \(\frac{(1+i)^{2}}{1-i}\) is

COMEDK 2020 Easy

Explore more questions on app

From COMEDK

If \(f(x) = \begin{cases} \dfrac{\sqrt{1+x} - \sqrt{1-x}}{\sin x}, & x \neq 0 \\ k, & x = 0 \end{cases}\) is continuous at \(x = 0\), then \(k =\)

COMEDK 2026 Medium
A thread is tied slightly loose to a wire frame as in figure and the frame is dipped into a soap solution and taken out. The frame is completely covered with the film. When the portion \(A\) is punctured with a pin, the thread

COMEDK 2019 Medium
If the vectors \(\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}\) and \(\mathbf{c}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\) are coplanar, then the value of \(\lambda\) is

COMEDK 2024 Easy
The number of common tangents to the circles \(x^{2}+y^{2}-2 x-4 y+1=0\) and \(x^{2}+y^{2}-12 x-16 y+91=0\) is

COMEDK 2012 Easy
A straight line passes through the point \(P(\log_2 16 , \log_3 27)\) such that the portion of the line intercepted between the co-ordinate axes is divided by \(P\) in the ratio \(1:2\) internally (starting from the \(x\)-axis). Then the equation of the line is:

COMEDK 2026 Easy
When 2-bromopentane is heated with alcoholic solution of potassium hydroxide, the major product obtained is

COMEDK 2014 Easy

Explore more questions on app