The set of points in an Argand diagram which satisfy both `abs(z)le4` and `0 le arg(z) le pi/3`, is

Options

A. (a)a circle and a line
B. (b)a radius of a circle
C. (c)a sector of a circle
D. (d)an infinite part line

Answer

To solve the problem of finding the set of points in an Argand diagram that satisfy both conditions \( |z| \leq 4 \) and \( 0 \leq \arg(z) \leq \frac{\pi}{3} \), we can follow these steps:

### Step 1: Understand the condition \( |z| \leq 4 \)
The condition \( |z| \leq 4 \) represents all points inside and on the boundary of a circle centered at the origin (0,0) with a radius of 4.

### Step 2: Understand the condition \( 0 \leq \arg(z) \leq \frac{\pi}{3} \)
The argument \( \arg(z) \) represents the angle that the line connecting the point \( z \) to the origin makes with the positive x-axis. The condition \( 0 \leq \arg(z) \leq \frac{\pi}{3} \) means we are considering points in the first quadrant that lie between the positive x-axis (0 radians) and the line that makes an angle of \( \frac{\pi}{3} \) radians (or 60 degrees) with the positive x-axis.

### Step 3: Visualize the region
The intersection of these two conditions gives us a sector of the circle. The sector is defined by:
- The radius of the circle is 4 (from the first condition).
- The angle of the sector is \( \frac{\pi}{3} \) (from the second condition).

### Step 4: Identify the sector
The points that satisfy both conditions lie in the sector of the circle with:
- A radius of 4.
- An angle of \( 60^\circ \) (or \( \frac{\pi}{3} \)) in the first quadrant.

### Conclusion
Thus, the set of points that satisfy both conditions is a sector of a circle with radius 4 and an angle of \( 60^\circ \), starting from the positive x-axis and extending to the line at \( 60^\circ \).

### Final Answer
The correct option is the one that describes this sector of the circle.

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