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the vaule of arg (x) when x lt 0 is (a)...

the vaule of arg (x) when `x lt 0` is (a) 0 (b) π /2 (c) π  (d) none of these

A

0

B

`(pi)/(2)`

C

`pi`

D

None of these

Text Solution

AI Generated Solution

The correct Answer is:
To find the value of \( \text{arg}(x) \) when \( x < 0 \), we can follow these steps: ### Step 1: Understand the Argument of a Complex Number The argument of a complex number \( z = x + iy \) is defined as the angle \( \theta \) that the line connecting the point \( (x, y) \) to the origin makes with the positive direction of the x-axis. ### Step 2: Identify the Case When \( x < 0 \) When \( x < 0 \), the complex number can be represented as \( z = x + i \cdot 0 \), which means it lies on the negative x-axis. Here, \( y = 0 \). ### Step 3: Visualize the Position of the Complex Number On the Cartesian plane, the negative x-axis is where all points have a negative x-coordinate and a y-coordinate of zero. The point \( (x, 0) \) where \( x < 0 \) is located to the left of the origin. ### Step 4: Determine the Angle The angle \( \theta \) that this point makes with the positive x-axis is \( \pi \) radians (or 180 degrees) because it is directly opposite to the positive x-axis. ### Step 5: Conclusion Thus, the value of \( \text{arg}(x) \) when \( x < 0 \) is \( \pi \). The correct answer is: (c) \( \pi \) ---
To find the value of \( \text{arg}(x) \) when \( x < 0 \), we can follow these steps: ### Step 1: Understand the Argument of a Complex Number The argument of a complex number \( z = x + iy \) is defined as the angle \( \theta \) that the line connecting the point \( (x, y) \) to the origin makes with the positive direction of the x-axis. ### Step 2: Identify the Case When \( x < 0 \) When \( x < 0 \), the complex number can be represented as \( z = x + i \cdot 0 \), which means it lies on the negative x-axis. Here, \( y = 0 \). ...
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