For two non-zero complex numbers z₁ and z₂, if Re(z₁z₂) = 0 and Re(z₁ + z₂), then which of the following are possible?
A. Im(z₁) > 0 and Im(z₂) > 0 |
B. Im(z₁) < 0 and Im(z₂) > 0
C. Im(z₁) > 0 and Im(z₂) < 0
D. Im(z₁) < 0 and Im(z₂) < 0
Choose the correct answer from the options given below

Question

If $50 \left( \frac{2x}{1 + 3i} + \frac{y}{1 - 2i} \right) = 31 + 17i$ where $x, y \in R$ & $i = \sqrt{-1}$ then value of $10(x + 3y)$ is ________

Answer 1

To solve this problem, we need to evaluate the given conditions for the complex numbers $ z_1 $ and $ z_2 $. Given the conditions:

Re($z_1 z_2$) = 0
Re($z_1 + z_2$)

Let's analyze these conditions:

Condition 1: Re($z_1 z_2$) = 0
- This implies the product of the two complex numbers $ z_1 $ and $ z_2 $ is purely imaginary.
Condition 2: Re($z_1 + z_2$)
- This condition is incomplete, but let us assume it should imply zero or another significant meaning.

We are given four options regarding the imaginary parts of $ z_1 $ and $ z_2 $:

A. Im($z_1$) > 0 and Im($z_2$) > 0
B. Im($z_1$) < 0 and Im($z_2$) > 0
C. Im($z_1$) > 0 and Im($z_2$) < 0
D. Im($z_1$) < 0 and Im($z_2$) < 0

Let's evaluate which options can satisfy the given conditions.

Evaluating Option A:
- If both imaginations are positive, $ z_1 z_2 $ would have a positive and a negative imaginary component, making the real part non-zero typically. Hence, this option cannot satisfy the condition Re($z_1 z_2$) = 0.
Evaluating Option B:
- For Im($z_1$) < 0 and Im($z_2$) > 0, the multiplying will result in Re($z_1 z_2$) = 0 more likely due to opposite imaginary signs. This option satisfies the conditions.
Evaluating Option C:
- If Im($z_1$) > 0 and Im($z_2$) < 0, again the real part of the product can be zero as the imaginations subtract off. So, this can satisfy the condition also.
Evaluating Option D:
- Both imaginary parts being negative would not typically lead to a real part of zero due to the product being purely imaginary condition. Thus, this option is not feasible.

Conclusion: The correct options are B and C, as these possibilities satisfy the condition Re($z_1 z_2$) = 0 where the product of their imaginary parts results in a zero real component.

Answer 2

To solve this problem, we need to evaluate the given conditions for the complex numbers $ z_1 $ and $ z_2 $. Given the conditions:

Re($z_1 z_2$) = 0
Re($z_1 + z_2$)

Let's analyze these conditions:

Condition 1: Re($z_1 z_2$) = 0
- This implies the product of the two complex numbers $ z_1 $ and $ z_2 $ is purely imaginary.
Condition 2: Re($z_1 + z_2$)
- This condition is incomplete, but let us assume it should imply zero or another significant meaning.

We are given four options regarding the imaginary parts of $ z_1 $ and $ z_2 $:

A. Im($z_1$) > 0 and Im($z_2$) > 0
B. Im($z_1$) < 0 and Im($z_2$) > 0
C. Im($z_1$) > 0 and Im($z_2$) < 0
D. Im($z_1$) < 0 and Im($z_2$) < 0

Let's evaluate which options can satisfy the given conditions.

Evaluating Option A:
- If both imaginations are positive, $ z_1 z_2 $ would have a positive and a negative imaginary component, making the real part non-zero typically. Hence, this option cannot satisfy the condition Re($z_1 z_2$) = 0.
Evaluating Option B:
- For Im($z_1$) < 0 and Im($z_2$) > 0, the multiplying will result in Re($z_1 z_2$) = 0 more likely due to opposite imaginary signs. This option satisfies the conditions.
Evaluating Option C:
- If Im($z_1$) > 0 and Im($z_2$) < 0, again the real part of the product can be zero as the imaginations subtract off. So, this can satisfy the condition also.
Evaluating Option D:
- Both imaginary parts being negative would not typically lead to a real part of zero due to the product being purely imaginary condition. Thus, this option is not feasible.

Conclusion: The correct options are B and C, as these possibilities satisfy the condition Re($z_1 z_2$) = 0 where the product of their imaginary parts results in a zero real component.

The Correct Option is A

Solution and Explanation

Top Questions on Complex Numbers and Quadratic Equations

Questions Asked in JEE Main exam

For two non-zero complex numbers z1 and z2, if Re(z1z2) = 0 and Re(z1 + z2), then which of the following are possible? A. Im(z1) > 0 and Im(z2) > 0 |B. Im(z1) < 0 and Im(z2) > 0 C. Im(z1) > 0 and Im(z2) < 0 D. Im(z1) < 0 and Im(z2) < 0 Choose the correct answer from the options given below

The Correct Option is A

Solution and Explanation

Top Questions on Complex Numbers and Quadratic Equations

Questions Asked in JEE Main exam