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If z1, z2, z3 are complex numbers such that
|z1| = |z2| = |z3| =|1/z1 +1/z2 + 1/z3| = 1, then find the value of z1+ z2 + z3 .
- a)equal to 1
- b)less than 1
- c)greater than 3
- d)equal to 3
Correct answer is option 'A'. Can you explain this answer?
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If z1, z2, z3are complex numbers such that|z1| = |z2| = |z3| =|1/z1+1/...
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If z1, z2, z3are complex numbers such that|z1| = |z2| = |z3| =|1/z1+1/...
Solution:
Given that |z1| = |z2| = |z3| = |1/z1 1/z2 1/z3| = 1.
To find the value of z1 * z2 * z3, we need to simplify the given condition step by step.
1. |z1| = |z2| = |z3| = 1
The modulus of complex number z1, z2, and z3 is 1. This means that all three complex numbers lie on the unit circle in the complex plane.
2. |1/z1 1/z2 1/z3| = 1
The modulus of the complex number 1/z1, 1/z2, and 1/z3 is also 1. This implies that the reciprocals of z1, z2, and z3 also lie on the unit circle.
Now, let's consider the product z1 * z2 * z3.
Since z1, z2, and z3 lie on the unit circle, they can be represented as:
z1 = e^(iθ1)
z2 = e^(iθ2)
z3 = e^(iθ3)
where θ1, θ2, and θ3 are the arguments (angles) of z1, z2, and z3 respectively.
Now, let's consider the product 1/z1 * 1/z2 * 1/z3.
Since the reciprocals of z1, z2, and z3 lie on the unit circle, they can be represented as:
1/z1 = e^(-iθ1)
1/z2 = e^(-iθ2)
1/z3 = e^(-iθ3)
Now, let's consider the product (1/z1) * (1/z2) * (1/z3):
(1/z1) * (1/z2) * (1/z3) = e^(-iθ1) * e^(-iθ2) * e^(-iθ3)
= e^(-i(θ1 + θ2 + θ3))
The product of the reciprocals of z1, z2, and z3 is equal to e^(-i(θ1 + θ2 + θ3)).
Since the modulus of this product is 1, it means that the argument of this product is an integer multiple of 2π.
Therefore, θ1 + θ2 + θ3 = 2nπ, where n is an integer.
Now, substituting the values of z1, z2, and z3:
z1 * z2 * z3 = e^(iθ1) * e^(iθ2) * e^(iθ3)
= e^(i(θ1 + θ2 + θ3))
= e^(i(2nπ))
= cos(2nπ) + i sin(2nπ)
= 1 + 0i
= 1
Therefore, the value of z1 * z2 * z3 is equal to 1.
Hence, the correct answer is option 'A'.
Given that |z1| = |z2| = |z3| = |1/z1 1/z2 1/z3| = 1.
To find the value of z1 * z2 * z3, we need to simplify the given condition step by step.
1. |z1| = |z2| = |z3| = 1
The modulus of complex number z1, z2, and z3 is 1. This means that all three complex numbers lie on the unit circle in the complex plane.
2. |1/z1 1/z2 1/z3| = 1
The modulus of the complex number 1/z1, 1/z2, and 1/z3 is also 1. This implies that the reciprocals of z1, z2, and z3 also lie on the unit circle.
Now, let's consider the product z1 * z2 * z3.
Since z1, z2, and z3 lie on the unit circle, they can be represented as:
z1 = e^(iθ1)
z2 = e^(iθ2)
z3 = e^(iθ3)
where θ1, θ2, and θ3 are the arguments (angles) of z1, z2, and z3 respectively.
Now, let's consider the product 1/z1 * 1/z2 * 1/z3.
Since the reciprocals of z1, z2, and z3 lie on the unit circle, they can be represented as:
1/z1 = e^(-iθ1)
1/z2 = e^(-iθ2)
1/z3 = e^(-iθ3)
Now, let's consider the product (1/z1) * (1/z2) * (1/z3):
(1/z1) * (1/z2) * (1/z3) = e^(-iθ1) * e^(-iθ2) * e^(-iθ3)
= e^(-i(θ1 + θ2 + θ3))
The product of the reciprocals of z1, z2, and z3 is equal to e^(-i(θ1 + θ2 + θ3)).
Since the modulus of this product is 1, it means that the argument of this product is an integer multiple of 2π.
Therefore, θ1 + θ2 + θ3 = 2nπ, where n is an integer.
Now, substituting the values of z1, z2, and z3:
z1 * z2 * z3 = e^(iθ1) * e^(iθ2) * e^(iθ3)
= e^(i(θ1 + θ2 + θ3))
= e^(i(2nπ))
= cos(2nπ) + i sin(2nπ)
= 1 + 0i
= 1
Therefore, the value of z1 * z2 * z3 is equal to 1.
Hence, the correct answer is option 'A'.
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If z1, z2, z3are complex numbers such that|z1| = |z2| = |z3| =|1/z1+1/z2+ 1/z3| = 1, then find the value of z1+ z2+ z3.a)equal to 1b)less than 1c)greater than 3d)equal to 3Correct answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If z1, z2, z3are complex numbers such that|z1| = |z2| = |z3| =|1/z1+1/z2+ 1/z3| = 1, then find the value of z1+ z2+ z3.a)equal to 1b)less than 1c)greater than 3d)equal to 3Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If z1, z2, z3are complex numbers such that|z1| = |z2| = |z3| =|1/z1+1/z2+ 1/z3| = 1, then find the value of z1+ z2+ z3.a)equal to 1b)less than 1c)greater than 3d)equal to 3Correct answer is option 'A'. Can you explain this answer?.
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