JEE Exam > JEE Questions > Let A, B, C be three complex numbers as defi...
Start Learning for Free
Let A, B, C be three complex numbers as defined below:
A = {z : Im z ≥ 1}
B = {z : |z - 2 - i| = 3}
C = {z : Re((1 - i)z) = 3√2}
The number of elements in the set A ∩ B ∩ C is
Correct answer is '1'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
Let A, B, C be three complex numbers as defined below:A = {z : Im z ≥...
Given Information:
Three complex numbers are defined as follows:
A = {z : Im z ≥ 1}
B = {z : |z - 2 - i| = 3}
C = {z : Re((1 - i)z) = 3√2}
Explanation:
To find the number of elements in the set A ∩ B ∩ C, we need to find the common complex numbers that satisfy all three conditions.
Step 1: Analyzing Set A
Set A is defined as {z : Im z ≥ 1}, which means the imaginary part of z is greater than or equal to 1. This represents a half-plane above the line y = 1 on the complex plane.
Step 2: Analyzing Set B
Set B is defined as {z : |z - 2 - i| = 3}, which means the distance between z and the point (2, 1) is 3. This represents a circle with center (2, 1) and radius 3 on the complex plane.
Step 3: Analyzing Set C
Set C is defined as {z : Re((1 - i)z) = 3√2}, which means the real part of ((1 - i)z) is equal to 3√2. Simplifying this equation, we get Re(z - iz) = 3√2, which further simplifies to Re(z) - Im(z) = 3√2. This represents a line in the complex plane.
Step 4: Finding the Common Elements
To find the common elements, we need to find the intersection of the half-plane, circle, and line.
The line defined by Set C intersects the circle defined by Set B at a single point. Let's call this point P.
Since the line intersects the circle at only one point, it means that the line does not intersect the circle at any other point.
Now, we need to check if this point P satisfies the condition of Set A. Since the imaginary part of P is equal to 1, it satisfies the condition of Set A.
Therefore, the number of elements in the set A ∩ B ∩ C is 1, which is the single common complex number that satisfies all three conditions.
Final Answer: The number of elements in the set A ∩ B ∩ C is 1.
Three complex numbers are defined as follows:
A = {z : Im z ≥ 1}
B = {z : |z - 2 - i| = 3}
C = {z : Re((1 - i)z) = 3√2}
Explanation:
To find the number of elements in the set A ∩ B ∩ C, we need to find the common complex numbers that satisfy all three conditions.
Step 1: Analyzing Set A
Set A is defined as {z : Im z ≥ 1}, which means the imaginary part of z is greater than or equal to 1. This represents a half-plane above the line y = 1 on the complex plane.
Step 2: Analyzing Set B
Set B is defined as {z : |z - 2 - i| = 3}, which means the distance between z and the point (2, 1) is 3. This represents a circle with center (2, 1) and radius 3 on the complex plane.
Step 3: Analyzing Set C
Set C is defined as {z : Re((1 - i)z) = 3√2}, which means the real part of ((1 - i)z) is equal to 3√2. Simplifying this equation, we get Re(z - iz) = 3√2, which further simplifies to Re(z) - Im(z) = 3√2. This represents a line in the complex plane.
Step 4: Finding the Common Elements
To find the common elements, we need to find the intersection of the half-plane, circle, and line.
The line defined by Set C intersects the circle defined by Set B at a single point. Let's call this point P.
Since the line intersects the circle at only one point, it means that the line does not intersect the circle at any other point.
Now, we need to check if this point P satisfies the condition of Set A. Since the imaginary part of P is equal to 1, it satisfies the condition of Set A.
Therefore, the number of elements in the set A ∩ B ∩ C is 1, which is the single common complex number that satisfies all three conditions.
Final Answer: The number of elements in the set A ∩ B ∩ C is 1.
Free Test
FREE
| Start Free Test |
Community Answer
Let A, B, C be three complex numbers as defined below:A = {z : Im z ≥...
A = |z : |mz ≥ 1|
B = {z : |z - 2 - i| = 3 }
C = {z : Re ((1 - i)2 = √2}
Let z = x + iy
Number of element in set A ∩ B ∩ C from the graph = 1
Attention JEE Students!
To make sure you are not studying endlessly, EduRev has designed JEE study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in JEE.
|
Explore Courses for JEE exam
|
|
Top Courses for JEEView all
Let A, B, C be three complex numbers as defined below:A = {z : Im z ≥ 1}B = {z : |z - 2 - i| = 3}C = {z : Re((1 - i)z) = 3√2}The number of elements in the set A ∩ B ∩ C isCorrect answer is '1'. Can you explain this answer?
Question Description
Let A, B, C be three complex numbers as defined below:A = {z : Im z ≥ 1}B = {z : |z - 2 - i| = 3}C = {z : Re((1 - i)z) = 3√2}The number of elements in the set A ∩ B ∩ C isCorrect answer is '1'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let A, B, C be three complex numbers as defined below:A = {z : Im z ≥ 1}B = {z : |z - 2 - i| = 3}C = {z : Re((1 - i)z) = 3√2}The number of elements in the set A ∩ B ∩ C isCorrect answer is '1'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let A, B, C be three complex numbers as defined below:A = {z : Im z ≥ 1}B = {z : |z - 2 - i| = 3}C = {z : Re((1 - i)z) = 3√2}The number of elements in the set A ∩ B ∩ C isCorrect answer is '1'. Can you explain this answer?.
Let A, B, C be three complex numbers as defined below:A = {z : Im z ≥ 1}B = {z : |z - 2 - i| = 3}C = {z : Re((1 - i)z) = 3√2}The number of elements in the set A ∩ B ∩ C isCorrect answer is '1'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let A, B, C be three complex numbers as defined below:A = {z : Im z ≥ 1}B = {z : |z - 2 - i| = 3}C = {z : Re((1 - i)z) = 3√2}The number of elements in the set A ∩ B ∩ C isCorrect answer is '1'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let A, B, C be three complex numbers as defined below:A = {z : Im z ≥ 1}B = {z : |z - 2 - i| = 3}C = {z : Re((1 - i)z) = 3√2}The number of elements in the set A ∩ B ∩ C isCorrect answer is '1'. Can you explain this answer?.
Solutions for Let A, B, C be three complex numbers as defined below:A = {z : Im z ≥ 1}B = {z : |z - 2 - i| = 3}C = {z : Re((1 - i)z) = 3√2}The number of elements in the set A ∩ B ∩ C isCorrect answer is '1'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning of Let A, B, C be three complex numbers as defined below:A = {z : Im z ≥ 1}B = {z : |z - 2 - i| = 3}C = {z : Re((1 - i)z) = 3√2}The number of elements in the set A ∩ B ∩ C isCorrect answer is '1'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Let A, B, C be three complex numbers as defined below:A = {z : Im z ≥ 1}B = {z : |z - 2 - i| = 3}C = {z : Re((1 - i)z) = 3√2}The number of elements in the set A ∩ B ∩ C isCorrect answer is '1'. Can you explain this answer?, a detailed solution for Let A, B, C be three complex numbers as defined below:A = {z : Im z ≥ 1}B = {z : |z - 2 - i| = 3}C = {z : Re((1 - i)z) = 3√2}The number of elements in the set A ∩ B ∩ C isCorrect answer is '1'. Can you explain this answer? has been provided alongside types of Let A, B, C be three complex numbers as defined below:A = {z : Im z ≥ 1}B = {z : |z - 2 - i| = 3}C = {z : Re((1 - i)z) = 3√2}The number of elements in the set A ∩ B ∩ C isCorrect answer is '1'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Let A, B, C be three complex numbers as defined below:A = {z : Im z ≥ 1}B = {z : |z - 2 - i| = 3}C = {z : Re((1 - i)z) = 3√2}The number of elements in the set A ∩ B ∩ C isCorrect answer is '1'. Can you explain this answer? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup