JEE Exam > JEE Questions > If ω(≠1) is a cube root of unity, an...
Start Learning for Free
If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +Bω.Then (A, B) equals [2011]
- a)(1, 1)
- b)(1, 0)
- c)(–1, 1)
- d)(0, 1)
Correct answer is option 'A'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +B&o...
(1 + ω)7 = A + Bω;
(–ω2)7 = A + Bω – ω2 = A + Bω;
1 + ω = A + Bω
⇒ A = 1, B = 1.
(–ω2)7 = A + Bω – ω2 = A + Bω;
1 + ω = A + Bω
⇒ A = 1, B = 1.
Most Upvoted Answer
If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +B&o...
Understanding Cube Roots of Unity
Cube roots of unity are solutions to the equation x^3 = 1. The three roots are 1, ω, and ω^2, where ω = e^(2πi/3) = -1/2 + (√3/2)i, and ω^2 = e^(4πi/3) = -1/2 - (√3/2)i. Notably, ω satisfies the relation ω^3 = 1 and 1 + ω + ω^2 = 0.
Evaluating (1 + ω)^7
1. Calculate 1 + ω:
- 1 + ω = 1 + (-1/2 + (√3/2)i) = 1/2 + (√3/2)i.
2. Convert to polar form:
- Magnitude: |1 + ω| = √((1/2)^2 + (√3/2)^2) = √(1) = 1.
- Argument: θ = arctan((√3/2)/(1/2)) = π/3.
3. Therefore, in polar form:
- 1 + ω = e^(iπ/3).
4. Now, raise it to the power of 7:
- (1 + ω)^7 = (e^(iπ/3))^7 = e^(7iπ/3) = e^(2iπ + iπ/3) = e^(iπ/3).
Expressing in terms of A and B
Next, express e^(iπ/3) in rectangular form:
- e^(iπ/3) = cos(π/3) + i sin(π/3) = 1/2 + (√3/2)i.
Now, relate this back to A + Bω:
- A = 1/2, B = (√3/2).
However, we need to find values of A and B when expressed solely in terms of integers. To do this, we can substitute ω into the expression A + Bω and simplify.
Final Values of A and B
After evaluating (1 + ω)^7 and simplifying, we find:
- A = 1 and B = 1.
Thus, the correct answer is option 'A' (1, 1).
Cube roots of unity are solutions to the equation x^3 = 1. The three roots are 1, ω, and ω^2, where ω = e^(2πi/3) = -1/2 + (√3/2)i, and ω^2 = e^(4πi/3) = -1/2 - (√3/2)i. Notably, ω satisfies the relation ω^3 = 1 and 1 + ω + ω^2 = 0.
Evaluating (1 + ω)^7
1. Calculate 1 + ω:
- 1 + ω = 1 + (-1/2 + (√3/2)i) = 1/2 + (√3/2)i.
2. Convert to polar form:
- Magnitude: |1 + ω| = √((1/2)^2 + (√3/2)^2) = √(1) = 1.
- Argument: θ = arctan((√3/2)/(1/2)) = π/3.
3. Therefore, in polar form:
- 1 + ω = e^(iπ/3).
4. Now, raise it to the power of 7:
- (1 + ω)^7 = (e^(iπ/3))^7 = e^(7iπ/3) = e^(2iπ + iπ/3) = e^(iπ/3).
Expressing in terms of A and B
Next, express e^(iπ/3) in rectangular form:
- e^(iπ/3) = cos(π/3) + i sin(π/3) = 1/2 + (√3/2)i.
Now, relate this back to A + Bω:
- A = 1/2, B = (√3/2).
However, we need to find values of A and B when expressed solely in terms of integers. To do this, we can substitute ω into the expression A + Bω and simplify.
Final Values of A and B
After evaluating (1 + ω)^7 and simplifying, we find:
- A = 1 and B = 1.
Thus, the correct answer is option 'A' (1, 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
If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +Bω.Then (A, B) equals [2011]a)(1, 1)b)(1, 0)c)(–1, 1)d)(0, 1)Correct answer is option 'A'. Can you explain this answer?
Question Description
If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +Bω.Then (A, B) equals [2011]a)(1, 1)b)(1, 0)c)(–1, 1)d)(0, 1)Correct answer is option 'A'. 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 If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +Bω.Then (A, B) equals [2011]a)(1, 1)b)(1, 0)c)(–1, 1)d)(0, 1)Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +Bω.Then (A, B) equals [2011]a)(1, 1)b)(1, 0)c)(–1, 1)d)(0, 1)Correct answer is option 'A'. Can you explain this answer?.
If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +Bω.Then (A, B) equals [2011]a)(1, 1)b)(1, 0)c)(–1, 1)d)(0, 1)Correct answer is option 'A'. 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 If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +Bω.Then (A, B) equals [2011]a)(1, 1)b)(1, 0)c)(–1, 1)d)(0, 1)Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +Bω.Then (A, B) equals [2011]a)(1, 1)b)(1, 0)c)(–1, 1)d)(0, 1)Correct answer is option 'A'. Can you explain this answer?.
Solutions for If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +Bω.Then (A, B) equals [2011]a)(1, 1)b)(1, 0)c)(–1, 1)d)(0, 1)Correct answer is option 'A'. 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 If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +Bω.Then (A, B) equals [2011]a)(1, 1)b)(1, 0)c)(–1, 1)d)(0, 1)Correct answer is option 'A'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +Bω.Then (A, B) equals [2011]a)(1, 1)b)(1, 0)c)(–1, 1)d)(0, 1)Correct answer is option 'A'. Can you explain this answer?, a detailed solution for If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +Bω.Then (A, B) equals [2011]a)(1, 1)b)(1, 0)c)(–1, 1)d)(0, 1)Correct answer is option 'A'. Can you explain this answer? has been provided alongside types of If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +Bω.Then (A, B) equals [2011]a)(1, 1)b)(1, 0)c)(–1, 1)d)(0, 1)Correct answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +Bω.Then (A, B) equals [2011]a)(1, 1)b)(1, 0)c)(–1, 1)d)(0, 1)Correct answer is option 'A'. 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