JEE Exam > JEE Questions > If a,b,c are integers not all equal andw is a...
Start Learning for Free
If a,b,c are integers not all equal andw is a cube root of unity (ω ≠ 1) then the minimum value of |a + bω+ cω2| is
- a)2
- b)a+b+c
- c)1
- d)0
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
If a,b,c are integers not all equal andw is a cube root of unity (&ome...
When a = b = 1, c = 2, it gives minimum value (since a,b,c not all equal)
Most Upvoted Answer
If a,b,c are integers not all equal andw is a cube root of unity (&ome...
If a, b, c are integers not all equal and w is a cube root of unity, we can express w as:
w = e^(2πi/3)
Let's consider the expression (a - bw)(a - cw)(b - cw). Expanding it, we have:
(a - bw)(a - cw)(b - cw) = a^2bc - ab^2w - ac^2w + abcw^2
Since w^2 = e^(4πi/3) = -e^(2πi/3), we can simplify the expression as follows:
(a - bw)(a - cw)(b - cw) = a^2bc - ab^2w - ac^2w + abcw^2
= a^2bc - ab^2w - ac^2w + abc(-w^2)
= a^2bc - ab^2w + ac^2w - abcw^2
= a^2bc - ab^2w + ac^2w + abcw^2
Now, let's consider the expression (a^2bc - ab^2w + ac^2w + abcw^2). We can see that it is a polynomial expression in terms of w. Since w is a cube root of unity, it satisfies the equation w^3 = 1. Therefore, we can substitute w^3 with 1 in the expression:
(a^2bc - ab^2w + ac^2w + abcw^2) = a^2bc - ab^2w + ac^2w + abcw^2
= a^2bc - ab^2(1) + ac^2(1) + abc(1)
= a^2bc - ab^2 + ac^2 + abc
Thus, the expression (a^2bc - ab^2w + ac^2w + abcw^2) is independent of the cube root of unity w and can be simplified to:
(a^2bc - ab^2w + ac^2w + abcw^2) = a^2bc - ab^2 + ac^2 + abc
w = e^(2πi/3)
Let's consider the expression (a - bw)(a - cw)(b - cw). Expanding it, we have:
(a - bw)(a - cw)(b - cw) = a^2bc - ab^2w - ac^2w + abcw^2
Since w^2 = e^(4πi/3) = -e^(2πi/3), we can simplify the expression as follows:
(a - bw)(a - cw)(b - cw) = a^2bc - ab^2w - ac^2w + abcw^2
= a^2bc - ab^2w - ac^2w + abc(-w^2)
= a^2bc - ab^2w + ac^2w - abcw^2
= a^2bc - ab^2w + ac^2w + abcw^2
Now, let's consider the expression (a^2bc - ab^2w + ac^2w + abcw^2). We can see that it is a polynomial expression in terms of w. Since w is a cube root of unity, it satisfies the equation w^3 = 1. Therefore, we can substitute w^3 with 1 in the expression:
(a^2bc - ab^2w + ac^2w + abcw^2) = a^2bc - ab^2w + ac^2w + abcw^2
= a^2bc - ab^2(1) + ac^2(1) + abc(1)
= a^2bc - ab^2 + ac^2 + abc
Thus, the expression (a^2bc - ab^2w + ac^2w + abcw^2) is independent of the cube root of unity w and can be simplified to:
(a^2bc - ab^2w + ac^2w + abcw^2) = a^2bc - ab^2 + ac^2 + abc
|
Explore Courses for JEE exam
|
|
Similar JEE Doubts
Top Courses for JEEView all
If a,b,c are integers not all equal andw is a cube root of unity (ω ≠ 1) then the minimum value of |a + bω+ cω2|isa)2b)a+b+cc)1d)0Correct answer is option 'C'. Can you explain this answer?
Question Description
If a,b,c are integers not all equal andw is a cube root of unity (ω ≠ 1) then the minimum value of |a + bω+ cω2|isa)2b)a+b+cc)1d)0Correct answer is option 'C'. 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 a,b,c are integers not all equal andw is a cube root of unity (ω ≠ 1) then the minimum value of |a + bω+ cω2|isa)2b)a+b+cc)1d)0Correct answer is option 'C'. 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 a,b,c are integers not all equal andw is a cube root of unity (ω ≠ 1) then the minimum value of |a + bω+ cω2|isa)2b)a+b+cc)1d)0Correct answer is option 'C'. Can you explain this answer?.
If a,b,c are integers not all equal andw is a cube root of unity (ω ≠ 1) then the minimum value of |a + bω+ cω2|isa)2b)a+b+cc)1d)0Correct answer is option 'C'. 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 a,b,c are integers not all equal andw is a cube root of unity (ω ≠ 1) then the minimum value of |a + bω+ cω2|isa)2b)a+b+cc)1d)0Correct answer is option 'C'. 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 a,b,c are integers not all equal andw is a cube root of unity (ω ≠ 1) then the minimum value of |a + bω+ cω2|isa)2b)a+b+cc)1d)0Correct answer is option 'C'. Can you explain this answer?.
Solutions for If a,b,c are integers not all equal andw is a cube root of unity (ω ≠ 1) then the minimum value of |a + bω+ cω2|isa)2b)a+b+cc)1d)0Correct answer is option 'C'. 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 a,b,c are integers not all equal andw is a cube root of unity (ω ≠ 1) then the minimum value of |a + bω+ cω2|isa)2b)a+b+cc)1d)0Correct answer is option 'C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
If a,b,c are integers not all equal andw is a cube root of unity (ω ≠ 1) then the minimum value of |a + bω+ cω2|isa)2b)a+b+cc)1d)0Correct answer is option 'C'. Can you explain this answer?, a detailed solution for If a,b,c are integers not all equal andw is a cube root of unity (ω ≠ 1) then the minimum value of |a + bω+ cω2|isa)2b)a+b+cc)1d)0Correct answer is option 'C'. Can you explain this answer? has been provided alongside types of If a,b,c are integers not all equal andw is a cube root of unity (ω ≠ 1) then the minimum value of |a + bω+ cω2|isa)2b)a+b+cc)1d)0Correct answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice If a,b,c are integers not all equal andw is a cube root of unity (ω ≠ 1) then the minimum value of |a + bω+ cω2|isa)2b)a+b+cc)1d)0Correct answer is option 'C'. Can you explain this answer? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup