JEE Exam > JEE Questions > If a, b and c are distinct positive numbers, ...
Start Learning for Free
If a, b and c are distinct positive numbers, then the expression (b + c – a)(c + a – b)(a + b – c) – abc is (1986 - 2 Marks)
- a)positive
- b)negative
- c)non-positive
- d)non -negative
- e)none of these
Correct answer is option 'B'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
If a, b and c are distinct positive numbers, then the expression (b + ...
We can simplify the given expression as follows:
(b + c - a)(c + a - b)(a + b - c)
Using the identity (x + y - z)^2 = x^2 + y^2 + z^2 + 2xy - 2xz - 2yz, we can expand each of the three factors in the expression:
(b + c - a)^2 = b^2 + c^2 + a^2 + 2bc - 2ab - 2ac
(c + a - b)^2 = c^2 + a^2 + b^2 + 2ca - 2cb - 2ab
(a + b - c)^2 = a^2 + b^2 + c^2 + 2ab - 2ac - 2bc
Adding these three expressions and simplifying, we get:
2(a^2 + b^2 + c^2) + 2(ab + ac + bc) - (a^2 + b^2 + c^2 + 2ab + 2ac + 2bc)
= a^2 + b^2 + c^2 - 2ab - 2ac - 2bc
= (a - b)^2 + (a - c)^2 + (b - c)^2
This is always non-negative, and is equal to zero if and only if a = b = c. Therefore, the given expression is positive for all distinct positive numbers a, b, and c.
(b + c - a)(c + a - b)(a + b - c)
Using the identity (x + y - z)^2 = x^2 + y^2 + z^2 + 2xy - 2xz - 2yz, we can expand each of the three factors in the expression:
(b + c - a)^2 = b^2 + c^2 + a^2 + 2bc - 2ab - 2ac
(c + a - b)^2 = c^2 + a^2 + b^2 + 2ca - 2cb - 2ab
(a + b - c)^2 = a^2 + b^2 + c^2 + 2ab - 2ac - 2bc
Adding these three expressions and simplifying, we get:
2(a^2 + b^2 + c^2) + 2(ab + ac + bc) - (a^2 + b^2 + c^2 + 2ab + 2ac + 2bc)
= a^2 + b^2 + c^2 - 2ab - 2ac - 2bc
= (a - b)^2 + (a - c)^2 + (b - c)^2
This is always non-negative, and is equal to zero if and only if a = b = c. Therefore, the given expression is positive for all distinct positive numbers a, b, and c.
Free Test
| FREE | Start Free Test |
Community Answer
If a, b and c are distinct positive numbers, then the expression (b + ...
First term is less than ABC so result will be negative
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
|
|
Similar JEE Doubts
Top Courses for JEEView all
If a, b and c are distinct positive numbers, then the expression (b + c – a)(c + a – b)(a + b – c) – abc is (1986 - 2 Marks)a)positiveb)negativec)non-positived)non -negativee)none of theseCorrect answer is option 'B'. Can you explain this answer?
Question Description
If a, b and c are distinct positive numbers, then the expression (b + c – a)(c + a – b)(a + b – c) – abc is (1986 - 2 Marks)a)positiveb)negativec)non-positived)non -negativee)none of theseCorrect answer is option 'B'. 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 a, b and c are distinct positive numbers, then the expression (b + c – a)(c + a – b)(a + b – c) – abc is (1986 - 2 Marks)a)positiveb)negativec)non-positived)non -negativee)none of theseCorrect answer is option 'B'. 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 a, b and c are distinct positive numbers, then the expression (b + c – a)(c + a – b)(a + b – c) – abc is (1986 - 2 Marks)a)positiveb)negativec)non-positived)non -negativee)none of theseCorrect answer is option 'B'. Can you explain this answer?.
If a, b and c are distinct positive numbers, then the expression (b + c – a)(c + a – b)(a + b – c) – abc is (1986 - 2 Marks)a)positiveb)negativec)non-positived)non -negativee)none of theseCorrect answer is option 'B'. 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 a, b and c are distinct positive numbers, then the expression (b + c – a)(c + a – b)(a + b – c) – abc is (1986 - 2 Marks)a)positiveb)negativec)non-positived)non -negativee)none of theseCorrect answer is option 'B'. 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 a, b and c are distinct positive numbers, then the expression (b + c – a)(c + a – b)(a + b – c) – abc is (1986 - 2 Marks)a)positiveb)negativec)non-positived)non -negativee)none of theseCorrect answer is option 'B'. Can you explain this answer?.
Solutions for If a, b and c are distinct positive numbers, then the expression (b + c – a)(c + a – b)(a + b – c) – abc is (1986 - 2 Marks)a)positiveb)negativec)non-positived)non -negativee)none of theseCorrect answer is option 'B'. 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 and c are distinct positive numbers, then the expression (b + c – a)(c + a – b)(a + b – c) – abc is (1986 - 2 Marks)a)positiveb)negativec)non-positived)non -negativee)none of theseCorrect answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
If a, b and c are distinct positive numbers, then the expression (b + c – a)(c + a – b)(a + b – c) – abc is (1986 - 2 Marks)a)positiveb)negativec)non-positived)non -negativee)none of theseCorrect answer is option 'B'. Can you explain this answer?, a detailed solution for If a, b and c are distinct positive numbers, then the expression (b + c – a)(c + a – b)(a + b – c) – abc is (1986 - 2 Marks)a)positiveb)negativec)non-positived)non -negativee)none of theseCorrect answer is option 'B'. Can you explain this answer? has been provided alongside types of If a, b and c are distinct positive numbers, then the expression (b + c – a)(c + a – b)(a + b – c) – abc is (1986 - 2 Marks)a)positiveb)negativec)non-positived)non -negativee)none of theseCorrect answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice If a, b and c are distinct positive numbers, then the expression (b + c – a)(c + a – b)(a + b – c) – abc is (1986 - 2 Marks)a)positiveb)negativec)non-positived)non -negativee)none of theseCorrect answer is option 'B'. 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
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup