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If A B and C are real numbers and A²+B²+C²=1 then AB+BC+CA=1 then AB+BC+CA can take values:- a) [-1,2] b) [-1/2,2] c) [-1,1] d [-1/2,1]?
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If A B and C are real numbers and A²+B²+C²=1 then AB+BC+CA=1 then AB+B...
Explanation:

Given:
A² + B² + C² = 1

To Find:
AB + BC + CA

Approach:
We can rewrite AB + BC + CA as 1/2[(A+B)² + (B+C)² + (C+A)² - A² - B² - C²].
From the given information, A² + B² + C² = 1, we can substitute this in the expression above to simplify.

Solution:
1. Start by expanding the expression:
AB + BC + CA = 1/2[(A+B)² + (B+C)² + (C+A)² - A² - B² - C²]
AB + BC + CA = 1/2[A² + B² + 2AB + B² + C² + 2BC + C² + A² + 2AC - A² - B² - C²]
AB + BC + CA = 1/2[2(A² + B² + C²) + 2(AB + BC + CA)]
AB + BC + CA = (A² + B² + C²) + (AB + BC + CA)
2. Substitute A² + B² + C² = 1:
AB + BC + CA = 1 + (AB + BC + CA)
3. Rearrange the equation:
AB + BC + CA - (AB + BC + CA) = 1
0 = 1
4. As the equation 0 = 1 is not true, the expression AB + BC + CA can take any real value.

Conclusion:
The expression AB + BC + CA can take any real value. Hence, the correct answer is:
d) [-1/2, 1]
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If A B and C are real numbers and A²+B²+C²=1 then AB+BC+CA=1 then AB+BC+CA can take values:- a) [-1,2] b) [-1/2,2] c) [-1,1] d [-1/2,1]?
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If A B and C are real numbers and A²+B²+C²=1 then AB+BC+CA=1 then AB+BC+CA can take values:- a) [-1,2] b) [-1/2,2] c) [-1,1] d [-1/2,1]? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If A B and C are real numbers and A²+B²+C²=1 then AB+BC+CA=1 then AB+BC+CA can take values:- a) [-1,2] b) [-1/2,2] c) [-1,1] d [-1/2,1]? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If A B and C are real numbers and A²+B²+C²=1 then AB+BC+CA=1 then AB+BC+CA can take values:- a) [-1,2] b) [-1/2,2] c) [-1,1] d [-1/2,1]?.
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