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Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x = cy + bz, y = az + cx, and z = bx + ay. Then a2 + b2 + c2 + 2abc is equal to
- a)2
- b)–1
- c)0
- d)1
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
Let a, b, c be any real numbers. Suppose that there are real numbers x...
The given equations are
–x + cy + bz = 0
cx –y + az = 0
bx + ay – z = 0
∵ x, y, z are not all zero
–x + cy + bz = 0
cx –y + az = 0
bx + ay – z = 0
∵ x, y, z are not all zero
∴ The above system should not have unique (zero) solution
⇒ –1(1– a2) – c(– c – ab) + b(ac + b) = 0
⇒–1 + a2 + b2 + c2 + 2abc = 0
⇒ a2 + b2 + c2 + 2abc = 1
⇒–1 + a2 + b2 + c2 + 2abc = 0
⇒ a2 + b2 + c2 + 2abc = 1
Most Upvoted Answer
Let a, b, c be any real numbers. Suppose that there are real numbers x...
Given Information:
a, b, c are real numbers.
x = cy + bz
y = az + cx
z = bx + ay
Solution:
Step 1: Finding a2 + b2 + c2 + 2abc
From the given equations, we can write:
x = cy + bz
=> x = c(az + cx) + b(bx + ay)
=> x = acz + c^2x + bbx + aby
=> (1 - ac)x = c^2x + b^2x
=> (1 - ac - c^2 - b^2)x = 0
Since x is not zero, we must have:
1 - ac - c^2 - b^2 = 0
=> a^2 + b^2 + c^2 + 2abc = 1
Therefore, a2 + b2 + c2 + 2abc = 1
Hence, the correct option is (D) 1.
a, b, c are real numbers.
x = cy + bz
y = az + cx
z = bx + ay
Solution:
Step 1: Finding a2 + b2 + c2 + 2abc
From the given equations, we can write:
x = cy + bz
=> x = c(az + cx) + b(bx + ay)
=> x = acz + c^2x + bbx + aby
=> (1 - ac)x = c^2x + b^2x
=> (1 - ac - c^2 - b^2)x = 0
Since x is not zero, we must have:
1 - ac - c^2 - b^2 = 0
=> a^2 + b^2 + c^2 + 2abc = 1
Therefore, a2 + b2 + c2 + 2abc = 1
Hence, the correct option is (D) 1.
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Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x = cy + bz, y = az + cx, and z = bx + ay. Then a2 + b2 + c2 + 2abc is equal toa)2b)–1c)0d)1Correct answer is option 'D'. Can you explain this answer?
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Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x = cy + bz, y = az + cx, and z = bx + ay. Then a2 + b2 + c2 + 2abc is equal toa)2b)–1c)0d)1Correct answer is option 'D'. 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 Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x = cy + bz, y = az + cx, and z = bx + ay. Then a2 + b2 + c2 + 2abc is equal toa)2b)–1c)0d)1Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x = cy + bz, y = az + cx, and z = bx + ay. Then a2 + b2 + c2 + 2abc is equal toa)2b)–1c)0d)1Correct answer is option 'D'. Can you explain this answer?.
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