If \(P{e^x} = Qe^{-x} \) for all real values of 𝑥, which one of the following…
2025
If \(P{e^x} = Qe^{-x} \) for all real values of 𝑥, which one of the following statements is true?
- A.
\(𝑃 = 𝑄 = 0\) - B.
\(𝑃 = 𝑄 = 1\) - C.
\(𝑃 = 1; 𝑄 = −1\) - D.
\(\frac{P}{Q} = 0 \)
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Correct answer: A
Reasoning: Multiply both sides of the given equation by e^x.
This gives P e^{2x} = Q for all real x.
If P ≠ 0 then e^{2x} would equal the constant Q/P for all x, but e^{2x} varies with x. This is impossible, so P must be 0.
With P = 0 the original equation becomes 0 = Q e^{-x} for all x, which forces Q = 0.
Conclusion: The only pair (P, Q) that satisfies the equation for every real x is P = Q = 0.
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