Question

If \( P e^{x} = Q e^{-x} \) 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. \( \dfrac{P}{Q} = 0 \)

Hint:

If an equation involving \( e^{x} \) must be true for all real \( x \), any multiplying constant must be zero

Answer 1

The Correct Option is A

Step 1: Start with the given equation.
We are given: \[ P e^{x} = Q e^{-x} \] This must hold for all real values of \( x \).
Step 2: Rearranging the equation.
Multiply both sides by \( e^{x} \): \[ P e^{2x} = Q \] Step 3: Apply the condition “for all real \( x \)”.
The term \( e^{2x} \) changes value for different \( x \). Therefore, the only way \( P e^{2x} = Q \) can remain true for all \( x \) is when: \[ P = 0 \] If \( P = 0 \), then from the original equation: \[ 0 = Q e^{-x} \] This is possible only if: \[ Q = 0 \] Step 4: Checking the options.
(A) \( P = Q = 0 \): Correct — satisfies the equation for all \( x \).
(B) \( P = Q = 1 \): Incorrect — does not hold for all \( x \).
(C) \( P = 1, Q = -1 \): Incorrect — violates the equation.
(D) \( \frac{P}{Q} = 0 \): Not sufficient — does not guarantee equality for all \( x \).
Step 5: Conclusion.
The only values that satisfy the equation for all real \( x \) are: \[ \boxed{P = Q = 0} \]