If xn - pyn + qzn is divisible by x+ abyz - bzx - axy, then what is \(\rm \frac{p}{a^n}-\frac{q}{b^n}\) equal to ?

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Answer (Detailed Solution Below)

Option 3 : 1
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Given:

xn - pyn + qzn is divisible by x+ abyz - bzx - axy

Calculation:

We have 

x+ abyz - bzx - axy

This can be written as

⇒ x2 - bzx - axy + abyz

⇒ x(x - bz) - ay(x - bz)

⇒ x2 + (bz - ay)x + abyz

⇒ (x - bz) (x - ay)

Therefore, bz & ay will be factor of (x+ abyz - bzx - axy)

As, x+ abyz - bzx - axy is factor of xn - pyn + qzn

We can write 

xn - pyn + qzn = (x - bz) (x - ay) × k    -----(1)

Put x = bz

⇒ bnzn - pyn + qzn = 0     -----(2)

Put x = ay

⇒ anyn - pyn + qzn = 0     -----(3)

Equation (3) - Equation (1) 

⇒ bnzn - anyn = 0

 bnzn = anyn

From equation (3)

⇒ bnzn - pyn + qzn = 0          (∵ bnzn = anyn)

⇒ 1 - (pyn/bnzn) + q/b= 0

⇒ 1 - (pyn/anyn) + q/b= 0    (∵ bnzn = anyn)

⇒ \(\rm \frac{p}{a^n}-\frac{q}{b^n}\) = 1

Hence, the value of \(\rm \frac{p}{a^n}-\frac{q}{b^n}\) = 1.

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