Question
Download Solution PDFIf xn - pyn + qzn is divisible by x2 + abyz - bzx - axy, then what is \(\rm \frac{p}{a^n}-\frac{q}{b^n}\) equal to ?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
xn - pyn + qzn is divisible by x2 + abyz - bzx - axy
Calculation:
We have
x2 + 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 (x2 + abyz - bzx - axy)
As, x2 + 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/bn = 0
⇒ 1 - (pyn/anyn) + q/bn = 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.
Last updated on May 29, 2025
-> The UPSC CDS 2 Notification has been released at upsconline.gov.in. for 453 vacancies.
-> Candidates can apply online from 28th May to 17th June 2025.
-> The CDS 2 Exam will be held on 14th September 2025.
-> Attempt UPSC CDS Free Mock Test to boost your score.
-> The selection process includes Written Examination, SSB Interview, Document Verification, and Medical Examination.
-> Refer to the CDS Previous Year Papers to enhance your preparation.