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IPMAT Indore 2023 - QA (MCQ) Q18 Explanation
December 30, 2024
Explanation:
(b)
logn (log2a) = 1
⇒ log2 a = n^1
⇒ a = (2)^n ……………(i)
logn (log2b) = 2
⇒ log2 b= n^2
⇒ b = (2)^n^2 ………….(ii)
logn (log2 c) = 3
⇒ log2 c = n^3
⇒ c = (2)^n^3……………(iii)
We have, 𝑎 = 2^𝑛, 𝑏 = 2^𝑛^2and c = 2^𝑛^3
Now, let us substitute these values of a, b and c in each option and check the equality.
Option (a) a^n + b^n = c^n
LHS = a^n +b^n = (2^𝑛)^𝑛 +(2^𝑛^2)^𝑛 = 2^𝑛^2 + 2%𝑛^3
RHS = 𝑐^𝑛 = (2^𝑛^3)^𝑛 = 2^𝑛^4
We can see LHS ≠ RHS
Option (b) (a^n + b)^n = ac
LHS = (a^n +b)^n = ((2^𝑛)^𝑛 +2^𝑛^2)^𝑛 = (2^𝑛^2 +2^𝑛^2)^𝑛
= (2. 2^𝑛^2)^𝑛 = 2^𝑛(𝑛^2+1)
RHS = ac = 2^𝑛. 2^𝑛^3= 2^𝑛+𝑛^3 = 2^𝑛(1+𝑛^2) = 2^𝑛(𝑛^2+1)
Here LHS = RHS
Thus option (b) is right. We need not to check the other options.
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