Let n ≥ 5 be an integer. If 9 ^ n - 8n - 1 = 64α and 6 ^ n - 5n - 1 = 25β , then α - β is equal to:

Correct answer: C. C3 n (8 - 5) + C4 n ( 8 ^ 2 - 5 ^ 2 ) + + Cn n ( 8 ^ n - 2 - 5 ^ n - 2 ) Using the binomial expansion (1 + x) ^ n = C0 n + C1 n x + C2 n x ^ 2 . . . . . . . + Cn n x ^ n We get, α = (1 + 8) ^ n - 8n - 1 64 = C2 n + C3 n 8 + C4 n 8 ^ 2 + Similarly β = C2 n + C3 n 5 + C4 n 5 ^ 2 + . So α - β = C3 n (8 - 5) + C4 n ( 8 ^ 2 - 5 ^ 2 ) + + Cn n ( 8 ^ n - 2 - 5 ^ n - 2 ) So, option (C) is the answer.

JEE Main 202229 JuneEvening Shift

Exam: JEE Main
Year: 2022
Date: 29 June
Shift: Evening Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2022 · 29 June · Evening Shift · Mathematics · Binomial Theorem

Let n ≥ 5 be an integer. If 9 ^ n - 8n - 1 = 64α and 6 ^ n - 5n - 1...

Let $n \ge 5$ be an integer. If ${9}^{n} - 8n - 1 = 64\alpha$ and ${6}^{n} - 5n - 1 = 25\beta$ , then $\alpha - \beta$ is equal to:

Options

A $1 + C2 n \left(8 - 5\right) + C3 n \left({8}^{2} - {5}^{2}\right) + \ldots + Cn n \left({8}^{n - 1} - {5}^{n - 2}\right)$
B $1 + C3 n \left(8 - 5\right) + C4 n \left({8}^{2} - {5}^{2}\right) + \ldots + Cn n \left({8}^{n - 2} - {5}^{n - 2}\right)$
C $C3 n \left(8 - 5\right) + C4 n \left({8}^{2} - {5}^{2}\right) + \ldots + Cn n \left({8}^{n - 2} - {5}^{n - 2}\right)$
D $C4 n \left(8 - 5\right) + C5 n \left({8}^{2} - {5}^{2}\right) + \ldots + Cn n \left({8}^{n - 3} - {5}^{n - 3}\right)$

Correct answer

C. $C3 n \left(8 - 5\right) + C4 n \left({8}^{2} - {5}^{2}\right) + \ldots + Cn n \left({8}^{n - 2} - {5}^{n - 2}\right)$

Step-by-step solution

Using the binomial expansion ${\left(1 + x\right)}^{n} = C0 n + C1 n x + C2 n {x}^{2} . . . . . . . + Cn n {x}^{n}$

We get,

$\alpha = \dfrac{\left(1 + 8\right){}^{n} - 8n - 1}{64} = C2 n + C3 n 8 + C4 n {8}^{2} + \ldots$ Similarly $\beta = C2 n + C3 n 5 + C4 n {5}^{2} + \ldots .$

So $\alpha - \beta =$ $C3 n \left(8 - 5\right) + C4 n \left({8}^{2} - {5}^{2}\right) + \ldots + Cn n \left({8}^{n - 2} - {5}^{n - 2}\right)$

So, option (C) is the answer.

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