If the 1011 ^ th term from the end in the binomial expansion of ((4x)/(5) - (5)/(2x)) ^ 2022 is 1024 times 1011 ^ th term from the beginning, then 32|x| is equal to

Correct answer: B. 10 Given, The 1011 ^ th term from the end in the binomial expansion of ((4x)/(5) - (5)/(2x)) ^ 2022 is 1024 times 1011 ^ th term from the beginning, So, according to question we have, T _ 1013 T _ 1011 = 1024 ⇒ C1012 2022 ((4x)/(5)) ^ 1010 · ((- 5)/(2x)) ^ 1012 C1010 2022 ((4x)/(5)) ^ 1012 ((- 5)/(2x)) ^ 1010 = 1024 ⇒ ((- 5)/(2x)) ^ 2 ((4x)/(5)) ^ 2 = 1024 ⇒ 5 ^ 4 2 ^ 6 x ^ 4 = 2 ^ 10 ⇒ ((5)/(x)) ^ 4 = 2 ^ 16 ⇒ |x| = (5)/(16) ⇒ 32|x| = 10 Note this question was bonus in Jee Main 2023 April session as answer key was not matching, so we have modified the question.

JEE Main 202311 AprilEvening Shift

Exam: JEE Main
Year: 2023
Date: 11 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2023 · 11 April · Evening Shift · Mathematics · Binomial Theorem

If the 1011 ^ th term from the end in the binomial expansion of...

If the

{1011}^{th}

term from the end in the binomial expansion of

{\left(\dfrac{4x}{5} - \dfrac{5}{2x}\right)}^{2022} is

1024

times

{1011}^{th}

term from the beginning, then

32\left|x\right| is equal to

Options

A $15$
B $10$
C $12$
D $8$

Correct answer

B. $10$

Step-by-step solution

Given,

The ${1011}^{th}$ term from the end in the binomial expansion of ${\left(\dfrac{4x}{5} - \dfrac{5}{2x}\right)}^{2022} is$ 1024 $times$ {1011}^{th} $term from the beginning,So, according to question we have,$ \dfrac{{T}_{1013}}{{T}_{1011}} = 1024 $ $ ⇒ \dfrac{C1012 2022 {\left(\dfrac{4x}{5}\right)}^{1010} \cdot {\left(\dfrac{- 5}{2x}\right)}^{1012}}{C1010 2022 {\left(\dfrac{4x}{5}\right)}^{1012}{\left(\dfrac{- 5}{2x}\right)}^{1010}} = 1024 $$ ⇒ \dfrac{{\left(\dfrac{- 5}{2x}\right)}^{2}}{{\left(\dfrac{4x}{5}\right)}^{2}} = 1024 $$ ⇒ \dfrac{{5}^{4}}{{2}^{6}{x}^{4}} = {2}^{10} $$ ⇒ {\left(\dfrac{5}{x}\right)}^{4} = {2}^{16} $$ ⇒ \left|x\right| = \dfrac{5}{16} $$ ⇒ 32\left|x\right| = 10$

Note this question was bonus in Jee Main 2023 April session as answer key was not matching, so we have modified the question.

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