If the fourth term in the binomial expansion of ( x ^ 1 1 + _ 10 x + x ^ (1)/(12) ) ^ 6 is equal to 200, and x > 1, then the value of x is

Correct answer: D. 10 General term for the given binomial is T _ r + 1 = ^ 6 C _ r ( x ^ 1 1 + _ 10 x ) ^ 6 - r ( x ^ (1)/(12) ) ^ r Given T _ 4 = 200, putting r = 3 ⇒ T _ 4 = ^ 6 C _ 3 ( x ^ 1 1 + _ 10 x ) ^ 3 ( x ^ (1)/(12) ) ^ 3 = 200 ⇒ ( x ^ 1 1 + _ 10 x ) ^ (3)/(2) ⋅ ( x ^ (1)/(4) ) = 10 ⇒ x ^ 3 2(1 + _ 10 x) + (1)/(4) = 10 ⇒ ( 3 2(1 + _ 10 x) + (1)/(4)) _ 10 x = 1 ⇒ x = 10, 10 ^ - 4 ⇒ x = 10 ( as x > 1 )

JEE Main 20198 AprilEvening Shift

Exam: JEE Main
Year: 2019
Date: 8 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2019 · 8 April · Evening Shift · Mathematics · Binomial Theorem

If the fourth term in the binomial expansion of ( x ^ 1 1 + _ 10 x...

If the fourth term in the binomial expansion of

{\left(\sqrt{{x}^{\dfrac{1}{1 + \log_{10}x}}} + {x}^{\dfrac{1}{12}}\right)}^{6} is equal to

200, and

x > 1,

then the value of

x

Options

A $100$
B ${10}^{4}$
C ${10}^{3}$
D $10$

Correct answer

D. $10$

Step-by-step solution

General term for the given binomial is

{T}_{r + 1} = {}^{6}{C}_{r}{\left(\sqrt{{x}^{\dfrac{1}{1 + \log_{10}x}}}\right)}

^{6 - r}{\left({x}^{\dfrac{1}{12}}\right)}^{r}

<br/>Given

{T}_{4} = 200,

putting

r = 3

<br/>

⇒ {T}_{4} = {}^{6}{C}_{3}{\left(\sqrt{{x}^{\dfrac{1}{1 + \log_{10}x}}}\right)}^{3}{\left({x}^{\dfrac{1}{12}}\right)}^{3} = 200

<br/>

⇒ {\left({x}^{\dfrac{1}{1 + \log_{10}x}}\right)}^{\dfrac{3}{2}} ⋅ \left({x}^{\dfrac{1}{4}}\right) = 10

<br/>

⇒ {x}^{\dfrac{3}{2\left(1 + \log_{10}x\right)} + \dfrac{1}{4}} = 10

<br/>

⇒ \left(\dfrac{3}{2\left(1 + \log_{10}x\right)} + \dfrac{1}{4}\right)\log_{10}x = 1

<br/>

⇒ x = 10, {10}^{- 4}

<br/>

⇒ x = 10 (

as

x > 1 )

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