The term independent of x in the expression of (1 - x ^ 2 + 3 x ^ 3 ) ((5)/(2) x ^ 3 - 1 5 x ^ 2 ) ^ 11 , x ≠ 0 is

Correct answer: B. (33)/(200) General term of ((5)/(2) x ^ 3 - 1 5 x ^ 2 ) ^ 11 is T _ r + 1 = Cr 11 ((5)/(2) x ^ 3 ) ^ 11 - r (- 1 5 x ^ 2 ) ^ r = Cr 11 (- 1) ^ r · 5 ^ 11 - 2r 2 ^ 11 - r · x ^ 33 - 5r The term independent of x in the expansion of (1 - x ^ 2 + 3 x ^ 3 ) ((5)/(2) x ^ 3 - 1 5 x ^ 2 ) ^ 11 will be the coefficient of x ^ 0 in ((5)/(2) x ^ 3 - 1 5 x ^ 2 ) ^ 11 - coefficient of x ^ - 2 in ((5)/(2) x ^ 3 - 1 5 x ^ 2 ) ^ 11 + 3 × coefficient of x ^ - 3 in ((5)/(2) x ^ 3 - 1 5 x ^ 2 ) ^ 11 = - C7 11 (- 1) ^ 7 · 5 ^ - 3 2 ^ 4 = 330 5 ^ 3 · 2 ^ 4 = (33)/(200)

JEE Main 202228 JuneEvening Shift

Exam: JEE Main
Year: 2022
Date: 28 June
Shift: Evening Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2022 · 28 June · Evening Shift · Mathematics · Binomial Theorem

The term independent of x in the expression of (1 - x ^ 2 + 3 x ^ 3...

The term independent of

x

in the expression of \left(1 - {x}^{2} + 3{x}^{3}\right){\left(\dfrac{5}{2}{x}^{3} - \dfrac{1}{5{x}^{2}}\right)}^{11}, x \ne 0$ is

Options

A $\dfrac{7}{40}$
B $\dfrac{33}{200}$
C $\dfrac{39}{200}$
D $\dfrac{11}{50}$

Correct answer

B. $\dfrac{33}{200}$

Step-by-step solution

General term of ${\left(\dfrac{5}{2}{x}^{3} - \dfrac{1}{5{x}^{2}}\right)}^{11} is </p><p>$ {T}_{r + 1} = Cr 11 {\left(\dfrac{5}{2}{x}^{3}\right)}^{11 - r}{\left(- \dfrac{1}{5{x}^{2}}\right)}^{r} = Cr 11 {\left(- 1\right)}^{r} \cdot \dfrac{{5}^{11 - 2r}}{{2}^{11 - r}} \cdot {x}^{33 - 5r} $</p><p>The term independent of$ x $in the expansion of \left(1 - {x}^{2} + 3{x}^{3}\right){\left(\dfrac{5}{2}{x}^{3} - \dfrac{1}{5{x}^{2}}\right)}^{11}$ will be the coefficient of ${x}^{0} in {\left(\dfrac{5}{2}{x}^{3} - \dfrac{1}{5{x}^{2}}\right)}^{11}-$ coefficient of ${x}^{- 2} in {\left(\dfrac{5}{2}{x}^{3} - \dfrac{1}{5{x}^{2}}\right)}^{11} + 3 \times$ coefficient of ${x}^{- 3} in {\left(\dfrac{5}{2}{x}^{3} - \dfrac{1}{5{x}^{2}}\right)}^{11}$

$= - C7 11 {\left(- 1\right)}^{7} \cdot \dfrac{{5}^{- 3}}{{2}^{4}} = \dfrac{330}{{5}^{3} \cdot {2}^{4}} = \dfrac{33}{200}$

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