Let β(m, n)=∫_0^1 x^ m-1 (1-x)^ n-1 ~d x, ~m, n>0 . If ∫_0^1(1-x^ 10 )^ 20 ~d x=a × β(b, c) , then 100(a+b+c) equals____

Correct answer: B. 2120 I=∫_0^1 1 ·(1-x^ 10 )^ 20 d x x^ 10 =t x=t^ 1 / 10 d x=(1)/(10)(t)^ -9 / 10 d t I=∫_0^1(1-t)^ 20 (1)/(10)(t)^ -9 / 10 d t I=(1)/(10) ∫_0^1 t^ -9 / 10 (1-t)^ 20 d t a=(1)/(10) b=(1)/(10) c=21

JEE Main 20245 AprilEvening Shift

Exam: JEE Main
Year: 2024
Date: 5 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Definite Integration

JEE Main 2024 · 5 April · Evening Shift · Mathematics · Definite Integration

Let β(m, n)=∫_0^1 x^ m-1 (1-x)^ n-1 ~d x, ~m, n>0 . If ∫_0^1(1-x^...

Let

\beta(\mathrm{m}, \mathrm{n})=\int_0^1 x^{\mathrm{m}-1}(1-x)^{\mathrm{n}-1} \mathrm{~d} x, \mathrm{~m}, \mathrm{n}>0

. If

\int_0^1\left(1-x^{10}\right)^{20} \mathrm{~d} x=\mathrm{a} \times \beta(\mathrm{b}, \mathrm{c})

, then

100(\mathrm{a}+\mathrm{b}+\mathrm{c})

equals____

Options

A1021
B2120
C2012
D1120

Correct answer

B. 2120

Step-by-step solution

I=\int_0^1 1 \cdot\left(1-x^{10}\right)^{20} d x

x^{10}=t

x=t^{1 / 10}

d x=\frac{1}{10}(t)^{-9 / 10} d t

I=\int_0^1(1-t)^{20} \frac{1}{10}(t)^{-9 / 10} d t

I=\frac{1}{10} \int_0^1 t^{-9 / 10}(1-t)^{20} d t

a=\frac{1}{10} \quad b=\frac{1}{10} \quad c=21

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