The value of k ∈ N for which the integral I_n=∫_0^1(1-x^k)^n d x, n ∈ ℕ , satisfies 147 I_ 20 =148 I_ 21 is

Correct answer: D. 7 aligned & I_n=∫_0^1(1-x^k)^n · 1 d x & I_n=(1-x^k)^n · x-n k ∫_0^1(1-x^k)^ n-1 · x^ k-1 · d x & I_n=n k ∫_0^1[(1-x^k)^n-(1-x^k)^ n-1 ] d x & I_n=n k I_n-n k I_n & I_n I_ n-1 =(n k)/(n k+1) & I_ 21 I_ 20 =(21 k)/(1+21 k) & =(147)/(148) k=7 aligned

JEE Main 20248 AprilMorning Shift

Exam: JEE Main
Year: 2024
Date: 8 April
Shift: Morning Shift
Subject: Mathematics
Chapter: Definite Integration

JEE Main 2024 · 8 April · Morning Shift · Mathematics · Definite Integration

The value of k ∈ N for which the integral I_n=∫_0^1(1-x^k)^n d x, n...

The value of

k \in \mathrm{N}

for which the integral

I_n=\int_0^1\left(1-x^k\right)^n d x, n \in \mathbb{N}

, satisfies

147 I_{20}=148 I_{21}

Options

Correct answer

D. 7

Step-by-step solution

\begin{aligned} & I_n=\int_0^1\left(1-x^k\right)^n \cdot 1 d x \\ & I_n=\left(1-x^k\right)^n \cdot x-n k \int_0^1\left(1-x^k\right)^{n-1} \cdot x^{k-1} \cdot d x \\ & I_n=n k \int_0^1\left[\left(1-x^k\right)^n-\left(1-x^k\right)^{n-1}\right] d x \\ & I_n=n k I_n-n k I_n \\ & \frac{I_n}{I_{n-1}}=\frac{n k}{n k+1} \\ & \frac{I_{21}}{I_{20}}=\frac{21 k}{1+21 k} \\ & =\frac{147}{148} \Rightarrow k=7\end{aligned}

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