Correct option is (4) 7
\(\mathrm{I}_{\mathrm{n}}=\int_{0}^{1}\left(1-\mathrm{x}^{\mathrm{k}}\right)^{\mathrm{n}} \cdot 1 \mathrm{dx}\)
\(I_{n}=\left(1-x^{k}\right)^{n} \cdot x-n k \int_{0}^{1}\left(1-x^{k}\right)^{n-1} \cdot x^{k-1} \cdot d x\)
\(\mathrm{I}_{\mathrm{n}}=\mathrm{nk} \int_{0}^{1}\left[\left(1-\mathrm{x}^{\mathrm{k}}\right)^{\mathrm{n}}-\left(1-\mathrm{x}^{\mathrm{k}}\right)^{\mathrm{n}-1}\right] \mathrm{dx}\)
\(\mathrm{I}_{\mathrm{n}}=\mathrm{nkI}_{\mathrm{n}}-\mathrm{nkI}_{\mathrm{n}}\)
\(\frac{I_{n}}{I_{n-1}}=\frac{n k}{n k+1}\)
\(\frac{\mathrm{I}_{21}}{\mathrm{I}_{20}}=\frac{21 \mathrm{k}}{1+21 \mathrm{k}}\)
\(=\frac{147}{148} \Rightarrow \mathrm{k}=7\)
