Question

For $U_n=\underset{0}{\overset{1}{\int }}{x}^n(2-x{)}^{n}dx;V_n=\underset{0}{\overset{1}{\int }}{x}^n(1-x{)}^{n}dxn\in N$, which of the following statement(s) is/are true?

• A. $U_n=2^n{V}_n$
• B. $U_n=2^{-n}{V}_n$
• C. $U_n=2^{2n}{V}_n$
• D. $U_n=2^{-2n}{V}_n$

Answer 1

Answer: (C)

Given $U_n=\underset{0}{\overset{1}{\int }}{x}^n\cdot (2-x{)}^{n}dx;V_n=\underset{0}{\overset{1}{\int }}{x}^n\cdot (1-x{)}^{n}dx$ in $U_n$
put $x=2t\Rightarrow dx=2dt$
$\therefore U_n=2\underset{0}{\overset{1/2}{\int }}{2}^n\cdot t^n{2}^n(1-t{)}^{n}dt$....(1)
Now $V_n=2\underset{0}{\overset{1/2}{\int }}{x}^n(1-x{)}^{n}dx$(Using Queen)....(2)
From (1) and (2)
$U_n=2^{2n}\cdot V_n\Rightarrow (C)$