Question

If \[\int\limits_0^1 f\left( x \right) dx = 1, \int\limits_0^1 xf\left( x \right) dx = a, \int\limits_0^1 x^2 f\left( x \right) dx = a^2 , then \int\limits_0^1 \left( a - x \right)^2 f\left( x \right) dx\] equals

Options

• 4a²

• 0

•  2a²

• none of these

Answer 1

\[\int_0^1 \left( a - x \right)^2 f\left( x \right) d x\]
\[ = a^2 \int_0^1 f\left( x \right)dx + \int_0^1 x^2 f\left( x \right) dx - 2a \int_0^1 x f\left( x \right)dx\]
\[ = a^2 \times 1 + a^2 - 2aa ...............\left( \text{As per given values} \right)\]
\[ = 2 a^2 - 2 a^2 \]
\[ = 0\]