The values of `a` for which the integral `int_0^2|x-a|dxgeq1` is satisfied are `(2,oo)` (b) `(-oo,0)` `(0,2)` (d) none of these

Options

A. `[2,oo)`
B. `(-oo,0]`
C. `(0,2)`
D. none of these

Answer

A, B, C

Explanation

For `ale0` given equation becomes
`int_(0)^(2)(x-a)dxge1` or `alt1/2`
So` ale0`
For `0 lt a lt 2`
`int_(0)^(2)|x-a|dxge1` or `int_(0)^(a)(a-x)dx+int_(a)^(2)(x-a)dxge1`
or `(a^(2))/2+2-2a+(a^(2))/2ge1`
or `a^(2)-2a+1ge0` or `(a-1)^(2)ge0`
So `0lt a lt 2`
For `a ge 2`.
`int_(0)^(2)|x-a|dxge1`
or `int_(0)^(2)(a-x)dxge1`
or `2a-2ge1` or `age3/2`
So `age2`