Which of the following is/are true ?

Options

A. `1000^(1000) gt 1002^(999)`
B. `1000^(1000) lt 1002^(999)`
C. `1000^(1002) lt 1002^(1000)`
D. `1000^(1002) gt 1002^(1000)`

Answer

A, D

Explanation

`((1002)/(1000))^(1000) xx 1/1002 = (1+2/1000)^(1000) xx 1/1002`
`lt (e^(2))/(1002)`
We know that `(1+2/n)^(n)` is monotanically increasing function of n and `underset(nrarroo)"lim"(1+2/n)^(n) = e^(2)`.
`rArr 1002^(999) lt 1000^(1000)`
Also, `((1002)/(1000))^(1000) xx 1/(1000^(2)) = (1+2/(1000))^(1000) xx (1)/(1000^(2)) lt (e^(2))/(1000^(2))`
`rArr 1002^(1000) lt 1000^(1002)`