$1$. $1001^2 = 1000^2 + 2.1000 + 1 = 1000000 + 2000 + 1 = 1002001$
$2$. $29,9 . 30,1 = (30 - 0,1).(30 + 0,1) = 30^2 - 0,1^2 = 900 - 0,01 = 899,99$
$3$. $201^2 = 200^2 + 2.200 + 1 = 40000 + 400 + 1 = 40401$
$4$. $37.43 = (40 - 3).(40 + 3) = 40^2 - 3^2 = 1600 - 9 = 1591$
$5$. $199^2 = (200-1)^2 = 200^2 - 2.200 + 1 = 40000 - 400 + 1 = 39601$