The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is _____________.

If $$A = \left[ {\matrix{ 0 & { - \tan \left( {{\theta \over 2}} \right)} \cr {\tan \left( {{\theta \over 2}} \right)} & 0 \cr } } \right]$$ and
$$({I_2} + A){({I_2} - A)^{ - 1}} = \left[ {\matrix{ a & { - b} \cr b & a \cr } } \right]$$, then $$13({a^2} + {b^2})$$ is equal to

Let A₁, A₂, A₃, ....... be squares such that for each n $$ \ge $$ 1, the length of the side of A_n equals the length of diagonal of A_n+1. If the length of A₁ is 12 cm, then the smallest value of n for which area of A_n is less than one, is __________.

If the system of equations

kx + y + 2z = 1

3x $$-$$ y $$-$$ 2z = 2

$$-$$2x $$-$$2y $$-$$4z = 3

has infinitely many solutions, then k is equal to __________.