If 2¹⁰ + 2⁹.3¹ + 2⁸ .3² +.....+ 2.3⁹ + 3¹⁰ = S - 2¹¹, then S is equal to :

If $${3^{2\sin 2\alpha - 1}}$$, 14 and $${3^{4 - 2\sin 2\alpha }}$$ are the first three terms of an A.P. for some $$\alpha $$, then the sixth terms of this A.P. is:

The minimum value of 2^sinx + 2^cosx is :

Let a₁, a₂, ..., an be a given A.P. whose
common difference is an integer and
S_n = a₁ + a₂ + .... + a_n. If a₁ = 1, a_n = 300 and 15 $$ \le $$ n $$ \le $$ 50, then
the ordered pair (S_n-4, a_n–4) is equal to: