If a₁, a₂, a₃, ..... are in A.P. such that a₁ + a₇ + a₁₆ = 40, then the sum of the first 15 terms of this A.P. is :

A value of $$\theta \in \left( {0,{\pi \over 3}} \right)$$, for which
$$\left| {\matrix{ {1 + {{\cos }^2}\theta } & {{{\sin }^2}\theta } & {4\cos 6\theta } \cr {{{\cos }^2}\theta } & {1 + {{\sin }^2}\theta } & {4\cos 6\theta } \cr {{{\cos }^2}\theta } & {{{\sin }^2}\theta } & {1 + 4\cos 6\theta } \cr } } \right| = 0$$, is :

The general solution of the differential equation (y² – x³)dx – xydy = 0 (x $$ \ne $$ 0) is : (where c is a constant of integration)

Let $$a \in \left( {0,{\pi \over 2}} \right)$$ be fixed. If the integral

$$\int {{{\tan x + \tan \alpha } \over {\tan x - \tan \alpha }}} dx$$ = A(x) cos 2$$\alpha $$ + B(x) sin 2$$\alpha $$ + C, where C is a

constant of integration, then the functions A(x) and B(x) are respectively :