Let $${T_n}$$ denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If $${T_{n + 1}} - {T_n} = 21$$, then n equals

Let $$\alpha $$, $$\beta $$ be the roots of $${x^2} - x + p = 0$$ and $$\gamma ,\delta $$ be the roots of $${x^2} - 4x + q = 0.$$ If $$\alpha ,\beta ,\gamma ,\delta $$ are in G.P., then the integral values of $$p$$ and $$q$$ respectively, are

If the sum of the first $$2n$$ terms of the A.P.$$2,5,8,......,$$ is equal to the sum of the first $$n$$ terms of the A.P.$$57,59,61,.....,$$ then $$n$$ equals

The number of distinct real roots of $$\left| {\matrix{ {\sin x} & {\cos x} & {\cos x} \cr {\cos x} & {\sin x} & {\cos x} \cr {\cos x} & {\cos x} & {\sin x} \cr } } \right|\,$$
$$\, = 0$$ in the interval $$ - {\pi \over 4} \le x \le {\pi \over 4}$$ is