Let $\alpha=3+4+8+9+13+14+\ldots$ upto 40 terms. If $(\tan \beta)^{\frac{\alpha}{1020}}$ is a root of the equation $x^2+x-2=0, \beta \in\left(0, \frac{\pi}{2}\right)$, then $\sin ^2 \beta+3 \cos ^2 \beta$ is equal to :

Consider the quadratic equation $\left(n^2-2 n+2\right) x^2-3 x+\left(n^2-2 n+2\right)^2=0, n \in \mathbf{R}$. Let $\alpha$ be the minimum value of the product of its roots and $\beta$ be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is $\alpha$ and the common ratio is $\frac{\alpha}{\beta}$, is :

The sum $1+\frac{1}{2}\left(1^2+2^2\right)+\frac{1}{3}\left(1^2+2^2+3^2\right)+\ldots$ upto 10 terms is equal to :

The value of $1^3-2^3+3^3-\ldots+15^3$ is: