Let 3,6,9,12,… upto 78 terms and 5,9,13,17,… upto 59 terms be two series. Then, the sum of the terms common to both the series is equal to .

If the quadratic equation $(\lambda+2)x^2-3\lambda x+4\lambda=0$, $\lambda \neq -2$, has two positive roots, then the number of possible integral values of $\lambda$ is:

The value of $\frac{{ }^{100} \mathrm{C}_{50}}{51}+\frac{{ }^{100} \mathrm{C}_{51}}{52}+\ldots.+\frac{{ }^{100} \mathrm{C}_{100}}{101}$ is :

Let $e_1$ and $e_2$ be two distinct roots of the equation $x^2 - ax + 2 = 0$. Let the sets $\{a \in \mathbb{R} : e_1 \text{ and } e_2 \text{ are the eccentricities of hyperbolas}\} = (\alpha, \beta)$, and $\{a \in \mathbb{R} : e_1 \text{ and } e_2 \text{ are the eccentricities of an ellipse and a hyperbola, respectively}\} = (\gamma, \infty)$. Then $\alpha^2 + \beta^2 + \gamma^2$ is equal to:

Among the statements : I: If $\left|\begin{array}{ccc}1 & \cos \alpha & \cos \beta \\ \cos \alpha & 1 & \cos \gamma \\ \cos \beta & \cos \gamma & 1\end{array}\right|=\left|\begin{array}{ccc}0 & \cos \alpha & \cos \beta \\ \cos \alpha & 0 & \cos \gamma \\ \cos \beta & \cos \gamma & 0\end{array}\right|$, then $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=\frac{3}{2}$, and II : If $\left|\begin{array}{ccc}x^{2}+x & x+1 & x-2 \\ 2 x^{2}+3 x-1 & 3 x & 3 x-3 \\ x^{2}+2 x+3 & 2 x-1 & 2 x-1\end{array}\right|=\mathrm{p} x+\mathrm{q}$, then $\mathrm{p}^{2}=196 \mathrm{q}^{2}$,

If $\left(\frac{1}{{ }^{15} \mathrm{C}_{0}}+\frac{1}{{ }^{15} \mathrm{C}_{1}}\right)\left(\frac{1}{{ }^{15} \mathrm{C}_{1}}+\frac{1}{{ }^{15} \mathrm{C}_{2}}\right) \cdots\left(\frac{1}{{ }^{15} \mathrm{C}_{12}}+\frac{1}{{ }^{15} \mathrm{C}_{13}}\right)=\frac{\alpha^{13}}{{ }^{14} \mathrm{C}_{0}{ }^{14} \mathrm{C}_{1} \cdots{ }^{14} \mathrm{C}_{12}}$, then $30 \alpha$ is equal to $\_\_\_\_$.