A committee of 11 members is to be formed from 8 males and 5 females. If $m$ is the number of ways the committee is formed with at least 6 males and $n$ is the number of ways the committee is formed with at least 3 females, then

If $y=[(x+1)(2 x+1)(3 x+1) \ldots \ldots \ldots(n x+1)]^4$ then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $x=0$ is

Let $\alpha$ and $\beta$ be two real roots of the equation $(k+1) \tan ^2 x-\sqrt{2} \lambda \tan x=(1-k)$ where $k(\neq-1)$ and $\lambda$ are real numbers. If $\tan ^2(\alpha+\beta)=50$, then a value of $\lambda$ is

Let $\overline{\mathrm{a}}=\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overline{\mathrm{c}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}}$. Then the vector $\overline{\mathrm{b}}$ satisfying $\overline{\mathrm{a}} \times \overline{\mathrm{b}}+\overline{\mathrm{c}}=\overline{0}$ and $\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=3$, is