Let $\quad \overline{\mathrm{a}}=\alpha \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}, \overline{\mathrm{b}}=3 \hat{\mathrm{i}}-\beta \hat{j}+4 \hat{\mathrm{k}} \quad$ and $\overline{\mathrm{c}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}$, where $\alpha, \beta \in \mathbb{R}$, be three vectors. If the projection of $\overline{\mathrm{a}}$ on $\overline{\mathrm{c}}$ is $\frac{10}{3}$ and $\overline{\mathrm{b}} \times \overline{\mathrm{c}}=-6 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}+7 \hat{\mathrm{k}}$, then the value of $2 \alpha+\beta$ is
If $4 a b=3 h^2$, then the ratio of the slope of lines represented by $a x^2+2 \mathrm{~h} x y+\mathrm{b} y^2=0$ is
If $\sin \left(\cot ^{-1}(x+1)\right)=\cos \left(\tan ^{-1} x\right)$ then considering positive square roots, $x$ has the value ___________
A random variable has the following probability distribution
| $\mathrm{X:}$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| $\mathrm{P}(x):$ | 0 | $\mathrm{2p}$ | $\mathrm{2p}$ | $\mathrm{3p}$ | $\mathrm{p^2}$ | $\mathrm{2p^2}$ | $\mathrm{7p^2}$ | $\mathrm{2p}$ |
Then the value of p is