If $\bar{a}=\frac{1}{\sqrt{10}}(3 \hat{i}+\hat{k})$ and $\bar{b}=\frac{1}{7}(2 \hat{i}+3 \hat{j}-6 \hat{k})$, then the value of $(2 \overline{\mathrm{a}}-\overline{\mathrm{b}}) \cdot((\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times(\overline{\mathrm{a}}+2 \overline{\mathrm{~b}}))=$

In a triangle with one of the angles $120^{\circ}$, the lengths of the sides form an A.P. If length of the greatest side is 7 m , then the area of the triangle is

If ${ }^{15} \mathrm{C}_4+{ }^{15} \mathrm{C}_5+{ }^{16} \mathrm{C}_6+{ }^{17} \mathrm{C}_7+{ }^{18} \mathrm{C}_8={ }^{19} \mathrm{C}_{\mathrm{r}}$, then the value of $r$ is equal to

If A and B are independent events such that $\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}^{\prime}\right)=\frac{3}{25}$ and $\mathrm{P}\left(\mathrm{A}^{\prime} \cap \mathrm{B}\right)=\frac{8}{25}$, then $P(A)=$