In $\triangle A B C$, if $r=3$ and $R=5$, then $\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c a}=$

An aeroplane is flying at a constant speed, parallel to the horizontal ground at a height of 5 kms . A person on the ground observed that the angle of elevation of the plane is changed from $15^{\circ}$ to $30^{\circ}$ in the duration of 50 seconds, then the speed of the plane (in kmph ) is

If the vector $\hat{\mathbf{i}}-7 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ is along the internal bisector of the angle between the vectors $\mathbf{a}$ and $-2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and the unit vector along $\mathbf{a}$ is $x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}}$ then, $x=$

If $\mathbf{a}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+6 \hat{\mathbf{k}} ; \mathbf{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{c}=3 \hat{\mathbf{j}}-\hat{\mathbf{k}}$, then $\mathbf{a} \times \mathbf{b} \times \mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a}=$