Let $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}, \overline{\mathrm{d}}$ are vectors such that $\overline{\mathrm{a}} \times \overline{\mathrm{b}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overline{\mathrm{c}} \times \overline{\mathrm{d}}=3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}$ and if $\left|\begin{array}{ll}\overline{\mathrm{a}} \cdot \overline{\mathrm{c}} & \overline{\mathrm{b}} \cdot \overline{\mathrm{c}} \\ \overline{\mathrm{a}} \cdot \overline{\mathrm{d}} & \overline{\mathrm{b}} \cdot \overline{\mathrm{d}}\end{array}\right|=0$, then $\lambda=$

In $\triangle \mathrm{ABC}$, with usual notations, if $\cos \frac{B}{2}=\sqrt{\frac{c+a}{2 a}}$, then $a^2=$

The joint equation of two lines passing through $(-2,3)$ and parallel to the bisectors of the angle between the co-ordinate axes is

Let the circle with centre at origin pass through the vertices of an equilateral triangle ABC . If $A \equiv(2,4)$, then the length of the median through A is