If $\mathbf{a , b , c}$ are three mutually perpendicular vectors such that the magnitudes of $\mathbf{b}$ and $\mathbf{c}$ are $1 / 2$ times and $\sqrt{3} / 2$ times that of $\mathbf{a}$, respectively, then the angle between the vectors $\mathbf{a}+\mathbf{b}+\mathbf{c}$ and $\mathbf{b}$ is

The locus of the point $P(\mathbf{r})$ which encloses a triangle $A B P$ of area 1 sq. unit with the fixed points $A(\hat{\mathbf{i}})$ and $B(\hat{\mathbf{j}})$ is

Let $A B C D$ be a parallelogram and $E$ be the mid-point of $A B$. If $P$ is the point of intersection of $D E$ and $A C$, then $\frac{D P}{P E}+\frac{A P}{P C}=$

A vector $\mathbf{a}$ has components $2 p$ and 1 with respect to a two dimensional rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise direction. If $\mathbf{a}$ has components $p+1$ and 1 with respect to the new system, then