If $P$ is a point on $C_1$ and $Q$ in another point on $\mathrm{C}_2$, then $\frac{\mathrm{PA}^2+\mathrm{PB}^2+\mathrm{PC}^2+\mathrm{PD}^2}{\mathrm{QA}^2+\mathrm{QB}^2+\mathrm{QC}^2+\mathrm{QD}^2}$ is equal to :

A circle touches the line $L$ and the circle $C_1$ externally such that both the circles are on the same side of the line, then the locus of center of the circle is:

Let ABCD be a square of side length 2 units. $\mathrm{C}_2$ is the circle through vertices $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ and $\mathrm{C}_1$ is the circle touching all the sides of the square ABCD . L is a line through A.

A line $M$ through $A$ is drawn parallel to $B D$. Point $S$ moves such that its distances from

the line BD and the vertex A are equal. If locus of S cuts M at $\mathrm{T}_2$ and $\mathrm{T}_3$ and AC at $\mathrm{T}_1$, then area of $\Delta T_1 T_2 T_3$ is :

Comprehension IV

$\mathrm{A}=\left[\begin{array}{lll}1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1\end{array}\right]$, if $\mathrm{U}_1, \mathrm{U}_2$ and $\mathrm{U}_3$ are columns matrices satisfying. $\mathrm{AU}_1=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right], \mathrm{AU}_2=\left[\begin{array}{l}2 \\ 3 \\ 0\end{array}\right], \mathrm{AU}_3=\left[\begin{array}{l}2 \\ 3 \\ 1\end{array}\right]$ and U is $3 \times 3$ matrix whose columns are $\mathrm{U}_1, \mathrm{U}_2, \mathrm{U}_3$ then answer the following questions

The sum of the elements of $\mathrm{U}^{-1}$ is: