The diagrams, given below, represent two inequations $\mathrm{A}$ and $\mathrm{B}$ on real number line.

Write down $\mathrm{A}$ and $\mathrm{B}$ in set builder notations.
 

The diagrams, given below, represent two inequations $\mathrm{A}$ and $\mathrm{B}$ on real number line.

Represent $\mathrm{A}\cap \mathrm{B}$ and $\mathrm{A}\text{'}\cap \mathrm{B}$ on two different number lines.
 

Solve for matrices $\mathrm{A}$ and $\mathrm{B},$ where

$2\mathrm{A}+\mathrm{B}=\left[\begin{array}{c}3 -4\\ 2 7 \end{array}\right]$ and $\mathrm{A}-2\mathrm{B}=\left[\begin{array}{c}4 3\\ 1 1\end{array}\right].$

As shown in the given figure; from an external point $\mathrm{P}$, a tangent $\mathrm{PT}$ and a line segment $\mathrm{PAB}$ are drawn to a circle with center $\mathrm{O}. \mathrm{ON}$ is perpendicular on the chord $\mathrm{AB}$. Prove that: $\mathrm{PA}·\mathrm{PB}={\mathrm{PN}}^2-{\mathrm{AN}}^2$.

As shown in the given figure; from an external point $\mathrm{P}$, a tangent $\mathrm{PT}$ and a line segment $\mathrm{PAB}$ are drawn to a circle with centre $\mathrm{O}. \mathrm{ON}$ is perpendicular on the chord $\mathrm{AB}$. Prove that: ${\mathrm{PN}}^2-{\mathrm{AN}}^2={\mathrm{OP}}^2-{\mathrm{OT}}^2$.