$A$ tangent $\overline{PM}$ is drawn from a point $P$ outside $\odot(O, 8)$. $\overline{OP}$ intersects the circle at $N$. If $NP = 2$,then $PM = \ldots$

Which of the following groups correctly matches the data of Part $I$ with the data of Part $II$?

Part $I$	Part $II$
$1.$ In $\Delta ABC, AB=3, BC=4, AC=5$	$a.$ In-radius $= 1$
$2.$ In $\Delta PQR, PQ=5, QR=12, PR=13$	$b.$ In-radius $= 2$
$3.$ In $\Delta XYZ, XY=8, YZ=15, XZ=17$	$c.$ In-radius $= 3$
$4.$ In $\Delta MNP, MN=20, NP=21, MP=29$	$d.$ In-radius $= 6$

Two concentric circles having radii $41$ and $40$ are given. The chord of the circle with larger radius touches the circle with smaller radius. Find the length of the chord.

$AB$ is a diameter and $AC$ is a chord of a circle with centre $O$ such that $\angle BAC = 30^{\circ}$. The tangent at $C$ intersects the extended $AB$ at a point $D$. Prove that $BC = BD$.

In $Fig.$,$PQ$ is a chord of a circle and $PT$ is the tangent at $P$ such that $\angle QPT = 60^{\circ}$. Then $\angle PRQ$ is equal to (in $^{\circ}$)

$\overrightarrow{ PA }$ and $\overrightarrow{ PB }$ are tangents to $\odot( O , 5) .$ If $OP = 13,$ then $PB = \ldots \ldots \ldots \ldots$