Question : Two chords AB and CD of a circle meet inside the circle at point P. If AP = 12 cm, AB = 20 cm, and CP = 16 cm,  find CD.

Option 1: 22 cm

Option 2: 15 cm

Option 3: 21 cm

Option 4: 24 cm

Question : Two chords $\mathrm{AB}$ and $\mathrm{CD}$ of a circle with centre $\mathrm{O}$, intersect each other at $\mathrm{P}$. If $\angle\mathrm{ AOD}=100^{\circ}$ and $\angle \mathrm{BOC}=70^{\circ}$, then the value of $\angle \mathrm{APC}$ is:

Option 1: $80^{\circ}$

Option 2: $75^{\circ}$

Option 3: $85^{\circ}$

Option 4: $95^{\circ}$

Question : AB and CD are two chords in a circle with centre O and AD is a diameter. AB and CD produced meet a point P outside the circle. If $\angle A P D=25^{\circ}$ and $\angle D A P=39^{\circ}$, then the measure of $\angle C B D$ is:

Option 1: 29°

Option 2: 26°

Option 3: 27°

Option 4: 32°

Question : In the given figure, $\mathrm{CD}$ and $\mathrm{AB}$ are the diameters of the circle and $\mathrm{AB}$ and $\mathrm{CD}$ are perpendicular to each other. $\mathrm{LQ}$ and $\mathrm{SR}$ are perpendiculars to $\mathrm{AB}$ and $\mathrm{CD}$, respectively. The radius of the circle is $5\;\mathrm{cm}$, $\mathrm{PB:PA = 2:3}$ and $\mathrm{CN:ND = 2:3}$. What is the length (in $\mathrm{cm}$) of $\mathrm{SM}$?

Option 1: $\left [ \left ( 5\sqrt{3} \right )-3 \right ]$

Option 2: $\left [ \left ( 4\sqrt{3} \right )-2 \right ]$

Option 3: $\left [ \left ( 2\sqrt{6} \right )-1 \right ]$

Option 4: $\left [ \left ( 5\sqrt{6} \right )-3 \right ]$

Question : Let O be the centre of the circle and P be a point outside the circle. If PAB is a secant of the circle which cuts the circle at A and B and PT is the tangent drawn from P, then find the length of PT, if PA = 3 cm and AB = 9 cm.

Option 1: $3 \sqrt{3} \mathrm{~cm}$

Option 2: $4 \sqrt{3} \mathrm{~cm}$

Option 3: 6 cm

Option 4: 8 cm