Question : Two men standing on the same side of a pillar, 75 metres high, observe the angles of elevation of the top of the pillar to be $30^{\circ}$ and $60^{\circ}$, respectively. The distance between the two men is:

Option 1: $100\sqrt{3}$ m

Option 2: $100$ m

Option 3: $50\sqrt{3}$ m

Option 4: $25\sqrt{3}$ m

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 : In a circle of radius 3 cm, two chords of length 2 cm and 3 cm lie on the same side of a diameter. What is the perpendicular distance between the two chords?

Option 1: $\frac{4 \sqrt{3}-3 \sqrt{2}}{2}$ cm

Option 2: $\frac{4 \sqrt{2}-3 \sqrt{3}}{2}$ cm

Option 3: $\frac{4 \sqrt{2}-3 \sqrt{3}}{3}$ cm

Option 4: $\frac{4 \sqrt{2}-3 \sqrt{3}}{4}$ cm

Question : AB is the chord of a circle such that AB = 10 cm. If the diameter of the circle is 20 cm, then the angle subtended by the chord at the centre is?

Option 1: $45^{\circ}$

Option 2: $60^{\circ}$

Option 3: $30^{\circ}$

Option 4: $90^{\circ}$

Question : If $\sin\left ( 2a+45^{\circ} \right )=\cos\left ( 30^{\circ}-a \right )$ where $0^{\circ}< a< 90^{\circ}$, then the value of a is:

Option 1: $0^{\circ}$

Option 2: $15^{\circ}$

Option 3: $45^{\circ}$

Option 4: $60^{\circ}$