51). ABC is an isosceles triangle with AB =AC. A circle through B touching AC at the middle point intersects AB at P. Then, AP : AB is
| A). 3 : 5 |
| B). 1 : 4 |
| C). 4 : 1 |
| D). 2 : 3 |
|
52). A chord AB of a circle \( C_{1} \) of radius \( \Large \left(\sqrt{3} + 1\right) \)cm touches a circle \( C_{2} \) of radius \( \Large \left(\sqrt{3} - 1\right) \)cm, then the length of AB is
| A). \( \Large 8\sqrt{3} \) |
| B). \( \Large 4 \sqrt[4]{3} \) |
| C). \( \Large 4\sqrt{3} \) |
| D). \( \Large 2 \sqrt[4]{3} \) |
|
53). P and Q are two points on a circle with centre at O. R is a point on the minor arc of the circle between the points P and Q. The tangents to the circle from the point S are drawn which touch the circle at P and Q. If \( \Large \angle \) PSQ = \( \Large 20 ^{\circ} \), then \( \Large \angle \) PRQ is equal to
| A). \( \Large 200 ^{\circ} \) |
| B). \( \Large 160 ^{\circ} \) |
| C). \( \Large 100 ^{\circ} \) |
| D). \( \Large 80 ^{\circ} \) |
|
54). The length of the diagonal of a square is 8 cm. A circle has been drawn circumscribing the square. The area of the portion between the circle and the square (in sq cm) is
| A). \( \Large 16\frac{2}{7} \) |
| B). \( \Large 18\frac{2}{7} \) |
| C). \( \Large 10\frac{2}{7} \) |
| D). \( \Large 12\frac{2}{7} \) |
|
55). In the given figure, AB and CD are two parallel chords of a circle with centre O and radius 5 cm. Also, AB = 8 cm and CD = 6 cm. If OP \( \Large \perp \) AB and OQ \( \Large \perp \) CD, then determine the length of PQ
| A). 7 cm |
| B). 10 cm |
| C). 8 cm |
| D). None of these |
|
56). Suppose AB is a diameter of a circle, whose centre is at O and C be any point on the circle. If CD \( \Large \perp \) AB and CD = 12 cm, AD = 16 cm, then BD is equal to
| A). 10 cm |
| B). 12 cm |
| C). 8 cm |
| D). 9 cm |
|
57). Three circles of radii 4 cm, 6 cm and 8 cm touch each other pairwise externally. The area of the triangle formed by the line segments joining the centres of the three circles is
| A). \( \Large 6\sqrt{6} \) sq cm |
| B). \( \Large 24\sqrt{6} \) sq cm |
| C). \( \Large 144\sqrt{13} \) sq cm |
| D). \( \Large 12\sqrt{105} \) sq cm |
|
58). A, B and C are three points on a circle. The tangent at C meets BA extended at T. Given, \( \Large \angle \) ATC = \( \Large 36 ^{\circ} \) and \( \Large \angle \) ACT = \( \Large 48 ^{\circ} \), the angle subtended by AB at the centre of the circle is
| A). \( \Large 84 ^{\circ} \) |
| B). \( \Large 48 ^{\circ} \) |
| C). \( \Large 96 ^{\circ} \) |
| D). \( \Large 72 ^{\circ} \) |
|
59). In the figure given below, AO = CD where O is the centre of the circle. What is the value of \( \Large \angle \) APB?
| A). \( \Large 60 ^{\circ} \) |
| B). \( \Large 50 ^{\circ} \) |
| C). \( \Large 45 ^{\circ} \) |
| D). \( \Large 30 ^{\circ} \) |
|
60). A circular ring with centre O is kept in the vertical position by two weightless thin strings TP and TQ attached to the ring at P and Q. The line OT meets the ring at E whereas a tangential string at E meets TP and TQ at A and B, respectively. If the radius of the ring is 5 cm and OT = 13 cm, then what is the length of AB?
| A). \( \Large \frac{10}{3} \) cm |
| B). \( \Large \frac{20}{3} \) cm |
| C). 10 cm |
| D). \( \Large \frac{40}{3} \) cm |
|
