1. Two tangents are drawn to a circle from points A and B on the circumference, and they intersect at point C. Another point P lies on the circumference, on the side opposite to where point C is located with respect to the center. If \(\angle\)APB = 35°, then what is the measure of \(\angle\)ACB?

(a) 145° (b) 55° (c) 110° (d) None of the above

2. A passenger on an airplane once saw Howrah Station on one side and Shaheed Minar on the opposite side at angles of depression of 60° and 30°, respectively. At that moment, if the airplane was flying at a height of \(545\sqrt{3}\) meters, what is the distance between Howrah Station and Shaheed Minar?

3. In a right-angled triangle, the difference between the two acute angles is \(\frac{2\pi}{5}\). Express the measures of these two angles in both radians and degrees.

4. In a right-angled triangle, the difference between the two acute angles is 30°. Express the measures of those two angles in both radians and degrees.

5. Durga was standing on a railway overbridge that is 5√3 meters high. She observed the engine of a moving passenger train at a depression angle of 30° on one side of the bridge. Two seconds later, she saw the same engine at a depression angle of 60° on the other side of the bridge. Durga's position was vertically above the railway track, which is assumed to be a straight line. Find the speed of the train.

6. In a right-angled triangle, if one of the acute angles is 30°, determine the measure of the other acute angle in sexagesimal system.

7. The difference between two given angles is 40°, and their sum is \(\frac{\pi}{2}\) radians. Find the radian measures of the two angles.

8. In a right-angled triangle, the hypotenuse is 6 cm longer than one of the other two sides and 12 cm longer than the other. Find the area of the triangle.

9. If one side of a rhombus is 6 cm and one of its angles measures 60°, then the area of the rhombus-shaped field is –

(a) \(9\sqrt{3}\) square cm (b) \(18\sqrt{3}\) square cm (c) \(36\sqrt{3}\) square cm (d) \(6\sqrt{3}\)square cm

10. O is the center of the circle, and AB is its diameter. ABCD is a cyclic quadrilateral. Given that \(\angle\)ABC = 65° and \(\angle\)DAC = 60°, find the measure of \(\angle\)BCD.

11. From the roof of a 15-meter-high house located at one end of a park, the base and top of a brick kiln chimney at the other end of the park are seen at angles of depression and elevation of 30° and 60°, respectively. Determine the height of the brick kiln chimney and the distance between the kiln and the house.

12. AOB is a diameter of the circle with center O, and C is a point on the circle. If \(\angle\)OBC = 60°, then find the measure of \(\angle\)OCA.

13. A right-angled triangle where the hypotenuse is 12 cm and one of the other sides is 5 cm. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the radius of the circumcircle. [Only drawing symbols required]

14. Draw a triangle where one side is 6.7 cm and the two adjacent angles are 75° and 55°. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the length of the circumradius (i.e., the radius of the circumcircle). [Only drawing symbols required]

15. The length of one side is 6.2 cm, and the measures of the two angles adjacent to that side are 50° and 75°. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

16. A right-angled triangle in which the hypotenuse is 9 cm and one of the other sides is 5.5 cm. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

17. In a right-angled triangle, the lengths of the two sides adjacent to the right angle are 4 cm and 3 cm. If the triangle is rotated once completely about the longer of these two sides as the axis, the solid formed is a cone. Calculate and write the lateral surface area, total surface area, and volume of the cone formed.

18. If the angle of elevation of the top of a chimney from a point on level ground at its base is 60°, and from another point on the same straight line 24 meters further away from the base, the angle of elevation is 30°, then calculate and write the height of the chimney. [Take the approximate value of \(\sqrt{3} = 1.732\) and find the result correct to three decimal places.]

19. Suppose a passenger on an airplane, at a certain moment, sees Howrah Station on one side and Shaheed Minar on the exact opposite side, with angles of depression of 60° and 30° respectively. At that time, if the airplane is flying at a height of \(545\sqrt{3}\) meters, then determine the distance between Howrah Station and Shaheed Minar.

20. A person is positioned along a horizontal straight line on the same plane as a chimney. From a certain point, the angle of elevation to the top of the chimney is 30°. After moving 50 meters closer along the same straight path, the angle of elevation becomes 60°. Calculate and write the height of the chimney.

21. From the roof of a 15-meter-high house situated at one end of a park, the foot and the top of a brick kiln chimney located at the other end of the park are seen at angles of depression and elevation of 30° and 60°, respectively. Find the height of the chimney and the distance between the brick kiln and the house.

22. "Prove that, in a triangle, a straight line drawn parallel to the second side through the midpoint of one side bisects the third side. [Prove using Thales's Theorem]"

23. If the ratio of three consecutive angles of a cyclic quadrilateral is 1:2:3, find the first and third angles in radians.

24. A rotating ray turns in the counterclockwise direction, completing one full revolution followed by a half revolution and then produces a \(30^\circ\) angle. What is the radian measure of the angle in trigonometric terms?

25. Here's the full translation of your math problem into English: > The height of one building forms a right angle with the window of another building situated directly across the street. From that window, the angle of elevation to the top of the first building is 30°. > If the width of the street is \(4\sqrt{3}\) meters, find: > (1) the height of the first building, and > (2) how high the window of the second building is above the street level.

26. Two circles touch each other externally at point C. AB is a common tangent to both circles and touches the circles at points A and B, respectively. Find the measure of \(\angle\)ACB.

(a) 60° (b) 45° (c) 30° (d) 90°

27. In a circle centered at O, the chords AB and CD have equal lengths. If \(\angle\)AOB = 60°, then the value of \(\angle\)COD is -

(a) 60° (b) 30° (c) 120° (d) 90°

28. Two circles touch each other externally at point \(C\). \(AB\) is a common tangent to the two circles, touching them at points \(A\) and \(B\), respectively. The measurement of \(\angle ACB\) is –

(a) 60° (b) 45° (c) 30° (d) 90°

29. If a triangle similar to one with sides 4 cm, 6 cm, and 8 cm has its largest side measuring 6 cm, what is the length of the smallest side of that triangle?

(a) 4 cm (b) 3 cm (c) 2 cm (d) 5 cm

30. In the cyclic quadrilateral ABCD, if AB = AD, \(\angle\)DAC = 70° and \(\angle\)BDC = 50°, then what is the measure of \(\angle\)ACD?

(a) 30\(^o\) (b) 40\(^o\) (c) 50\(^o\) (d) 70\(^o\)