1. "A, B, C, and D are respectively the midpoints of the sides PQ, QR, RS, and SP of the parallelogram PQRS. If the area of the parallelogram PQRS is 36 square centimeters, then what is the area of the quadrilateral ABCD?"

(a) 24 square cm (b) 18 square cm (c) 30 square cm (d) 36 square cm

2. In triangle \(\triangle ABC\), D and E are the midpoints of sides AB and AC respectively. If the area of triangle ABC is 36 square centimeters, then what is the area of triangle ADE?

(a) 6 square centimeters (b) 9 square centimeters (c) 12 square centimeters (d) 16 square centimeters

3. In triangle ABC, E and F are the midpoints of sides AB and AC respectively. If the area of triangle AEF is 50 square cm, then what is the area of triangle ABC?

(a) 100 sq cm (b) 200 sq cm (c) 150 sq cm (d) 300 sq cm

4. A straight line parallel to side BC of triangle ∆ABC intersects sides AB and AC at points D and E respectively. If AD : BD = 3 : 5, then what is the ratio of the area of triangle ∆ADE to the area of trapezium DBCE?

5. In the parallelogram ABCD, point P is the midpoint of side CD. If the area of triangle \( \triangle APD \) is 25 square centimeters, then what is the area of the parallelogram?

(a) 100 square cm (b) 75 square cm (c) 150 square cm (d) 50 square cm

6. In triangle \(\triangle ABC\), AB = AC. Points E and F are the midpoints of sides AB and AC respectively. AD is perpendicular to BC, and AD = 4 cm. If EF = 3 cm, then what is the length of BD?

(a) 4 cm (b) 3 cm (c) 6 cm (d) 7 cm

7. If the quadrilateral formed by joining the midpoints of the sides of parallelogram ABCD has an area of 100 square cm, then what is the area of the parallelogram ABCD?

(a) 400 sq cm (b) 200 sq cm (c) 600 sq cm (d) 800 sq cm

8. If the length, breadth, and height of a right rectangular prism are in the ratio 3:2:1 and its total surface area is 88 square centimeters, then what is its volume?

(a) 120 cubic cm (b) 64 cubic cm (c) 48 cubic cm (d) 24 cubic cm

9. A hollow cylindrical iron pipe open at both ends has an inner radius and outer radius of 4 cm and 5 cm respectively. If the total surface area of the pipe is 1188 square centimeters, what is the length of the pipe?

10. In a right-angled triangle, the lengths of the two sides adjacent to the right angle are 4 cm and 3 cm respectively. If the triangle is rotated once completely around the longer of the two adjacent sides, find the total surface area and the volume of the solid formed.

11. If the curved surface area of a right circular cylinder is 264 square meters and its volume is 924 cubic meters, then what is the radius of the base?

12. In triangle ABC, let X and Y be the midpoints of sides AB and AC respectively. If \(BC + XY = 12\) units, then what is the value of \(BC - XY\)?

13. In triangle ABC, the midpoints of sides BC, CA, and AB are D, E, and F respectively. BE and DF intersect at point X, and CF and DE intersect at point Y. Then, the length of XY is equal.

(a) \(\cfrac{1}{2}\)BC (b) \(\cfrac{1}{4}\)BC (c) \(\cfrac{1}{3}\)BC (d) \(\cfrac{1}{8}\)BC

14. Point O is any point inside parallelogram ABCD. If \(\triangle AOB + \triangle COD = 16\) square cm, then what is the area of parallelogram ABCD?

(a) 8 square cm (b) 4 square cm (c) 32 square cm (d) 64 square cm

15. ABC is a right-angled triangle with \(\angle ABC = 90^\circ\); if the coordinates of points A and C are (0, 4) and (3, 0) respectively, then the area of triangle ABC is:

(a) 12 square units (b) 6 square units (c) 24 square units (d) 8 square units

16. If the height of a right circular cylinder is 14 cm and the curved surface area is 264 square cm, then what is the volume of the cylinder?

17. ABC is an acute-angled triangle. AP is the diameter of the circumcircle of triangle ABC. BE and CF are perpendiculars dropped to sides AC and AB respectively, and they intersect at point Q. Prove that BPCQ is a parallelogram.

18. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points P and Q respectively. If \( AB = 3 \times PB \) and \( BC = 18 \) cm, then what is the length of \( PQ \)?

(a) 10 cm (b) 9 cm (c) 12 cm (d) 8 cm

19. The length of a right rectangular prism is three times its width and five times its height. If its volume is 14,400 cubic centimeters, then what is its total surface area?

(a) 4300 square cm (b) 4320 square cm (c) 4500 square cm (d) 4520 square cm

20. If the volume of a cube is \(V\) cubic centimeters, the total surface area is \(S\) square centimeters, and the length of the diagonal is \(d\) centimeters, then prove that \(Sd = 6\sqrt{3}V\).

21. Here’s the English translation of your sentence: "A right rectangular prism has its length twice the breadth and its height half the breadth. If the total surface area of the prism is 448 square centimeters, find its volume."

22. A hollow vertical cylindrical iron pipe has an outer radius of 5 cm and an inner radius of 4 cm. If the total surface area of the pipe is 1188 square cm, what is the length of the pipe?

23. If the curved surface area of a sphere is \(S\) square units and its volume is \(V\) cubic units, then determine the relationship between \(S\) and \(V\).

24. A rectangular playground is 45 meters long and 40 meters wide. Around it, there is a road of uniform width along all four sides. The area of the road is 450 square meters. What is the width of the road?

(a) 5 meter (b) 2.5 meter (c) 1.5 meter (d) 4.5 meter

25. If the base area of a cone is \(x\) square units, its height is \(y\) units, and its volume is \(z\) cubic units, then what is the value of \(\frac{xy}{z}\)?

26. The volume of a right circular cone is \(V\) cubic units. If the area of the base is \(A\) square units and the height is \(H\) units, then find the value of \(\frac{AH}{V}\).

27. If the bases of two triangles lie on the same straight line and the other vertex of both triangles is common, then the ratio of their areas is ______to the ratio of the lengths of their bases.

28. If the volume of a right circular cone is \(X\) cubic units, the area of its base is \(Y\) square units, and its height is \(Z\) units, then what is the value of \(\frac{YZ}{X}\)?

29. 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.

30. A classroom needs to be built for 40 students in such a way that each student gets 5 square meters of floor area and 25 cubic meters of space. If the length of the room is 20 meters, what should be its height?

(a) 5 meters (b) 7 meters (c) 9 meters (d) 11 meters