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. If E and F are the midpoints of sides AB and AD respectively in parallelogram ABCD, and the area of parallelogram ABCD is 1600 square centimeters, then what is the area of triangle AEF?

(a) 400 sq cm (b) 200 sq cm (c) 300 sq cm (d) None of the above

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. 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

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 \(\triangle ABC\), 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

7. 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?

8. A ring-shaped copper plate has an area of 269.5 square centimeters. If the outer radius of the ring is 28 cm, what is its inner radius?

(a) 10.5 cm (b) 21 cm (c) 12 cm (d) 24 cm

9. 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?

10. If a right circular cone has a vertex angle of 60° and a base radius of 14 cm, then what is the curved surface area of the cone?

(a) 1680 square cm (b) 1232 square cm (c) 1080 square cm (d) 1442 square cm

11. A rectangular piece of soft cardboard has a length that is four times its width. When rolled along its length, it forms a right circular cone whose lateral surface area is 100 square centimeters. What is the height of the cone?

12. 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

13. In a right-angled quadrilateral, if the number of vertices is denoted by \(x\), the number of sides by \(y\), and the number of diagonals by \(z\), then what is the value of \(x + 3y - 5z\)?

(a) 14 (b) 44 (c) 20 (d) 24

14. If a right circular cone has a base diameter of 7 cm and a slant height of 10 cm, what is the total surface area of the cone?

(a) 184.5 square cm (b) 185.4 square cm (c) 148.5 square cm (d) 184.6 square cm

15. If the radius of a sphere is increased by 2 cm, its curved surface area increases by 352 square centimeters. What was the original radius of the sphere?

(a) 5 cm (b) 6 cm (c) 7 cm (d) 5.6 cm

16. 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

17. 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?

18. If a right circular cone has a height of 12 cm and a volume of \(100\pi\) cubic cm, then what is the radius of the cone?

19. 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

20. 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?

21. 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 \(\frac{xy}{z}\)?

22. If the area of the square drawn on one side of any triangle is equal to the sum of the areas of the squares drawn on the other two sides, then prove that the angle opposite to the first side is a right angle.

23. Draw a rectangle with sides of 8 cm and 6 cm, and then draw a square having the same area as that rectangle. (Only construction marks are required.)

24. An open cylindrical vessel has a total surface area of 2002 square centimeters. If the radius of its base is 7 cm, how many liters of water can it hold? (1 liter = 1 cubic decimeter)

25. To make a vertical circular conical tent, 77 square meters of canvas is used. If the slant height of the tent is 7 meters, then what is the area of the base of the tent?

26. Prove that in a triangle, if the area of the square constructed on one side is equal to the sum of the areas of the squares constructed on the other two sides, then the angle opposite the longest side is a right angle.

27. 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 \(\frac{YZ}{X}\)?

28. The length of a rectangular field is \(\frac{3}{2}\) times its breadth. If the length is reduced by 1200 cm and the breadth is increased by 1200 cm, the field becomes a square. Find the area of the field.

29. A right-angled triangular prism has a base area of 100 square centimeters, and the areas of the two adjacent faces to the base are 40 square centimeters and 160 square centimeters. Its volume will be—

(a) 800 cubic cm (b) 880 cubic cm (c) 890 cubic cm (d) 900 cubic cm

30. If the diagonal of a rectangle is 10 cm and its area is 62.5 square cm, then the sum of the length and breadth of the rectangle is —

(a) 12 cm (b) 15 cm (c) 20 cm (d) 25 cm