21). The lateral surface of a cylinder is developed in a square whose diagonal is \( \Large \sqrt{2} \) cm long. Find the volume of the cylinder in cubic centimeters is, if height is 1 cm.
| A). \( \Large \frac{1}{4 \pi } \) |
| B). \( \Large \frac{2}{ \pi } \) |
| C). \( \Large 3\frac{ \pi }{4} \) |
| D). \( \Large 2 \pi \) |
|
22). A plane is drawn parallel to the base of a right circular cone dividing altitude in the ratio of 2 : 1 to chop off the conical part. The chopped off conical part is taller than the remaining portion. The ratio of the volumes of the chopped off part and the remaining part is
| A). 1 : 26 |
| B). 2 : 25 |
| C). 4 : 23 |
| D). 8 : 19 |
|
23). Three cubes of metal whose edges are 3cm, 4cm, 5cm, respectively are melted and a new cube is formed. The diagonal of the new cube is
| A). \( \Large 4\sqrt{3} cm \) |
| B). 6 cm |
| C). \( \Large 6\sqrt{3} cm \) |
| D). 8 cm |
|
24). The length, breadth and height of a rectangular solid are 10 cm, 5 cm and 2 cm respectively. Its whole surface area in sq. cms is
| A). 7 |
| B). 70 |
| C). 160 |
| D). 280 |
|
25). A gold bar in the shape of a rectangular solid, 36 cms long, 9 cms broad and 6 cms high, is to be melted and cast into two different cubes; volume of the bigger being eight times that of the smaller. The surface area of the smaller cube, in \( cm^{2} \) is
| A). 36 |
| B). 216 |
| C). 72 |
| D). 144 |
|
26). A hollow right circular cylinder of radius r and height 4r is standing vertically on a plane. If a solid right circular cone of radius 2r and height 6r is placed with its vertex down in the cylinder, then volume of the portion of the cone outside the cylinder is
| A). \( \Large 7 \pi r^{3} \) |
| B). \( \Large \frac{8}{3} \pi r^{3} \) |
| C). \( \Large \frac{9}{8} \pi r^{3} \) |
| D). \( \Large \frac{14}{3} \pi r^{3} \) |
|
27). A right pyramid on a regular hexagonal base is of height 60m. Each side of the base is 10m. The volume of the pyramid is
| A). \( \Large 4500 m^{3} \) |
| B). \( \Large 5000 m^{3} \) |
| C). \( \Large 5196 m^{3} \) |
| D). \( \Large 6196 m^{3} \) |
|
28). If a sphere of maximum volume is placed inside a hollow right circular cone with radius r and slant height I such that base of the cone touches the sphere, then volume of the sphere is
| A). \( \Large \frac{4}{3} \pi \left(\frac{l+r}{l-r}\right)^{3} \) |
| B). \( \Large \frac{4}{3} \pi r^{3} \left(\frac{l-r}{l+r}\right)^{\frac{3}{2}} \) |
| C). \( \Large \frac{4}{3} \pi \left(\frac{l-r}{l+r}\right)^{3} \) |
| D). \( \Large \frac{4}{3} \pi r^{3} \left(\frac{l+r}{l-r}\right)^{\frac{3}{2}} \) |
|
29). If diameter of the base of a right circular cylinder is r and its height is equal to radius of the base, then its volume is
| A). \( \Large \frac{1}{8} \pi r^{3} \) |
| B). \( \Large \pi r^{3} \) |
| C). \( \Large 2 \pi r^{3} \) |
| D). \( \Large 4 \pi r^{3} \) |
|
30). If numerical value of the curved surface area of a right circular cylinder is equal to numerical value of its volume, then numerical value of the radius of the base of the cylinder is
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