Question : The length of the largest possible rod that can be placed in a cubical room is $35\sqrt3$ metre. The surface area of the largest possible sphere that fits within the cubical room (assuming $\pi=\frac{22}{7}$) (in sq. metres) is:
Option 1: 3,500
Option 2: 3,850
Option 3: 2,450
Option 4: 4,250
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Correct Answer: 3,850
Solution : Given: The length of the largest possible rod that can be placed in a cubical room is $35\sqrt3$ metres. The length of the largest rod = diagonal of the cube = side × $\sqrt3$ Here, side × $\sqrt3=35\sqrt3$ Or, the side of the cube = 35 metres Now, the radius of the largest possible sphere that fits within the cubical room = $\frac{35}{2}$ metres Therefore, the surface area of the sphere: = $4\pi (\frac{35}{2})^2$ = $4×\frac{22}{7}×\frac{35}{2}×\frac{35}{2}$ = 3850 sq. metres Hence, the correct answer is 3850.
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Question : The length of two parallel sides of a trapezium are 15 metres and 20 metres. If its height is 22 metres, then what is the area of the trapezium?
Option 1: 295 metre2
Option 2: 385 metre2
Option 3: 425 metre2
Option 4: 315 metre2
Question : A train can completely cross a 1000-metre-long bridge in 15 seconds. It can completely cross another 1300-metre-long bridge in 19 seconds. What is the length of the train?
Option 1: 125 metres
Option 2: 200 metres
Option 3: 150 metres
Option 4: 175 metres
Question : The ratio of the area of an equilateral triangle and that of its circumcircle is:
Option 1: $2\sqrt3:2\pi$
Option 2: $4:\pi$
Option 3: $3\sqrt3:4\pi$
Option 4: $7\sqrt2:2\pi$
Question : A 1000-metre-long train is running at the speed of 72 km/hr. If it crosses a bridge in 85 seconds, then what is the length of the bridge?
Option 1: 800 metres
Option 2: 700 metres
Option 3: 950 metres
Option 4: 750 metres
Question : The surface area of a sphere is 2464 cm². Find the radius of the sphere. [Use $\pi=\frac{22}{7}$ ]
Option 1: 14 cm
Option 2: 28 cm
Option 3: 42 cm
Option 4: 40 cm
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