Mensuration Test-1

When a 5 cc cube is sliced into 1 cc cubes, we will get 5*5*5 = 125 1 cc cubes.

In each side of the larger cube, the smaller cubes on the edges will have more than one of their sides painted.
Therefore, the cubes which are not on the edge of the larger cube and that lie on the facing sides of the larger cube will have exactly one side painted.

In each face of the larger cube, there will be 5*5 = 25 cubes. Of these, there will be 16 cubes on the edge and 3*3 = 9 cubes which are not on the edge.
Therefore, there will be 9 1-cc cubes per face that will have exactly one of their sides painted.
In total, there will be 9*6 = 54 such cubes.

Let ‘a ’meters be the length of a side of the square field.

Therefore, its area = a²  square meters. --- (1)

We know that the length of the diagonal ‘d ’of a square whose side is ‘a ’meters = √2---(2)
From (1) and (2), we can deduce that the square of the diagonal = d²  = 2a ²

The time taken to cross a length of 220 meters while traveling at 6.6 kmph is given by
 (converting 1 km = 1000 meters and 1 hour = 60 minutes).

The circumference of the front wheel is 30 ft and that of the rear wheel is 36 feet.

Let the rear wheel make n revolutions. At this time, the front wheel should have made n+5 revolutions.

As both the wheels would have covered the same distance, n*36 = (n+5)*30

36n = 30n + 150

6n = 150

n = 25.

Distance covered = 25*36 = 900 ft.

The sides of the triangle happen to be a Pythagorean triplet. Hence, the triangle is a right angled triangle. 

For a right angled triangle, 

The measure of in radius = (product of perpendicular sides) / (perimeter of triangle)

= (72 * 21) / (72 + 21 +75) = 1512 / 168 = 9

The volume of the larger cube = 5³  = 125 cm³ .

The volume of each of the smaller cubes = 1³  = 1 cm³ . Therefore, one would get 125 smaller cubes.

The surface area of the larger cube = 6a²  = 6(5² ) = 6 * 25 = 150

The surface area of each of the smaller cubes = 6 (1² ) = 6.

Therefore, surface area of all of the 125, 1 cm³  cubes = 125 * 6 = 750.

Therefore, the required ratio = 150 : 750 = 1 : 5

The sum of the exterior angle and interior angle of a regular polygon is 360⁰ .

As the measure of each interior angle is 150, the measure of each exterior angle will be 30⁰ .

The number of sides of a regular polygon is given by the following relationship n = 360/exterior angle

As the value of each exterior angle is 30⁰ , the number of sides = 12.

The length of the rope in which the horses tied should be equal to half of the side of the square plot so that they just cannot reach one another.

Therefore, the length of the rope is 35m (70/2).

The area covered by each horse should be equal to the area of sector with radius of 70/2 = 35m(length of the rope).

Total area covered by the four horses = 4* area of sector of radius 35 metres = Area of circle of radius 35m.

Area left ungrazed by the horses = Area of square field - Area covered by four horses.

= 70²  - (22/7)*35*35 = 4900 - 3850 = 1050 sq.m.

Surface area of the cylinder = Surface area of the square  2 πh = a²
Here height 'h 'of the cylinder = side of the square since it is rolled along its length. 2 πa = a²
Base radius r = a / 2 π
Ratio of base radius to side of square = 

If l, b, h be the dimensions of the cuboid, then volume of the cuboid = l X b X h

The diagonal  d  = 25m. and area A = 168 m² .
Let 'l 'be the length and 'b 'be the width of the rectangle.
Therefore, l²  + b²  = d² . and  lb  = A
We can therefore write (l  + b )²  = d²  + 2A and (l  - b )²  = d²  - 2A.
Substituting and solving we get, l  + b  = 31 and  l  - b  = 17. Hence  l  = 24 and  b  = 7