Previous Year Topic Wise Questions With Solutions: Mensuration-Volume, Surface Area & Solid Figures | CSAT Preparation - UPSC PDF Download

Ans: (b)
Let n be the number of cubes on each edge of the cube. n³ = 64
or n = 4
So, there are 4 cubes on each edge of the cube. 
For this to happen we need to cut each edge 3 times.
This can be achieved by cutting the cube 3 times along each of the three axes – x, y and z. 
So, the total number of minimum possible cuts required = 3 + 3 + 3 = 9

Ans: (a)
We have to find the number of internal cubes, i.e. the number of cubes that are not exposed
There are 125 cubes. Now, 5³ = 125. So, n = 5 
Number of internal cubes = (n - 2)³ = (5 - 2)³ = 3³ = 27 
Hence, option (a) is correct.

Ans: (a)
(57242)^{9 × 7 × 5 × 3 × 1} = (57242)945
The unit digit of the resultant number only depends on the unit digit of the given number 57242, i.e. 2. 
Now, we know that:
2¹ = 2
2² = 4
2³ = 8
2⁴ = 16 (unit digit 6)
2⁵ = 32 (unit digit 2)
.
.
.
And so on.
So, 2 has a cyclicity of 4. Exponent of any number ending in 2 will produce a number that will end in 2, 4, 8, or 6.
Now, 945 = 944 + 1
944 is divisible by 4. So, the last digit of (57242)⁹⁴⁵ will be the same as that of (57242)¹, which is 2. 
Hence, option (a) is correct.

Ans: (c)
The rectangle is of dimensions 20 cm × 8 cm. 
Statement I: You may think that as the area of rectangle is not a perfect square, it is not possible to cut it into exactly 4 square sheets. 
But there’s a catch. The statement never says that the 4 squares have to be equal in area. 
We can do so as follows:
Statement II: You may think that we can cut the rectangle into 8 triangles of equal area (as shown below), but not in 10.

But again, this can be done. It has been shown below:

Hence, both statements 1 and 2 are correct.

Ans: (d)
Volume of the cube = Side × Side × Side = 1 × 1 × 1 = 1 cubic meter
Now, 1 cubic meter = 1000000 mililiters