1. Observe the dots on a dice (one to six dots) in the following figures. How many dots are contained on the face opposite to that containing four dots?

A. 2 B. 3 C. 5 D. Cannot be determined

Answer - Option A
Explanation -
We shall assume the dice in fig. (ii) to be rotated so that the 5 dots appear at the same position as in fig. (i) i.e. on RHS face (i.e. on face II as per activity 1) and 1 dot appears at the same position as in fig; (i) i.e. on Front face (i.e. on face I). Then, from the, two figures, 2 dots appear on the top face (i.e. on face V) and 4 dots appear on the Bottom face (i.e. on face VI).
Since, these two faces are opposite to each other, therefore, two dots are contained on the face opposite to that containing four dots.
2. Three different positions of a dice are shown below. How many dots lie opposite 2 dots?

A. 1 B. 3 C. 5 D. 6

Answer - Option C
Explanation -
From figures (ii) and (iii), we conclude that 1, 6, 3 and 4 dots lie adjacent to 5 dots. Therefore, 2 dots must lie opposite 5 dots. Conversely, 5 dots must lie opposite 2 dots.
3. The six faces of a dice have been marked with alphabets A, B, C, D, E and F respectively. This dice is rolled down three times. The three positions are shown as: Find the alphabet opposite A.

A. C B. D C. E D. F

Answer - Option C
Explanation - From figures (ii) and (iii), we conclude that the alphabets C, D, B and F appear adjacent to the alphabet E. Therefore, the alphabet A appears opposite E. Conversely, E appears opposite A.
4. Three positions of a dice are given. Based on them find out which number is found opposite the number 2 in the given cube.

A. 6 B. 5 C. 3 D. 1

Answer - Option A
Explanation -
From figures (i) and (ii), we conclude that the numbers 1, 4, 3 and 5 lie adjacent to the number 6. Clearly, the number 2 lies opposite 6 and conversely 6 lies opposite 2.
5. A dice is thrown four times and its four different positions are shown below. Find the number on the face opposite the face showing 2.

A. 3 B. 4 C. 5 D. 6

Answer - Option C
Explanation -
From figures (i), (ii) and (iv) We conclude that 6, 4, 3 and 1 lie adjacent to 2. Hence, 5 must lie opposite 2.

1. Two positions of a dice are shown. When 4 is at the bottom, what number will be on the top?

A. 1 B. 2 C. 5 D. 6

Answer - Option A
Explanation -
From figures (i) and (ii), we conclude that 2, 3, 5 and 6 he adjacent to 1. Therefore, 4 lies opposite 1. Hence, when 4 is at the bottom, then1 must be on the top.
2. A dice is rolled twice and the two positions are shown in the figure below. What is the number of dots at the bottom face when the dice is in position (i)?

A. 1 B. 5 C. 6 D. Cannot be determined

Answer - Option C
Explanation -
From figures (i) and (ii) we conclude that 3, 4, 1 and 5 dots appear adjacent to 2 dots. Therefore, 6 dots must appear opposite 2 dots. Since, there are 2 dots on the top face when the dice is in position (i), therefore, the number of dots at the bottom face must be 6.
3. Below are depicted the three different positions of a dice. Find the number of dots on the face opposite to the face with one dot.

A. 2 B. 3 C. 4 D. 6

Answer - Option D
Explanation -
From figures (i), (ii) and (iii), we conclude that 1, 3, 5 and 6 dots appear adjacent to the face with 2 dots. Therefore, 4 dots will appear opposite to 2 dots. Now, from figures (i) and (ii), we conclude that 2, 3 and 5 dots appear adjacent to 1 dot Therefore, either 4 or 6 dots will appear opposite to 1 dot. But since, 4 dots appear opposite to 2 dots it follows that 6 dots will appear opposite 1 dots.
4. Two positions of a paralleled piped are shown below. When the number 3 will be on the top side, then which number will be at the bottom?

A. 1 B. 4 C. 5 D. 6

Answer - Option C
Explanation -
The number 2 is common to both the figures. We assume the parallelepiped in fig. (ii) to be rotated so that 2 appears at the same position as in fig. (i) i.e. on the RHS face and the numbers 6 and 3 move to the faces hidden behind the numbers 1 and 5 respectively [in fig. (i)]. Then, the combined figure will have 1 opposite 6 and 5 opposite 3. Thus, when 3 will be on the top, then 5 will appear at the bottom.
5. A dice is numbered from 1 to 6 in different ways. If 1 is adjacent to 2, 3 and 5, then which of the following statements is necessarily true?

A. 4 is adjacent to 6 B. 2 is adjacent to 5 C. 1 is adjacent to 6 D. 1 is adjacent to 4

Answer - Option A
Explanation -
If 1 is adjacent to 2, 3 and 5, then either 4 or 6 lies opposite to 1. So, the numbers 4 and 6 cannot lie opposite to each other. Hence, 4 necessarily lies adjacent to 6.

Direction [1 - 5] : The sheet of paper shown in the figure (X) given on the left hand side, in each problem, is folded to form a box. Choose from amongst the alternatives (1), (2), (3) and (4), the boxes that are similar to the box that will be formed.
1. Choose the box that is similar to the box formed from the given sheet of paper (X).

A. 1 and 2 only B. 2 and 4 only C. 2 and 3 only D. 1 and 4 only

Answer - Option C
2. Choose the box that is similar to the box formed from the given sheet of paper (X).

A. 1 and 4 only B. 3 and 4 only C. 1 and 2 only D. 2 and 3 only

Answer - Option A
Explanation -
So, when the sheet shown in fig. (X) is folded to form a cube then one of the two half-shaded faces lies opposite to one of the blank faces and the other half-shaded face lies opposite to another blank face. The two remaining blank faces lie opposite to each other. Thus, both the cubes shown in figures (1).and (4) can be formed when the sheet shown in fig. (X) is folded. Also, though the cubes shown in figures (2) and (3) have faces that can appear adjacent to each other but the cube formed by folding the sheet in fig. (X) cannot be rotated to form either of the two. Hence, the cubes in figures (2) and (3) cannot be formed.
3. How many dots lie opposite to the face having three dots, when the given figure is folded to form a cube?

A. 2 B. 4 C. 5 D. 6

Answer - Option D
Explanation -
Therefore, when this figure is folded to form a cube then the face bearing six dots will lie opposite the face bearing three dots.
4. Choose the box that is similar to the box formed from the given sheet of paper (X).

A. 1 and 3 only B. 1 and 4 only C. 2 and 4 only D. 3 and 4 only

Answer - Option A
Explanation -
So, when the sheet shown in fig. (X) is folded to form a cube then the two half-shaded faces lie opposite to each other, the face bearing a circle lies opposite to one of the two blank faces and the two remaining blank faces lie opposite to each other. Therefore, the cubes shown in fig. (4) which has the two half-shaded faces adjacent to each other, cannot be formed by folding the sheet shown in fig. (X). Also, the cube shown in fig. (2) has the face bearing a circle adjacent to two blank faces. This is not possible since there is one blank face opposite to the circle and one blank face opposite to the third blank face. Hence, only the cubes in figures (1) and (3) can be formed.
5. Choose the box that is similar to the box formed from the given sheet of paper (X).

A. 1 and 2 only B. 2 and 3 only C. 2 and 4 only D. 1, 2, 3 and 4

Answer - Option D
Explanation -
So, when a cube is formed by folding the sheet shown in fig. (X), then the two half-shaded faces lie opposite to each other and one. of the three blank faces appears opposite to the face bearing a dot. Clearly, each one of the four cubes shown in figures (1), (2), (3) and (4) can be formed by folding the sheet shown in fig. (X).