1. A cube has six faces. How many minimum colours are required to paint all faces of a cube such that no two adjacent faces have the same colour?
A) 2
B) 3
C) 4
D) 6
Answer: B) 3
Explanation: Opposite faces can have the same colour; hence, minimum 3 colours are needed.
2. How many faces are there on a cube?
A) 4
B) 5
C) 6
D) 8
Answer: C) 6
Explanation: A cube or dice has 6 square faces.
3. How many edges does a cube have?
A) 8
B) 12
C) 6
D) 10
Answer: B) 12
Explanation: A cube has 6 faces, 8 vertices, and 12 edges.
4. How many vertices are there in a cube?
A) 4
B) 6
C) 8
D) 12
Answer: C) 8
Explanation: Each corner of a cube is a vertex; total 8 vertices exist.
5. On a dice, opposite faces always:
A) Have consecutive numbers
B) Have same colour
C) Have specific relation (sum often 7)
D) Have same number
Answer: C) Have specific relation (sum often 7)
Explanation: In standard dice, opposite faces add up to 7.
6. In a dice, if the face opposite 1 is 6, then opposite 2 is:
A) 3
B) 5
C) 4
D) Cannot be determined
Answer: D) Cannot be determined
Explanation: Unless full dice faces are shown, opposite faces cannot be identified directly.
7. A dice shows 2 opposite 5, and 3 opposite 6. Which is opposite 1?
A) 4
B) 2
C) 3
D) 5
Answer: A) 4
Explanation: Each face has only one opposite face. So if (2–5), (3–6), then (1–4).
8. In a standard dice, which of the following pairs are opposite faces?
A) 1–2
B) 1–3
C) 1–6
D) 1–5
Answer: C) 1–6
Explanation: Standard dice opposite faces are (1–6), (2–5), (3–4).
9. A dice is rolled once. What is the probability of getting an even number?
A) 1/2
B) 1/3
C) 2/3
D) 1/6
Answer: A) 1/2
Explanation: Even numbers = {2, 4, 6}, total outcomes = 6 → 3/6 = 1/2.
10. Which of these statements about dice is false?
A) Opposite faces are equal.
B) A dice has 12 edges.
C) Adjacent faces share a common edge.
D) Opposite faces share a common edge.
Answer: D) Opposite faces share a common edge.
Explanation: Opposite faces never share an edge.
11. In a dice, how many faces are adjacent to any one face?
A) 2
B) 3
C) 4
D) 5
Answer: C) 4
Explanation: Each face is adjacent to 4 faces and opposite to 1.
12. If the top and front faces are known, how many faces can be hidden?
A) 3
B) 4
C) 2
D) 1
Answer: B) 4
Explanation: Since 2 are visible, 4 are hidden (as total = 6).
13. When a cube is cut into smaller cubes of equal size, number of smaller cubes is given by:
A) n³
B) n²
C) 6n
D) n⁴
Answer: A) n³
Explanation: If cube edge is divided into n parts, smaller cubes = n³.
14. On a dice, if 2 is opposite 6 and 3 is opposite 4, which face is opposite 5?
A) 1
B) 2
C) 3
D) 6
Answer: A) 1
Explanation: Every face has one opposite face only, so (1–5).
15. In a cube, how many pairs of opposite faces exist?
A) 2
B) 3
C) 4
D) 6
Answer: B) 3
Explanation: 6 faces → 3 pairs of opposite faces.
16. Two positions of a dice are shown:
(1) 2 on top, 4 in front
(2) 4 on top, 6 in front.
Which number will be on the bottom face when 2 is on top?
A) 4
B) 6
C) 1
D) 5
Answer: D) 5
Explanation: By rotation rules, opposite of 2 is 5.
17. When opposite faces of a dice are added, the sum is 7. If top is 3, bottom is:
A) 2
B) 4
C) 5
D) 4
Answer: C) 4
Explanation: 3 + 4 = 7, hence opposite face is 4.
18. A dice is numbered 1–6. Which number is opposite 2, if opposite pairs are (1–6), (3–5)?
A) 4
B) 3
C) 5
D) 6
Answer: A) 4
Explanation: Remaining number 2’s opposite is 4.
19. If opposite faces of a cube are painted differently, how many colours are needed?
A) 2
B) 3
C) 4
D) 6
Answer: B) 3
Explanation: Opposite faces can repeat, so 3 colours suffice.
20. A cube painted on all faces is cut into 64 small cubes. How many small cubes have 3 faces painted?
A) 4
B) 8
C) 12
D) 24
Answer: B) 8
Explanation: Only corner cubes have 3 faces painted → 8 corners.
21. In a cube of side divided into 3 parts (27 small cubes), how many will have exactly two faces painted?
A) 6
B) 8
C) 12
D) 24
Answer: C) 12
Explanation: Edge cubes excluding corners → 12 such cubes.
22. A cube is painted red on all faces and cut into 27 small cubes. How many cubes will have only one face painted?
A) 9
B) 12
C) 6
D) 24
Answer: A) 9
Explanation: Center cubes on each face (3×3 – corners and edges removed) = 9.
23. In a cube of side 4 cm painted on all sides and cut into 1 cm cubes, how many cubes will be completely unpainted?
A) 8
B) 16
C) 27
D) 64
Answer: C) 27
Explanation: Inner unpainted layer forms (n−2)³ = (4−2)³ = 8 cubes.
24. Opposite faces of a cube are painted red, blue, and green. How many different colour pairs exist?
A) 2
B) 3
C) 4
D) 6
Answer: B) 3
Explanation: Each pair of opposite faces has one colour pair → 3 total.
25. In dice reasoning, opposite faces never appear:
A) Adjacent
B) Together in one view
C) In same direction
D) All of these
Answer: D) All of these
Explanation: Opposite faces can never be adjacent or visible at the same time.
26. Two different positions of a dice are shown:
1st: 1 is on top and 4 is in front.
2nd: 4 is on top and 6 is in front.
Which number will be on the bottom when 1 is on top?
A) 3
B) 5
C) 6
D) 2
Answer: B) 5
Explanation: Since 4 is adjacent to both 1 and 6, 6 cannot be opposite 1. The only number left is 5.
27. Two positions of a dice are shown:
(1) 2 is on top and 6 in front
(2) 4 is on top and 2 in front.
Which number is on the bottom when 6 is on top?
A) 4
B) 2
C) 5
D) 3
Answer: D) 3
Explanation: Faces adjacent in both positions can’t be opposite; remaining face (3) is opposite to 6.
28. If two adjacent faces of a cube are red and blue, what will be the colour of the face opposite to red?
A) Blue
B) Green
C) Yellow
D) Cannot be determined
Answer: D) Cannot be determined
Explanation: Without seeing the cube fully, opposite faces cannot be decided from one pair only.
29. A dice has numbers 1, 2, 3 on adjacent faces. What number must be opposite to 1?
A) 2
B) 3
C) 4
D) 5
Answer: C) 4
Explanation: Opposite faces can’t be adjacent; so opposite of 1 = 4.
30. Two positions of a dice are shown below:
(1) 3 is on top, 1 in front
(2) 5 is on top, 3 in front.
Which number is opposite 1?
A) 2
B) 4
C) 6
D) 5
Answer: B) 4
Explanation: The common number (3) helps determine rotation; the face opposite 1 is 4.
31. In a dice, 2, 3, and 5 are adjacent to 6. Which number is opposite 6?
A) 1
B) 2
C) 3
D) 5
Answer: A) 1
Explanation: Opposite face of 6 is the one not adjacent to it → 1.
32. Two dice show numbers as:
(1) 5, 3, 1
(2) 3, 1, 6
If 5 is opposite 6, what number is opposite 3?
A) 1
B) 2
C) 4
D) 5
Answer: C) 4
Explanation: Common numbers (3, 1) are adjacent in both; remaining pair (5–6) are opposite → opposite of 3 = 4.
33. Two different dice show numbers as below:
(1) 1 opposite to 6
(2) 2 opposite to 5
(3) 3 opposite to 4
What is the sum of numbers on opposite faces?
A) 6
B) 7
C) 8
D) 9
Answer: B) 7
Explanation: Standard dice rule – sum of opposite faces = 7.
34. A dice has 2 on top, 4 in front, and 6 at right. What number is on the face opposite 6?
A) 1
B) 3
C) 5
D) 4
Answer: C) 5
Explanation: Adjacent faces to 6 are 2 and 4; opposite will be 5.
35. A dice has numbers 1, 2, 3 adjacent to 4. Which number is opposite 4?
A) 5
B) 6
C) 2
D) Cannot be determined
Answer: D) Cannot be determined
Explanation: We cannot find opposite face with partial adjacency data.
36. If two adjacent faces of a cube are red and blue, and the third adjacent is green, the face opposite to green is:
A) Red
B) Blue
C) Yellow
D) Cannot be determined
Answer: D) Cannot be determined
Explanation: Opposite colour can’t be decided without knowing full arrangement.
37. A dice shows numbers 4, 2, 6 on three adjacent faces. What number is opposite to 4?
A) 3
B) 5
C) 2
D) 6
Answer: B) 5
Explanation: The only number not adjacent to 4 is 5.
38. Two positions of a dice are given:
(1) 3 at top, 4 in front
(2) 4 at top, 2 in front.
Which number will be opposite to 2?
A) 3
B) 4
C) 5
D) 6
Answer: D) 6
Explanation: Common face 4 helps determine orientation; 2 and 6 are opposite.
39. A cube has faces numbered 1–6. If opposite faces are (1–6), (2–5), (3–4), then what number is opposite to 5?
A) 2
B) 4
C) 6
D) 1
Answer: A) 2
Explanation: Given opposite face pair (2–5).
40. If in a dice 1 is opposite 3, and 2 is opposite 4, then which number is opposite 5?
A) 1
B) 2
C) 6
D) 3
Answer: C) 6
Explanation: Only remaining number (6) will be opposite to 5.
41. Two positions of a dice are given:
(1) 2 opposite 6,
(2) 3 opposite 4.
Which is opposite to 1?
A) 5
B) 4
C) 3
D) 2
Answer: A) 5
Explanation: After using given pairs, 1 remains opposite to 5.
42. If two dice have 3 on top and 5 in front in one, and 5 on top and 2 in front in the other, which number is opposite 2?
A) 1
B) 3
C) 4
D) 6
Answer: D) 6
Explanation: Common face logic — 5 is common → opposite to 2 is 6.
43. In a dice, 2 and 3 are adjacent to 6, and 4 is opposite to 3. What is opposite to 2?
A) 4
B) 1
C) 5
D) 6
Answer: C) 5
Explanation: As per adjacency, 2 cannot be opposite 6, 3, or 4 → hence opposite is 5.
44. When a dice is rolled twice, what is the maximum possible sum?
A) 6
B) 9
C) 12
D) 11
Answer: C) 12
Explanation: Maximum on each roll = 6; total = 6 + 6 = 12.
45. When a dice is rolled once, what is the probability of getting 1 or 6?
A) 1/2
B) 1/3
C) 1/6
D) 2/6
Answer: D) 2/6
Explanation: Favourable outcomes = {1, 6}, total = 6 → 2/6 = 1/3.
46. If three different positions of a dice show (2, 4, 6), (4, 6, 5), (6, 5, 3), then opposite of 2 is:
A) 1
B) 3
C) 4
D) 5
Answer: B) 3
Explanation: Only 2 and 3 never appear together, hence they are opposite.
47. Two dice show the following faces:
(1) 1, 2, 3
(2) 2, 3, 4
Which number is opposite 1?
A) 2
B) 3
C) 4
D) Cannot be determined
Answer: C) 4
Explanation: 1 and 4 never appear together → they are opposite.
48. Two positions of a dice are given:
(1) 5, 3, 2
(2) 3, 2, 6
Find the number opposite to 5.
A) 6
B) 3
C) 2
D) 4
Answer: A) 6
Explanation: 5 and 6 never appear together → they are opposite.
49. A dice shows numbers 1, 3, 5 on its three adjacent faces. What is the sum of numbers on the faces opposite to these?
A) 9
B) 12
C) 15
D) 18
Answer: B) 12
Explanation: Opposite faces sum to 7 → (7–1)+(7–3)+(7–5) = 6+4+2 = 12.
50. Two dice show faces as:
(1) 1, 2, 3
(2) 3, 4, 1
Which face is opposite 2?
A) 1
B) 3
C) 4
D) Cannot be determined
Answer: C) 4
Explanation: 2 and 4 never appear together → they must be opposite.
51. Two views of a dice are shown:
View-A: Top = 1, Front = 2, Right = 3.
View-B: Top = 2, Front = 5, Right = 1.
Which number is opposite to 3?
A) 1
B) 4
C) 5
D) 6
Answer: B) 4
Explanation: From View-A, 1,2,3 are mutually adjacent, so opposite of 3 is one of {4,5,6}. In View-B, 1 and 2 are adjacent, and 5 is adjacent to 2 — so 5 is not opposite 3. Also 1 is adjacent to 3 (from View-A), so cannot be opposite. Remaining possibility is 4.
52. Three positions of a dice are given:
(1) faces visible: 2, 3, 5
(2) faces visible: 3, 4, 6
(3) faces visible: 5, 6, 1
Which face is opposite 2?
A) 3
B) 4
C) 1
D) Cannot be determined
Answer: B) 4
Explanation: From (1) 2 is adjacent to 3 and 5. From (2) 3 adjacent to 4 and 6. Since 2 and 4 never appear together in any view, they are likely opposite. Check other faces: 1 appears with 5 & 6 (3), so 1 is adjacent to 5/6, not necessarily opposite to 2. The consistent opposite for 2 is 4.
53. Two views:
View-1: Top = 6, Front = 2
View-2: Top = 3, Front = 6
What is on the bottom when the face showing 2 is on top?
A) 3
B) 4
C) 5
D) 1
Answer: C) 5
Explanation: From View-1, 6 opposite ? and adjacent to 2. From View-2, 6 adjacent to 3, so 3 and 2 are adjacent to 6 meaning opposite pairs are (6–?), (2–?), (3–?). Standard opposite set that fits common dice is (1–6),(2–5),(3–4). Thus when 2 is top, bottom = 5.
54. A dice shows 1, 2, 3 on three faces that meet at one corner. What is the sum of numbers on the three faces opposite to these?
A) 6
B) 9
C) 12
D) 15
Answer: C) 12
Explanation: Opposites sum to 7, so opposites are 6,5,4 → sum = 6+5+4 = 15? Wait — check: If faces are 1,2,3 then opposites are (6,5,4) whose sum is 15. But exam-standard dice have opposite pairs summing to 7: 1↔6, 2↔5, 3↔4 → sum = 6+5+4 = 15.
Corrected Answer: D) 15.
(Note: careful arithmetic — opposites of 1,2,3 are 6,5,4 → 6+5+4 = 15.)
55. Two positions:
(1) 4 (top), 1 (front)
(2) 1 (top), 3 (front)
Which number is on the right face in position (1)?
A) 2
B) 3
C) 5
D) 6
Answer: A) 2
Explanation: From (1), 4 adjacent to 1. From (2), 1 adjacent to 3. So around the corner the triplet 4-1-3 appear cyclically; the remaining numbers adjacent to these are 2 and 5/6. Using standard opposite pairs (1–6),(2–5),(3–4), if 3 opposite 4, then 2 must be right of 4.
56. Three views:
A: visible faces 1,4,5
B: visible faces 2,3,1
C: visible faces 3,6,5
Which number is opposite 1?
A) 2
B) 3
C) 6
D) 4
Answer: C) 6
Explanation: From B, 1 adjacent to 2 & 3, so opposite of 1 is not 2 or 3. From A, 1 adjacent to 4 & 5, so opposite is not 4 or 5. Hence remaining number is 6.
57. A standard dice is rolled twice. Given the first roll is 4, what is the probability the second roll is greater than the first?
A) 1/6
B) 1/3
C) 1/2
D) 1/4
Answer: B) 1/3
Explanation: Numbers greater than 4 are {5,6} → 2 favourable out of 6 → 2/6 = 1/3.
58. A dice has faces 1–6. If 2 is adjacent to 3, 3 adjacent to 4, and 4 adjacent to 5, which number is opposite to 3?
A) 1
B) 2
C) 6
D) Cannot be determined
Answer: C) 6
Explanation: Adjacent to 3 are 2 and 4, so opposite must be one of {1,5,6}. But 4 adjacent to 5 means 5 is adjacent to 4 (so 5 likely not opposite to 3), 1 hasn’t been paired yet. Using standard sums (if assumed), 3↔4 would be inconsistent. Best consistent opposite is 6 if we assume standard pairing (3–4 are opposites in standard dice? Actually standard is 3–4 opposite). Wait — check consistency: standard opposite pairs are (1–6),(2–5),(3–4). Given 3 adjacent to 4 contradicts standard. The puzzle’s given adjacency implies 3 cannot be opposite 4. Without full info, result is D) Cannot be determined.
Corrected Answer: D) Cannot be determined.
59. A cube is painted on all six faces, then cut into 27 equal small cubes (3×3×3). How many small cubes have exactly two painted faces?
A) 6
B) 12
C) 8
D) 0
Answer: B) 12
Explanation: In 3×3×3, edge-center small cubes (not corners) have exactly two painted faces. Each of 12 edges contributes 1 such cube → 12.
60. Two views of a dice:
(1) faces seen {1, 5, 6}
(2) faces seen {2, 5, 3}
Which number is opposite 5?
A) 4
B) 1
C) 6
D) 2
Answer: A) 4
Explanation: 5 appears in both views adjacent to {1,6} and {2,3}, so 4 never appears with 5 → 4 is opposite 5.
61. A dice is placed so that 2 is top and 3 is front. After a 90° clockwise rotation about the vertical axis (looking from top), which face will be front?
A) 1
B) 4
C) 5
D) 6
Answer: B) 4
Explanation: Clockwise rotation moves right face to front. If initially front = 3 and top = 2, right face must be 4 (consistent with standard orientation), so front becomes 4.
62. Three views: (i) {1,2,6} (ii) {2,3,4} (iii) {4,5,6}. Which number is opposite 2?
A) 4
B) 5
C) 1
D) 3
Answer: B) 5
Explanation: 2 appears with 1 & 6 in (i) and with 3 & 4 in (ii) — thus 2 is adjacent to 1,6,3,4 leaving only 5 to be opposite.
63. Two positions: (A) top=1, front=2, right=3. (B) after rotation, top=3, front=1. What is the right face in position B?
A) 2
B) 4
C) 5
D) 6
Answer: D) 6
Explanation: Track rotations: rotating the cube so that 3 becomes top and 1 becomes front implies 6 becomes right (since 1 opp 6 in standard orientation).
64. A painted cube of side split into 64 small cubes (4×4×4). How many small cubes have no painted faces?
A) 8
B) 27
C) 64
D) 8
Answer: A) 8
Explanation: Inner unpainted cubes = (n−2)³ = (4−2)³ = 2³ = 8.
65. If opposite pairs are (1–6),(2–5),(3–4), which triplet cannot be faces meeting at a corner?
A) 1,2,3
B) 1,5,3
C) 6,2,4
D) 2,3,4
Answer: D) 2,3,4
Explanation: 3 and 4 are opposites and cannot be adjacent; triplet containing both cannot meet at one corner.
66. Two views: (i) {4,2,1} (ii) {1,3,5}. Which face is opposite 1?
A) 6
B) 4
C) 2
D) 3
Answer: A) 6
Explanation: 1 is adjacent to {4,2} in (i) and {3,5} in (ii), so only remaining number 6 is opposite 1.
67. A dice shows 6 on top and 4 in front. After rotating the dice 180° around the horizontal axis through front-back (i.e., flipping forward), which number will be on top?
A) 1
B) 2
C) 3
D) 5
Answer: A) 1
Explanation: Opposite of 6 is 1, so flipping forward swaps top and bottom: top becomes previous bottom = 1.
68. Three positions: (A) {2,4,1} (B) {4,6,3} (C) {6,5,2}. Which number is opposite to 4?
A) 2
B) 3
C) 5
D) 1
Answer: B) 3
Explanation: 4 appears with 2 &1 in (A) and with 6 &3 in (B). 3 never appears with 4 in A, but appears with 4 in B as adjacent — wait: check carefully. From views: 4 adjacent to {2,1,6,3} so only remaining number is 5 → opposite is 5.
Corrected Answer: C) 5.
69. A cube is painted on one face only, then cut into 27 small cubes (3×3×3). How many small cubes will have exactly one painted face?
A) 9
B) 6
C) 1
D) 12
Answer: A) 9
Explanation: For 3×3×3, the face centers of the painted face (a 3×3 grid) give 9 small cubes with exactly one painted face.
70. Two views: (i) {1,4,2} (ii) {4,2,5}. Which number is opposite 1?
A) 3
B) 6
C) 5
D) 2
Answer: B) 6
Explanation: 1 is adjacent to 4 & 2; it is also not present in (ii) so its opposite must be the remaining number 6.
71. On a standard dice, if the sum of numbers on three mutually adjacent faces is 9, which are the three faces?
A) 1,2,6
B) 1,3,5
C) 2,3,4
D) 1,4,4 (invalid)
Answer: B) 1,3,5
Explanation: Check sums: 1+3+5 = 9 and none are opposite to each other (1–6,3–4,2–5), so they can meet at a corner.
72. A dice shows 2, 5, 3 on three visible faces. What is the face opposite to 5?
A) 2
B) 3
C) 4
D) 1
Answer: D) 1
Explanation: Opposite pairs standard: 5 opposite 2? No standard is 2–5; but if 2 and 5 are visible together they are adjacent → impossible in standard dice. For puzzles, if 2 and 5 appear together, they cannot be opposite. Using standard pairs (1–6,2–5,3–4) but 2 & 5 cannot be adjacent in standard. So if the visible triple is valid, opposite of 5 must be 2? This is inconsistent.
Better approach: Assume standard opposite pairs (2–5), if 2 and 5 are visible together, the configuration is non-standard. The safe answer: C) 4? — this is messy.
Correction & Final: Replace with clear valid data.
New Q72: A dice shows 2, 3, 6 on three adjacent faces. What is opposite 3?
A) 4
B) 5
C) 1
D) 2
Answer: A) 4
Explanation: Standard pair 3↔4 → opposite is 4.
73. Three views: (1) {1,2,3} (2) {2,4,6} (3) {6,5,1}. Which face is opposite 2?
A) 3
B) 5
C) 4
D) None of the above
Answer: B) 5
Explanation: From (1) 2 adjacent to 1 & 3; from (2) 2 adjacent to 4 & 6. So 2 adjacent to {1,3,4,6} leaving 5 opposite.
74. A dice is rolled. What is the probability that the number is a prime? (Prime numbers on dice: 2,3,5)
A) 1/6
B) 1/2
C) 1/3
D) 2/3
Answer: C) 1/2? Wait calculate: primes = {2,3,5} → 3 outcomes out of 6 = 1/2.
Final Answer: B) 1/2.
Explanation: 3 favourable outcomes/6 total = 1/2.
75. A cube is painted on three mutually adjacent faces (meeting at one corner) and then cut into 27 small cubes (3×3×3). How many small cubes have exactly one painted face?
A) 3
B) 6
C) 9
D) 12
Answer: C) 9
Explanation: Each painted face (3×3) contributes its center small cube (excluding edges and corner) → 3 faces × 3 centers per face = 9.
76. A cube (dice) is painted on all 6 faces and then cut into 5×5×5=1255\times5\times5=1255×5×5=125 small equal cubes. How many small cubes have exactly two painted faces?
A) 12
B) 48
C) 60
D) 88
Answer: B) 48
Explanation: For an n×n×nn\times n\times nn×n×n cut, cubes with exactly two painted faces = 12×(n−2)12\times (n-2)12×(n−2). Here n=5n=5n=5: 12×(5−2)=12×3=3612\times(5-2)=12\times3=3612×(5−2)=12×3=36. Wait — recalc carefully: standard formula = 12(n−2)12(n-2)12(n−2). So 12×3=3612\times3=3612×3=36. But 36 is not an option. Check: yes formula is 12*(n-2)=36. None of options match — choose closest? We must correct options.
Corrected final: Answer should be 36 (not listed).
(Note: standard result — for n=5, exactly-two-painted = 12×(5−2)=36.)
77. A cube is painted on all faces and cut into 4×4×44\times4\times44×4×4 small cubes. How many small cubes have no painted face?
A) 8
B) 27
C) 64
D) 16
Answer: A) 8
Explanation: Unpainted (interior) = (n−2)3=(4−2)3=23=8.(n-2)^3=(4-2)^3=2^3=8.(n−2)3=(4−2)3=23=8.
78. A standard dice is rolled three times. What is the probability the results are strictly increasing (each roll larger than previous)?
A) 1/61/61/6
B) 1/41/41/4
C) 1/361/361/36
D) 1/201/201/20
Answer: A) 1/61/61/6
Explanation: Total outcomes 63=2166^3=21663=216. Number of strictly increasing triples = number of 3-combinations from 6 (order forced) = (63)=20\binom{6}{3}=20(36)=20. But each combination corresponds to exactly 1 increasing sequence, so probability = 20/216=5/5420/216=5/5420/216=5/54. Wait—compute: 20/216=5/5420/216 = 5/5420/216=5/54, ≈0.0926. None of options equal 5/54 (~0.0926). Option A: 1/6 = 0.1667 — wrong. Correct answer is 5/545/545/54.
Corrected final: Probability = 554\dfrac{5}{54}545 (≈0.0926).
79. A dice shows 1, 2, 3 on three faces meeting at one corner. If opposite faces sum to 7, what is the product of the numbers on the three faces opposite to 1,2,3?
A) 120
B) 30
C) 120 (typo?)
D) 6
Answer: B) 30
Explanation: Opposites are 6,5,4 → product 6×5×4=1206\times5\times4=1206×5×4=120. Wait—product is 120. Option B is 30; mismatch. Correct product = 120.
Corrected final: 120.
80. Two identical dice are thrown. What is the probability that the sum is 7?
A) 1/61/61/6
B) 1/121/121/12
C) 1/91/91/9
D) 1/31/31/3
Answer: A) 1/61/61/6
Explanation: Favourable pairs: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) → 6 outcomes out of 36 → 6/36=1/66/36=1/66/36=1/6.
81. A dice is rolled once. Given the face shown is odd, what is the probability it is 5?
A) 1/61/61/6
B) 1/31/31/3
C) 1/21/21/2
D) 1/51/51/5
Answer: B) 1/31/31/3
Explanation: Odd faces = {1,3,5} (3 outcomes). Probability = 1 favourable /3 = 1/31/31/3.
82. You stack three identical dice one above another. If the bottom shows 2 and the top shows 3, what can be the number on the middle die’s top face (visible between dice) if opposite pairs are (1–6),(2–5),(3–4)?
A) 1 only
B) 4 only
C) 1 or 4
D) 5 only
Answer: C) 1 or 4
Explanation: Bottom shows 2 — its top (face contacting middle die bottom) must be 5 (opposite). Top shows 3 — its bottom contacting middle’s top must be 4. Middle die’s top could be any face not conflicting with its bottom; multiple orientations possible → middle top could be 1 or 4 depending on rotations. (Multiple possibilities exist.)
83. A cube is painted on three mutually adjacent faces and cut into n=3n=3n=3 layers (3×3×3). How many small cubes have exactly two painted faces?
A) 3
B) 6
C) 9
D) 12
Answer: B) 6
Explanation: For three adjacent painted faces, edge-centre cubes (not corners) on painted edges that are shared by two painted faces: there are 3 such edges with one cube each = 3? Wait carefully: painted faces meet at one corner — there are 3 painted faces; edges shared between painted faces: 3 edges; on each of these edges (excluding corner) there is (n−2)=1 cube → total =3. But there are also cubes on opposite edges? No. So exactly-two-painted = 3. Options mismatch. Correct answer = 3.
Corrected final: 3.
84. Two dice are placed side by side showing faces: left die shows (front) 2, (top) 6; right die shows (front) 3, (top) 6. What number is touching between them on the adjacent faces?
A) 1
B) 4
C) 5
D) 2
Answer: B) 4
Explanation: For each die, top 6 means bottom 1. If fronts are 2 and 3, the faces that touch are the right face of left die and left face of right die. Using standard orientation (with 1–6,2–5,3–4 opposites), the face adjacent to 6 and 2 would be 4 on left die, and similarly matching 4 on right die — so touching faces are 4.
85. A dice net is shown with numbers: three faces in a row 2–3–4 and attached to 3 is 1, to 2 is 5, to 4 is 6. If folded into a cube, which face is opposite 3?
A) 1
B) 5
C) 6
D) 4
Answer: B) 5
Explanation: Visualize net: center face 3 — faces attached around it are 2 (left),4 (right),1 (top),? bottom is 5 — opposite 3 is the face not adjacent in net = 5.
86. In a standard dice, which of these pairs cannot appear together on adjacent faces?
A) (1,2)
B) (2,6)
C) (3,4)
D) (5,2)
Answer: C) (3,4)
Explanation: 3 and 4 are opposites (in standard dice), so they cannot be adjacent.
87. A dice is rolled twice. What is the probability that the second roll is strictly less than the first?
A) 5/125/125/12
B) 1/21/21/2
C) 1/31/31/3
D) 1/61/61/6
Answer: A) 5/125/125/12
Explanation: Symmetry: probability of 2nd < 1st equals probability 2nd > 1st; ties probability = 6/36 =1/6. So favourable probability = (1 − 1/6)/2 = (5/6)/2 = 5/12.
88. Three dice are thrown. What is the probability that the sum is 10?
A) 27/21627/21627/216
B) 27/3627/3627/36
C) 27/21627/21627/216 simplifies to ?
D) 3/83/83/8
Answer: A) 27/21627/21627/216 (which simplifies to 1/81/81/8)
Explanation: Number of ways to get sum 10 with 3 dice = 27 (can be enumerated). Total = 216 → 27/216=1/827/216=1/827/216=1/8.
89. A cube is painted on faces except one (i.e., 5 faces painted), then cut into 3×3×33\times3\times33×3×3. How many small cubes have exactly three painted faces?
A) 8
B) 1
C) 0
D) 3
Answer: D) 3
Explanation: Only corner small cubes at corners where three painted faces meet will have three painted faces. Since one face is unpainted, the three corners touching the unpainted face will not have three painted faces. A 3×3×3 cube has 8 corners; corners that are at the painted-face-corner count = number of corners not touching the unpainted face. The unpainted face removes 4 corners, leaving 4 corners with 3 painted faces? Wait — compute carefully: If exactly one face is unpainted, painted faces are five — corners that touch only painted faces are those not on the unpainted face. The unpainted face has 4 corners; remaining 4 corners have three painted faces. So correct answer = 4.
Corrected final: 4.
90. If you randomly pick one face from a fair dice, what is the expected value (mean) of the face number?
A) 3.5
B) 4
C) 3
D) 7
Answer: A) 3.5
Explanation: Mean = (1+2+3+4+5+6)/6=21/6=3.5(1+2+3+4+5+6)/6 = 21/6 = 3.5(1+2+3+4+5+6)/6=21/6=3.5.
91. Two dice are thrown. The product of the numbers is 12. What is the probability that the sum equals 7?
A) 1/31/31/3
B) 1/41/41/4
C) 1/21/21/2
D) 1/61/61
Answer: B) 1/41/41/4
Explanation: Pairs with product 12: (2,6),(3,4),(4,3),(6,2) → 4 outcomes. Among them, sum=7 occurs for (3,4) and (4,3): 2 favourable/4 = 1/2. Wait — calculation: that’s 2/4 = 1/2. So correct = 1/2. Options include 1/2 as C.
Corrected final: 1/2.
92. A dice is rolled and shows an even number. Given this, probability that it shows 4 is:
A) 1/61/61/6
B) 1/31/31/3
C) 1/21/21/2
D) 1/21/21/2 repeated
Answer: B) 1/31/31/3
Explanation: Evens = {2,4,6} (3 outcomes). Probability = 1/31/31/3.
93. From a standard dice, three faces meeting at a corner are removed (i.e., those three corner small unit cubes). If remaining small cubes are 23 (after removing 3), how many faces were painted originally on the big cube? — (trick: reverse painting/cutting)
A) 3
B) 4
C) 6
D) Cannot be determined
Answer: D) Cannot be determined
Explanation: Problem as stated is ambiguous without cut size or painting info; insufficient data.
94. A dice shows 1 on top and 2 in front. After rotating the dice 90° anticlockwise about a vertical axis, which number will be at front? (Assume standard pairings.)
A) 3
B) 4
C) 5
D) 6
Answer: B) 4
Explanation: A 90° anticlockwise rotation moves the left face to front. With top 1 and front 2, left face will be 4 (standard orientation), so front becomes 4.
95. A cube is sliced into n3n^3n3 small cubes. The number of small cubes having exactly one painted face equals 54. What is nnn?
A) 5
B) 6
C) 7
D) 8
Answer: A) 5
Explanation: Exactly-one-painted = 6×(n−2)26\times (n-2)^26×(n−2)2. So 6(n−2)2=546(n-2)^2=546(n−2)2=54 → (n−2)2=9(n-2)^2=9(n−2)2=9 → n−2=3n-2=3n−2=3 → n=5n=5n=5.
96. You roll two dice. What is the probability that at least one shows a 6?
A) 11/3611/3611/36
B) 1/61/61/6
C) 35/3635/3635/36
D) 1/31/31/3
Answer: A) 11/3611/3611/36
Explanation: Probability none shows 6 = 5/6×5/6=25/365/6\times5/6=25/365/6×5/6=25/36. So at least one 6 = 1−25/36=11/361-25/36=11/361−25/36=11/36.
97. A dice is numbered in such a way that opposite faces sum to 7. If face 1 is opposite 6 and face 2 is opposite 5, which of these triples can be mutually adjacent at a corner?
A) 1,2,6
B) 1,2,3
C) 2,5,4
D) 3,4,6
Answer: B) 1,2,3
Explanation: A triple is valid if no two numbers are opposites. 1,2,3 are pairwise non-opposite so can meet at a corner.
98. Three dice are thrown. What is the probability that all three show the same number?
A) 1/361/361/36
B) 1/61/61/6
C) 1/2161/2161/216
D) 1/361/361/36 repeated
Answer: A) 1/361/361/36
Explanation: Choose the common face (6 choices). For that face, probability all three show it = (1/6)3(1/6)^3(1/6)3 for a fixed face; multiply by 6 gives 6×(1/216)=6/216=1/366\times(1/216)=6/216=1/366×(1/216)=6/216=1/36.
99. A 4×4×44\times4\times44×4×4 cube is painted on all faces and then cut. How many small cubes have exactly one painted face?
A) 24
B) 32
C) 24? check
D) 16
Answer: A) 24
Explanation: Exactly-one-painted = 6×(n−2)2=6×(4−2)2=6×4=24.6\times(n-2)^2 = 6\times(4-2)^2 = 6\times4 = 24.6×(n−2)2=6×(4−2)2=6×4=24.
100. A dice is rolled. What is the probability of getting a number that is either a square or a prime? (Squares on dice: 1,4. Primes: 2,3,5.)
A) 5/65/65/6
B) 2/32/32/3
C) 4/64/64/6
D) 1/21/21/2
Answer: A) 5/65/65/6
Explanation: Numbers that are square or prime: {1,2,3,4,5} — 5 outcomes out of 6 → 5/65/65/6.