A Square and A Cube — Mastery

Q1. In the locker puzzle, locker number 36 is opened or closed by the kth servant if k is a factor of 36. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. After all servants pass, locker 36 will be

• A. closed, because 36 has 9 factors and 9 is odd
• B. open, because 36 ends in 6
• C. closed, because 36 = 6 × 6
• D. open, because 36 has 9 factors and 9 is odd

Q2. How many perfect squares lie between 1 and 200 (both inclusive)?

• A. 13
• B. 16
• C. 15
• D. 14

Q3. Look at the sequence of perfect squares 1, 4, 9, 16, 25, 36, … . The first differences are 3, 5, 7, 9, 11, … . The second differences (differences of these) are

• A. always 1
• B. they keep increasing
• C. always 3
• D. always 2

Q4. Sahil claims: 'If a number's square ends in 1, the number itself must end in 1.' Evaluate his claim.

• A. Partially correct — numbers ending in 1 or 9 both give squares ending in 1, so his claim is too narrow.
• B. Correct — only numbers ending in 1 can have squares ending in 1.
• C. Incorrect — numbers ending in 1 always give squares ending in 9.
• D. Cannot be decided without checking each number.

Q5. Square root is defined as the inverse of squaring. Which of the following pairs of operations is analogous to the (square, square root) pair?

• A. (addition, multiplication)
• B. (multiplication, division)
• C. (addition, subtraction)
• D. (multiplication, addition)

Q6. Every positive number has two square roots — one positive and one negative. The two square roots of 64 are

• A. 8 and 16
• B. +4 and −4
• C. +8 only
• D. +8 and −8

Q7. Without using a calculator, estimate √250 between two consecutive whole numbers and decide which it is closer to.

• A. between 14 and 15, closer to 15
• B. between 15 and 16, closer to 15
• C. between 15 and 16, closer to 16
• D. between 16 and 17, closer to 16

Q8. How many 1 cm unit cubes are needed to build a solid cube of edge 3 cm?

• A. 9
• B. 18
• C. 27
• D. 36

Q9. Using the notation n³ = n × n × n from, evaluate 11³.

• A. 33
• B. 121
• C. 1100
• D. 1331

Q10. Between which pair of consecutive whole numbers is there NO perfect cube strictly between them?

• A. 1 and 8
• B. 8 and 27
• C. 10 and 26
• D. 27 and 64

Q11. Consider two statements about perfect cubes: Statement I: In the prime factorisation of a perfect cube, every prime appears a number of times that is a multiple of 3. Statement II: 2² × 3³ × 5⁶ is a perfect cube. Which is correct?

• A. Statement I is true; Statement II is false.
• B. Statement I is true; Statement II is true.
• C. Statement I is false; Statement II is true.
• D. Both statements are false.

Q12. Take the sequence of cubes 1, 8, 27, 64, 125, 216, … . First differences: 7, 19, 37, 61, 91. Second differences: 12, 18, 24, 30. Third differences are

• A. always 2
• B. they keep increasing
• C. always 12
• D. always 6

Q13. Using the pattern that expresses each n³ as the sum of n consecutive odd numbers, the value of 21 + 23 + 25 + 27 + 29 is

• A. 5³ = 125
• B. 5² = 25
• C. 25³ = 15625
• D. 125²

Q14. A teacher is starting the chapter A Square and A Cube in a Class 8 Hindi-medium classroom. Which is the BEST first activity to introduce the meaning of n²?

• A. Give students 1 cm grid paper and ask them to draw squares of side 1, 2, 3, 4, 5 cm and count the unit cells inside each.
• B. Ask students to memorise the squares of 1 to 20.
• C. Write the formula n² = n × n on the board and ask students to substitute n = 1, 2, 3.
• D. Show students a video of a building's square floor.

Q15. Rina argues: '26 ends in 6, and the square of every number ending in 6 ends in 6 , so 26 must itself be a perfect square.' The best diagnosis of Rina's reasoning error is that she

• A. uses the correct property in the correct direction.
• B. confuses 26 with 36.
• C. miscalculates the units digit of 26.
• D. reverses the implication — 'square ends in 6' does not mean 'number ending in 6 is a square'.

Q16. A special symbol for the positive square root. The symbol √ is called the

• A. factor symbol
• B. summation symbol
• C. integral symbol
• D. radical symbol

Q17. A cube is described as a solid figure with three special properties. Which of the following correctly lists those properties?

• A. Six equal square faces, all edges equal, all angles right angles
• B. Six rectangular faces, opposite edges equal, all angles right angles
• C. Four equal triangular faces, three edges per vertex, no right angles
• D. Six faces of any shape as long as opposite faces are parallel

Q18. Squares can end only in 0, 1, 4, 5, 6 or 9 . Looking at the cubes 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, which of the following is correct about the units digits of cubes?

• A. Cubes also end only in 0, 1, 4, 5, 6 or 9
• B. Cubes always end in the same digit as the original number
• C. Cubes end only in 0, 1 or 8
• D. Cubes can end in any digit 0 through 9

Q19. Extends cubing to fractions: (a/b)³ = a³/b³. The value of (4/6)³ is

• A. 12/18
• B. 16/36
• C. 64/216
• D. 8/216

Q20. The cube root is defined as the inverse operation of cubing. Which statement best captures this definition?

• A. The cube root of n is the number which, when added to itself three times, equals n.
• B. The cube root of n is the number which, when multiplied by itself three times, equals n.
• C. The cube root of n is one-third of n.
• D. The cube root of n is the same as the square root of n.

Q21. A symbol for the cube root that looks like the square-root sign with a tiny 3 above it. The symbol ∛125 stands for

• A. the square root of 125
• B. the number whose cube is 125
• C. 125 divided by 3
• D. 125 multiplied by itself three times

Q22. One of the two ways expresses 1729 as a sum of two cubes uses 1 and 12. The value of 1³ + 12³ is

• A. 1331
• B. 1728
• C. 1729
• D. 1735

Q23. The second way expresses 1729 as a sum of two cubes uses 9 and 10. The value of 9³ + 10³ is

• A. 1000
• B. 1729
• C. 1900
• D. 729

Q24. 1729 is the smallest 'taxicab' number — expressible as the sum of two cubes in two ways. The next such number is 4104. Which TWO pairs of cubes both add to 4104?

• A. 1³ + 12³ and 9³ + 10³
• B. 2³ + 16³ and 9³ + 15³
• C. 3³ + 12³ and 4³ + 11³
• D. 5³ + 14³ and 7³ + 13³

Q25. The 'Pinch of History' panel notes which ancient civilisation had already written lists of square numbers on clay tablets around 1700 BCE?

• A. Babylonians
• B. Greeks
• C. Romans
• D. Egyptians

Q26. The Sanskrit word 'varga' originally referred to a square geometric figure and was later extended to the second power of a number. Following the same pattern, 'varga-mula' literally translates as

• A. the area of a square
• B. the root of a square (i.e., square root)
• C. the side of a square
• D. the diagonal of a square

Q27. The great Indian mathematician-astronomer Aryabhata, who in 499 CE used the term 'varga' in his work to mean what we today call

• A. a triangular number
• B. the second power of a number
• C. a prime number
• D. the cube of a number

Q28. The same Sanskrit word 'mula' (root, as of a plant) is used in 'varga-mula' and 'ghana-mula'. This shared word reveals that ancient Indian mathematicians saw square root and cube root as

• A. completely unrelated ideas
• B. the inverse 'roots' from which a square or cube number 'grows'
• C. approximations rather than exact operations
• D. operations that work only for whole numbers

Q29. A Class 8 teacher wants students to decide whether 1764 is a perfect square. Which sequence of teaching steps best builds understanding before computing?

• A. Hand out a calculator and ask for √1764 to two decimal places.
• B. First check the units digit (4 — possible); then estimate between 40² = 1600 and 45² = 2025; then prime-factorise to confirm 1764 = 2² × 3² × 7² = 42².
• C. Ask students to memorise the squares of 30 to 50 and match.
• D. Tell students the answer is 42 and ask them to verify by squaring.

Q30. A teacher in a Hindi-medium Class 8 plans to introduce the cube using only the modern symbol n³ and no historical or geometric link. A peer suggests also using the Sanskrit word 'ghana' and the unit-cube stacking picture from. Evaluate the peer's suggestion.

• A. Reject — Sanskrit terms confuse students learning modern notation.
• B. Accept — 'ghana' carries the meaning 'solid/thrice multiplied', and the unit-cube picture connects symbol to a tangible solid, deepening conceptual understanding.
• C. Accept only the Sanskrit word; the unit-cube picture is unnecessary.
• D. Reject — unit-cube picture is decorative, not conceptual.