Question Bank — A Square and A Cube

NCERT Class 8 Mathematics · Chapter 1 · CBSE Board Pattern

Class 8 Mathematics Chapter 1 Question Bank CBSE Board Pattern

QUESTION BANK

Chapter: A Square and A Cube (NCERT Ganita Prakash, Class 8)

Section A — MCQs (1 mark each)

  1. The number of times a locker is toggled equals the number of its
    (a) multiples (b) factors (c) digits (d) prime factors

  2. A locker remains open at the end only if its number is
    (a) even (b) prime (c) a perfect square (d) a perfect cube

  3. Which of the following has an odd number of factors?
    (a) 6 (b) 10 (c) 36 (d) 15

  4. The units digit of a perfect square can never be
    (a) 0 (b) 4 (c) 2 (d) 5

  5. If a number ends with three zeros, its square ends with
    (a) three zeros (b) four zeros (c) six zeros (d) nine zeros

  6. The sum of the first n odd natural numbers is
    (a) 2n (b) n (c) n² (d) 2n+1

  7. Which of the following is a perfect cube?
    (a) 64 (b) 72 (c) 80 (d) 90

  8. Assertion (A): Every perfect square has an odd number of factors.
    Reason (R): In a perfect square, one factor is repeated (the square root itself) and does not have a distinct partner factor.
    (a) Both A and R are true and R is the correct explanation of A.
    (b) Both A and R are true but R is not the correct explanation of A.
    (c) A is true but R is false.
    (d) A is false but R is true.

  9. Assertion (A): 156 is not a perfect square.
    Reason (R): The prime factors of 156 cannot be grouped into identical pairs.
    (a) Both A and R are true and R is the correct explanation of A.
    (b) Both A and R are true but R is not the correct explanation of A.
    (c) A is true but R is false.
    (d) A is false but R is true.

  10. The positive square root of 576 is
    (a) 24 (b) 26 (c) 28 (d) 32

Section B — Very Short Answer (2 marks each)

  1. Write the locker numbers (from 1 to 100) that remain open at the end of the process.
  2. State whether 327 is a perfect square. Give reason using the units-digit rule.
  3. How many numbers lie between 16² and 17²?
  4. Using prime factorisation, show that 324 is a perfect square.
  5. What is the cube root of 3375? Show the grouping of prime factors.
  6. Give one reason why a cube cannot end with exactly two zeros.

Section C — Short Answer (3 marks each)

  1. Find the smallest square number that is divisible by 4, 9 and 10.
  2. Using the pattern of consecutive odd numbers, find 36² given that 35² = 1225.
  3. By prime factorisation, check whether 1156 is a perfect square. If yes, find its square root.
  4. Estimate √250 (between two consecutive integers) using the method shown in the chapter.
  5. Find the cube root of 27000 by prime factorisation.

Section D — Long Answer (5 marks each)

  1. A square cloth has area 125 cm².
    (i) Can a handkerchief of side 15 cm be cut from it?
    (ii) What is the largest integer side length of a square handkerchief that can be cut? Justify using nearest perfect squares.
  2. Prove that only perfect squares have an odd number of factors. Use the idea of factor pairs and the special case when a factor repeats (as explained for locker numbers).
  3. Find the smallest number by which 9408 must be multiplied so that the product is a perfect square. Hence find the square root of the resulting number. Show all prime-factor grouping.

Section E — Case/Source-Based (4 marks each)

25. In the queen’s mansion, 100 lockers are toggled by 100 persons exactly as described in the will.
(a) Which lockers remain open? (1 mark)
(b) Why do perfect-square numbered lockers stay open while others close? (1 mark)
(c) Name the five smallest prime-numbered lockers that are toggled exactly twice. (1 mark)
(d) If the passcode is formed by these five numbers separated by hyphens, write the code. (1 mark)

26. Akhil has a square piece of cloth of area 125 cm² and wants to cut the largest possible square handkerchief with integer side length.
(a) Is 125 a perfect square? How do you know? (1 mark)
(b) Between which two consecutive perfect squares does 125 lie? (1 mark)
(c) What is the side length of the largest integer-sided square he can cut? (1 mark)
(d) What area of cloth will be left unused? (1 mark)

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Section A

  1. (b) 2. (c) 3. (c) 4. (c) 5. (c) 6. (c) 7. (a) 8. (a) 9. (a) 10. (a)

Section B

  1. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (1 mark for listing squares, 1 mark for stating they have odd factors).
  2. 327 ends with 7 → cannot be a square (2 marks).
  3. 33 numbers (17² – 16² – 1) (2 marks).
  4. 324 = 2² × 3⁴ = (2 × 3²)² → perfect square (3 marks: factorisation 1, pairing 1, conclusion 1).
  5. 3375 = 3³ × 5³ → cube root 15 (2 marks).
  6. Cubes of numbers ending with 0 end with at least three zeros (2 marks).

Section C

  1. LCM of 4,9,10 = 180; 180 × 5 = 900 (3 marks: LCM 1, multiply to square 1, answer 1).
  2. 36th odd number = 71; 1225 + 71 = 1296 (3 marks).
  3. 1156 = 2² × 17² → √1156 = 34 (3 marks).
  4. 15² = 225, 16² = 256 → √250 lies between 15 and 16 (approx. 15.8) (3 marks).
  5. 27000 = 3³ × 5³ × 10³ → cube root 30 (3 marks).

Section D

  1. (i) No (125 not square) (ii) 11 cm (121 cm²) (marking: nearest squares 2, conclusion 2, side 1).
  2. Full explanation of factor pairs + repeated factor in squares (5 marks).
  3. 9408 × 3 = 28224 = (168)² (5 marks: prime factors 2, grouping 2, answer 1).

Section E

  1. (a) Square numbers (1) (b) Odd number of factors (1) (c) 2-3-5-7-11 (1) (d) 2-3-5-7-11 (1).
  2. (a) No (ends with 5 but check) (1) (b) 11² & 12² (1) (c) 11 cm (1) (d) 4 cm² (1).
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All questions are answerable from the NCERT chapter text. Reviewed by GFIS faculty.