Class-8 Ganita prakash ASSERTION & REASONING Maths ch-1 a square and a cube

Class-8 Ganita prakash ASSERTION & REASONING Maths ch-1 a square and a cube

Assertion & Reasoning Questions (15)

For each question:

• A = Assertion

• R = Reason
  Choose the correct option:
  

  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.

• 
  

1. (A): All perfect squares have an odd number of factors.
   (R): They have one factor that is repeated in the middle.
   Answer: Both A and R are true, and R is the correct explanation.

2. (A): If a number ends in 2, 3, 7, or 8, it can never be a perfect square.
   (R): The units digit of a perfect square is always 0, 1, 4, 5, 6, or 9.
   Answer: Both A and R are true, and R is the correct explanation.

3. (A): 36 is a perfect square.
   (R): It can be expressed as the product of two identical numbers, 6 × 6.
   Answer: Both A and R are true, and R is the correct explanation.

4. (A): Every perfect square can be written as the sum of consecutive odd numbers starting from 1.
   (R): The sum of the first n odd numbers equals n².
   Answer: Both A and R are true, and R is the correct explanation.

5. (A): All prime numbers are perfect squares.
   (R): Prime numbers have exactly two factors: 1 and itself.
   Answer: A is false, R is true.

6. (A): The cube of a number ending in 0 will always have exactly two zeros at the end.
   (R): Cubes can have multiple zeros, depending on the number of 10s in the factors.
   Answer: A is false, R is true.

7. (A): 729 is a perfect cube.
   (R): It can be written as 9 × 9 × 9.
   Answer: Both A and R are true, and R is the correct explanation.

8. (A): A number is a perfect cube if all prime factors occur in triples in its prime factorisation.
   (R): Triplet grouping of prime factors ensures equal grouping in three identical sets.
   Answer: Both A and R are true, and R is the correct explanation.

9. (A): 1729 is called the Hardy–Ramanujan number.
   (R): It is the smallest number expressible as the sum of two cubes in two different ways.
   Answer: Both A and R are true, and R is the correct explanation.

10. (A): The cube of an even number is always even.
    (R): Multiplying an even number three times still results in an even number.
    Answer: Both A and R are true, and R is the correct explanation.

11. (A): The square root of a perfect square is always a natural number.
    (R): A perfect square is the product of an integer with itself.
    Answer: Both A and R are true, and R is the correct explanation.

12. (A): A cube can end in exactly two zeros.
    (R): Each zero in the cube comes from a factor of 10³.
    Answer: Both A and R are false.

13. (A): 500 is a perfect cube.
    (R): Its prime factors cannot be arranged into triplets of identical factors.
    Answer: A is false, R is true.

14. (A): Every perfect square is also a perfect cube.
    (R): Both are powers of integers.
    Answer: A is false, R is true.

15. (A): The positive square root of 64 is 8.
    (R): 8 × 8 = 64.
    Answer: Both A and R are true, and R is the correct explanation.