Question 1

Q1: Which of the following numbers are not perfect cubes? (i) 216 (ii) 128 (iii) 1000 (iv) 100 (v) 46656

Accepted Answer

To Find: Which of the given numbers are not perfect cubes. Concept: A number is a perfect cube if all its prime factors can be grouped into triplets (i.e., each prime factor appears a number of times that is a multiple of 3). Solution: (i) 216 Prime factorisation of 216: $$216 = 2 	imes 108 = 2 	i...

Question 2

Q2: Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube. (i) 243 (ii) 256 (iii) 72 (iv) 675 (v) 10...

Accepted Answer

Concept: To find the smallest number to multiply to make a number a perfect cube, first find its prime factorisation. Then, identify the prime factors that are not in a group of three. The required multiplier is the product of the factors needed to complete the triplets. Solution: (i) 243 $$243 = 3...

Question 3

Q3: Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube. (i) 81 (ii) 128 (iii) 135 (iv) 192 (v) 704

Accepted Answer

Concept: To find the smallest number to divide to make a number a perfect cube, first find its prime factorisation. Then, identify the prime factors that are left over after grouping the factors into triplets. The required divisor is the product of these leftover factors. Solution: (i) 81 $$81 = 3 \...

Question 4

Q4: Parikshit makes a cuboid of plasticine of sides 5 cm, 2 cm, 5 cm. How many such cuboids will he need to form a cube?

Accepted Answer

Given: Sides of the cuboid are 5 cm, 2 cm, and 5 cm. To Find: The number of such cuboids needed to form a cube. Solution: First, we find the volume of the given cuboid. Volume of cuboid = length $	imes$ breadth $	imes$ height $$V = 5 	ext{ cm} 	imes 2 	ext{ cm} 	imes 5 	ext{ cm} = 50 	ext{ c...

Question 5

Q1: Find the cube root of each of the following numbers by prime factorisation method. (i) 64 (ii) 512 (iii) 10648 (iv) 27000 (v) 15625 (vi) 13824 (vii) 1...

Accepted Answer

Concept: To find the cube root of a number by prime factorisation, we first find all its prime factors. Then, we group the identical factors into triplets. The cube root is the product of one factor taken from each triplet. Solution: (i) 64 $$64 = \underline{2 	imes 2 	imes 2} 	imes \underline{2...

Question 6

Q2: State true or false. (i) Cube of any odd number is even. (ii) A perfect cube does not end with two zeros. (iii) If square of a number ends with 5, the...

Accepted Answer

Solution: (i) Cube of any odd number is even. Answer: False. Reason: The product of odd numbers is always odd. For example, $3^3 = 3 	imes 3 	imes 3 = 27$, which is an odd number. (ii) A perfect cube does not end with two zeros. Answer: True. Reason: For a number to end with zeros, it must have fa...

NCERT Solutions