Worksheet Solutions: Cube - 1 | Mental Mathematics for Class 8 PDF Download

Q1: Which of the following numbers is a perfect cube?
(a) 256
(b) 243
(c) 1331
(d) 250
Ans: (c)
Here 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 243 = 3 × 3× 3 × 3 × 3
1331 = 11 × 11 × 11
250 = 2 × 5 × 5 × 5
Clearly 1331 is a perfect cube.

Q2: What will be the volume of a cube having edge length 12m? (in m³)
(a) 1728
(b) 1628
(c) 2248
(d) 1848
Ans: (a)
Volume of cube = (12)³ = 1728 m³

Q3: What is the smallest number by which 576 is divided that quotient is a perfect cube?
(a) 8
(b) 9
(c) 4
(d) 72
Ans: (b)
We have
576 = 2 × 2 × 2 × 3 × 2 × 2 × 2 × 3
∴ 576 should be divided by 9, to get a prefect cube.

Q4: If, 1³ + 2³ + 3³ = (1 + 2 + 3)², 1³ + 2³ + 3³ + 4³ = (1 + 2 + 3 + 4)², then, 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³
(a) 900
(b) 441
(c) 784
(d) 484
Ans: (c)

= 784

Q5: Observing the pattern 1³ + 2³ + 3³ = (1 + 2 + 3)², 1³ + 2³ + 3³ + 4³ = (1 + 2 + 3 + 4)² , Find the sum: 1³ + 3³ + 5³ + 7³ + 9³
(a) 1225
(b) 2025
(c) 825
(d) 1625
Ans: (a)
1³ + 2³ + 3³ + …… + 9³ = (1 + 2 + 3 …… + 9)²
⇒ (1³ + 3³ + 5³ + 7³ + 9³) + 2³ (1³ + 2³ + 3³ + 4³)
= (45)² = 2025
⇒ x + 2³ (1 + 2 + 3 + 4)² = 2025
⇒ x = 2025 – 8 × 100 = 1225

Q6: Simplify :
(a) 400
(b) 8000
(c) 64000
(d) 512000
Ans: (b)
Here 12² + 16² = 400


Q7: Simplify: (64/125)^2/3
(a) 4/5
(b) 8/25
(c) 16/25
(d) 4/25
Ans: (c)


Q8: A natural number is of the form (3n + 2). Its cube will be of the form :
(a) 3n
(b) 3n + 1
(c) 3n + 2
(d) None of these
Ans: (c)
(3n + 2)³ = 27n³ + 8 + 3 × 3n × 2(3n + 2)
= 27n³ + 8 + 18n (3n + 2)
= 27n³ + 54n² + 36n + 8
= 3 (9n³ + 18n² + 12n + 2) + 2
= 3n + 2

Q9: A rational number, p < 1, then,
(a) p³ > 1
(b) p³ < 0
(c) p³ < p
(d) p³ > p
Ans: (c)
If, 0 < p < 1,
then, p³  < p.

Q10: A real number ‘p’ is such that p > 1, then
(a) p³ < 1
(b) p³ > p
(c) p³ < p
(d) p³ <  0
Ans: (b)
If,  p > 1
then,  p – 1 > 0
∴   p³ > p

Q.11: Find the cube of 3.5.
Sol: 3.53 = 3.5 x 3.5 x 3.5
= 12.25 x 3.5
= 42.875

Q.12: Find the smallest number by which 128 must be divided to obtain a perfect cube.
Sol: The prime factorisation of 128 gives:
128 = 2×2×2×2×2×2×2
Now, if we group the factors in triplets of equal factors,
128 = (2×2×2)×(2×2×2)×2
Here, 2 cannot be grouped into triples of equal factors.
Therefore, we will divide 128 by 2 to get a perfect cube

Q.13: Find the cube root of 13824 by prime factorisation method.
Sol: First let us prime factorise 13824:
13824 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
= 23 × 23 × 23 × 33
3√13824 = 2 × 2 × 2 × 3 = 24

Q.14: Is 392 a perfect cube? If not, find the smallest natural number by which 392 should be multiplied so that the product is a perfect cube.
Sol: The prime factorisation of 392 gives:
392 = 2 x 2 x 2 x 7 x 7
Since, we can see, number 7 cannot be paired in a group of three. Therefore, 392 is not a perfect cube.
To make it a perfect cube, we have to multiply the 7 by the original number.
Thus,
2 x 2 x 2 x 7 x 7 x 7 = 2744, which is a perfect cube, such as 23 x 73 or 143.
Hence, the smallest natural number which should be multiplied to 392 to make a perfect cube is 7.

Q.15: 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?
Sol: Given, side of the cube is 5 cm, 2 cm and 5 cm.
Therefore, volume of cube = 5×2×5 = 50
The prime factorisation of 50 = 2×5×5
Here, 2, 5 and 5 cannot be grouped into triples of equal factors.
Therefore, we will multiply 50 by 2×2×5 = 20 to get perfect square.
Hence, 20 cuboid is needed.