Solutions of chapter 4. Cubes and Cube Roots | RD Sharma - Mathematics

1

Find the cubes of the following numbers:

(i) 7 (ii) 12

(iii) 16 (iv) 21

(v) 40 (vi) 55

(vii) 100 (viii) 302

(ix) 301

2

Write the cubes of all natural numbers between 1 and 10 and verify the following statements:

(i) Cubes of all odd natural numbers are odd.

(ii) Cubes of all even natural numbers are even.

3

Observe the following pattern:

Write the next three rows and calculate the value of by the above pattern.

4

Write the cubes of 5 natural numbers which are multiples of 3 and verify the followings:

“The cube of a natural number which is a multiple of 3 is a multiple of 27’

5

Write the cubes of 5 natural numbers which are of the form and verify the following:

“The cube of a natural number of the form 3n+1 is a natural number of the same from i.e. when divided by 3 it leaves the remainder 1’

6

Write the cubes 5 natural numbers of the from 3n+2(i.e.5,8,11….) and verify the following:

“The cube of a natural number of the form 3n+2 is a natural number of the same form i. e. when it is dividend by 3 the remainder is 2’

7

Write the cubes 5 natural numbers of which are multiples of 7 and verity the following:

“The cube of a multiple of 7 is a multiple of 7³.

8

Which of the following are perfect cubes?

(i) 64 (ii) 216

(iii) 243 (iv) 1000

(v) 1728 (vi) 3087

(vii) 4608 (viii) 106480

(ix) 166375 (x) 456533

9

Which of the following are cubes of even natural numbers?

216, 512, 729, 1000, 3375, 13824

10

Which of the following are cubes of odd natural numbers?

125, 343, 1728, 4096, 32768, 6859

11

What is the smallest number by which the following numbers must be multiplied, so that the products are perfect cubes?

(i) 675 (ii) 1323

(iii) 2560 (iv) 7803

(v) 107811 (vi) 35721

12

By which smallest number must the following numbers be divided so that the quotient is a perfect, cube?

(i) 675 (ii) 8640

(iii) 1600 (iv) 8788

(v) 7803 (vi) 107811

(vii) 35721 (viii) 243000

13

Prove that if a number is trebled then its cube is 27 time the cube of the given number.

14

What happens to the cube of a number if the number is multiplied by

(i) 3?

(ii) 4?

(iii) 5?

15

Find the value of a cube, one face of which has an area of 64m².

16

Find the volume of a cube whose surface area is 384m².

17

Evaluate the following:

(i)

(ii)

18

Write the units digit of the cube of each of the following numbers:

31, 109, 388, 4276, 5922, 77774, 44447, 125125125

19

Find the cubes of the following numbers by column method:

(i) 35

(ii) 56

(iii) 72

20

Which of the following numbers are not perfect cubes?

(i) 64

(ii) 216

(iii) 243

(iv) 1728

21

For each of the non-pefectcubbes in Q. No 20 find the smallest number by which it must be

(a) Multiplied so that the product is a perfect cube.

(b) Divided so that the quotient is a perfect cube.

22

By taking three different, values of n verify the truth of the following statements:

(i) If n is even, then n³ is also even.

(ii) If n is odd, then n³ is also odd.

(ii) If n leaves remainder 1 when divided by 3, then n³ also leaves 1 as remainder when divided by 3.

(iv) If a natural number n is of the form 3p+2 then n³ also a number of the same type.

23

Write true (T) or false (F) for the following statements:

(i) 392 is a perfect cube.

(ii) 8640 is not a perfect cube.

(iii) No cube can end with exactly two zeros.

(iv) There is no perfect cube which ends in 4.

(v) For an integer a, a³ is always greater than a².

(vi) If a and b are integers such that a²>b², then a³>b³.

(vii) If a divides b, then a³ divides b³.

(viii) If a² ends in 9, then a³ ends in 7.

(ix) If a² ends in an even number of zeros, then a³ ends in 25.

(x) If a² ends in an even number of zeros, then a³ ends in an odd number of zeros.

1

Find the cubes of:

(i) -11

(ii) -12

(iii) -21

2

Which of the following integers are cubes of negative integers

(i) -64

(ii) -1056

(iii) -2197

(iv) -2744

(v) -42875

3

Show that the following integers are cubes of negative integers. Also, find the integer whose cube is the given integer.

(i) -5832

(ii) -2744000

4

Find the cube of:

(i) (ii)

(iii) (iv)

(v) (vi)

(vii) 0.3 (viii) 1.5

(ix) 0.08 (x) 2.1

5

Find which of the following numbers are cubes of rational numbers:

(i)

(ii)

(iii) 0.001331

(iv) 0.04

1

Find the cube roots of the following numbers by successive subtraction of numbers:

1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397,….

(i) 64

(ii) 512

(iii) 1728

2

Using the method of successive subtraction examine whether or not the following numbers are perfect cubes:

(i) 130

(ii) 345

(iii) 792

(iv) 1331

3

Find the smallest number that must be subtracted from those of the numbers in question 2 which are not perfect cubes, to make them perfect cubes. What are the corresponding cube roots?

4

Find the cube root of each of the following natural numbers:

(i) 343 (ii) 2744

(iii) 4913 (iv) 1728

(v) 35937 (vi) 17576

(vii) 134217728 (viii) 48228544

(ix) 74088000 (x) 157464

(xi) 1157625 (xii) 33698267

5

Find the smallest number which when multiplied with 3600 will make the product a perfect cube. Further, find the cube root of the product.

6

Multiply 210125 by the smallest number so that the product is a perfect cube. Also, find out the cube root of the product.

7

What is the smallest number by which 8192 must be divided so that quotient is a perfect cube? Also, find the cube root of the quotient so obtained.

8

Three numbers are in the ratio 1:2:3. The sum of their cubes is 98784. Find the numbers.

9

The volume of a cube is 9261000 m³. Find the side of the cube.

1

Find the cube roots of each of the following integers:

(i)-125 (ii) -5832

(iii)-2744000 (iv) -753571

(v) -32768

2

Show that:

(i)

(ii)

(iii)

(iv)

3

Find the cube root of each of the following numbers:

(i) (ii)

(iii) (iv)

4

Evaluate:

(i)

(ii)

(iii)

(iv)

5

Find the cube root of each of the following rational numbers:

(i)

(ii)

(iii)

(iv)

(v)

6

Find the cube root of each of the following rational numbers:

(i) 0.001728

(ii) 0.003375

(iii) 0.001

(iv) 1.331

7

Evaluate each of the following:

(i)

(ii)

(iii)

(iv)

(v)

8

Show that:

(i)

(ii)

9

Fill in the blanks:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

10

The volume of a cubical box is 474. 552 cubic metres. Find the length of each side of the box.

11

Three numbers are to one another 2:3:4. The sum of their cubes is 0.334125. Find the numbers.

12

Find the side of a cube whose volume is

13

Evaluate:

(i)

(ii)

(iii)

(iv)

14

Find the cube roots of the numbers 3048625, 20346417, 210644875, 57066625 using the fact that

(i)

(ii)

(iii)

(iv)

15

Find the unit of the cube root of the following numbers:

(i) 226981

(ii) 13824

(iii) 571787

(iv) 175616

16

Find the tens digit of the cube root of each of the numbers in Q.No.15.

(i) 226981

(ii) 13824

(iii) 571787

(iv) 175616

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