Solutions of chapter 3. Squares and Square Roots | RD Sharma - Mathematics

1

Which of the following numbers are perfect squares?

(i) 484 (ii) 625

(iii) 576 (iii) 576

(iv) 941 (v) 961

(vi) 2500

2

Show that each of the following numbers is a perfect square. Also find the number whose square is the given number in each case:

(i) 1156

(ii) 2025

(iii)14641

(iv) 4761

3

Find the smallest number by which the given number must be multiplied so that the product is a perfect square:

(i) 23805

(ii) 12150

(iii) 7688

4

Find the smallest number by which the given number must be divided so that the resulting number is a perfect square:

(i) 12283

(ii) 1800

(iii) 2904

5

Which of the following numbers are perfect squares?

11, 12, 16, 32, 36, 50, 64, 79, 81, 111, 121

6

Using prime factorization method, find which of the following numbers are perfect squares?

189, 225, 2048, 343, 441, 2961, 11025, 3549

7

By what number should each of the following numbers by multiplied to get a perfect square in each case? Also find the number whose square is the new number.

(i) 8820 (ii) 3675

(iii) 605 (iv) 2880

(v) 4056 (vi) 3468

_{(vii) 7776}

8

By What numbers should each of the following be divided to get a perfect square in each case? Also, find the number whose square is the new number.

(i) 16562

(ii) 3698

(iii) 5103

(iv) 3174

(v) 1575

9

Find the greatest number of two digits which is a pefect square.

10

Find the least number of three disgits which is perfect square.

11

Find the smallest number by which 4851 must be multiplied so that the product becomes a perfect square.

12

Find the smallest number by which 28812 must be divided so that it becomes a perfect square. Also find the number whose square is the resulting number.

13

Find the smallest number by which 1152 must be divided so that it becomes a perfect square. Also find the number whose square is the resulting number.

1

The following numbers are not perfect squares. Give reason.

(i) 1547 (ii) 45743

(iii)8948 (iv) 333333

2

Show that the following numbers are not, perfect squares:

(i) 9327 (ii) 4058

(iii)22453 (iv) 743522

3

The square of which of the following numbers would be an old number?

(i) 731 (ii) 3456

(iii)5559 (iv) 42008

4

What will be the units digit of the squares of the following numbers?

(i) 52 (ii) 977

(iii) 4583 (iv) 78367

(v) 52698 (vi) 99880

(vii) 12796 (viii) 55555

(ix) 53924

5

Observe the following pattern

And write the value of 1+3+5+7+9+……… upto n terms.

6

Observe the following pattern

And find the value of

(i) 100²-99² (ii) 111²-109²

(iii)99²-96²

7

Which of the following triplets are Pythagorean?

(i) (8, 15, 17)

(ii) (18, 80, 82)

(iii) (14, 48, 51)

(iv) (10, 24, 26)

(v) (16, 63, 65)

(vi) (12, 35, 38)

8

Observe the following pattern

And find the value of

9

Observe the following pattern

And find the values of each of the following:

(i) 1+2+3+4+5+……….+50

(ii) 31+32+……..+50

10

Observe the following pattern

And find the values of each of the following.

(i)

(ii)

11

Which of the following numbers are squares of even numbers?

121, 225, 256, 324, 1296, 6561, 5476, 4489, 373758

12

By just examining the units digits, can you tell which of the following cannot be whole squares?

(i) 1026 (ii) 1028

(iii)1024 (iv) 1022

(v) 1023 (vi) 1027

13

Which of the numbers for which you cannot decide whether they are squares.

14

Write five numbers which you cannot decide whether they are square just by looking at the unit’s digit.

15

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

(i) The number of digits in a square number is even.

(ii) The square of a prime number is prime

(iii) The sum of two square numbers is a square number.

(iv) The difference of two square numbers is a square number.

(v) The product of two square numbers is a square number.

(vi) No square number is negative.

(vii) There is no square number between 50 and 56.

(viii) There are fourteen square number upto 200.

1

Find the squares of the following numbers using column method. Verify the result by finding the square using the usual multiplication:

(i) 25

(ii) 37

(iii) 54

(iv) 71

(v) 96

2

Find the squares of the following numbers using diagonal method:

(i) 98

(ii) 273

(iii)348

(iv) 295

(v) 171

3

Find the squares of the following numbers:

(i) 127 (ii) 503

(iii) 450 (iv) 862

(v) 265

4

Find the squares of the following numbers:

(i) 425 (ii) 575

(iii)405 (iv) 205

(v) 95 (vi) 745

(vii) 512 (viii) 995

5

Find the squares of the following numbers using the identify :

(i) 405

(ii) 510

(iii) 1001

(iv) 209

(v) 605

6

Find the squares of the following numbers using the identity

(i) 395 (ii) 995

(iii)495 (iv) 498

(v) 99 (vi) 999

(vii)599

7

Find the squares of the following numbers by visual method:

(i) 52 (ii) 95

(iii) 505 (iv) 702

(v) 99

1

Write the possible unit’s digits of the square root of the following numbers. Which of these numbers are odd square roots?

(i) 9801

(ii) 99856

(iii) 998001

(iv) 657666025

2

Find the square root of each of the following by prime factorization.

(i) 441 (ii) 196

(iii) 529 (iv) 1764

(v) 1156 (vi) 4096

(vii) 7056 (viii) 8281

(ix) 11664 (x) 47089

(xi) 24336 (xii) 190969

(xiii) 586756 (xiv) 27225

(xv) 3013696

3

Find the smallest number by which 180 must be multiplied so that it becames a perfect square. Also, find the square root of the perfect square so obtained.

4

Find the smallest number by which 147 must be multiplied so that it becomes a perfect square. Also, find the square root of the number so obtained.

5

Find the smallest number by which 3645 must be divided so that it becomes a perfect square. Also, find the square root of the resulting number.

6

Find the smallest number by which 1152 must be divided so that it becomes a square. Also, find the square root of the number so obtained.

7

The product of two numbers is 1296. If one number is 16 times the other, find the numbers.

8

A welfare association collected Rs 202500 as donation from the residents. If each paid as many rupees as there were residents, find the number of residents.

9

A society collected Rs 92.16. Each member collected as many paise as there were members. How many members were there and how much did each contribute?

10

A society collected Rs 2304 as fees from its students. If each student paid as many paise as there were students in the school, how many students were there in the school?

11

The area of a square field is 5184 m². A rectangular field, whose length is twice its breadth has its perimeter equal to the perimeter of the square field. Find the area of the rectangular field.

12

Find the least square number, exactly divisible by each one of the numbers: (i) 6,9, 15 and 20) (ii) 8,12, 15 and 20

13

Find the square roots of 121 and 169 by the method of repeated subtraction.

14

Write the prime factorization of the following numbers and hence find their square roots.

(i) 7744

(ii) 9604

(iii) 5929

(iv) 7056

15

The students of class VIII of a school donated Rs 2401 for PM’s National Relief Fund. Each student donated as many rupees as the number of students in the class, Find the number of students in the class.

16

A PT teacher wants to arrange maximum possible number of 6000 students in a field such that the number of rows is equal to the number of columns. Find the number of rows if 71 were left out after arrangement.

1

Find the square root of each of the following by long division method:

(i) 12544 (ii) 97344

(iii) 286225 (iv) 390625

(v) 363609 (vi) 974169

(vii) 120409 (viiii) 1471369

(ix) 291600 (x) 9653449

(xi) 1745041 (xii) 4008004

(xiii) 20657025 (xiv) 152547201

(xv) 20421361 (xvi)62504836

(xvii) 82264900 (xviii) 3226694416

(xix) 6407522209 (xx) 3915380329

2

Find the least number which must be subtracted from the following numbers to make them a perfect square:

(i) 2361

(ii) 194491

(iii) 26535

(iv) 161605

(v) 4401624

3

Find the least number which must be added to the following numbers to make them a perfect square:

(i) 5607

(ii)4931

(iii) 4515600

(iv) 37460

(v) 506900

4

Find the greatest number of 5 digits which is a perfect square.

5

Find the least number of 4 digits which is a perfect square.

6

Find the least number of six digits which is a perfect square.

7

Find the greatest number of 4 digits which is a perfect square.

8

A General arranges his soldiers in rows to form a perfect square. He finds that in doing so, 60 soldiers are left out. If the total number of soldiers be 8160, find the number of soldiers in each row.

9

The area of a square field is 60025m². A man cycles along its boundary at 18 Km/hr. In how much time will he return at the starting point?

10

The cost of leveling and turning a square lawn at Rs 2.50 per m² is Rs13322.50 Find the cost of fencing it at Rs 5 per metre.

11

Find the greatest number of three digits which is a perfect square.

12

Find the smallest number which must be added to 2300 so that it becomes a perfect square.

1

Find the square root of:

(i) (ii)

(iii) (iv)

(v) (vi)

(vii) (viii)

(ix) (x)

(xi) (xii)

(xiii) (xiv)

(xv)

2

Find the value of:

(i)

(ii)

(iii)

(iv)

(v)

3

The area of a square field is square metres. Find the length of each side of the field.

4

The area of a square field is . Calculate the length of the side of the square.

5

Find the length of a side of a square playground whose area is equal to the area of a rectangular field of dimensions 72m and 338 m.

1

Find the square root of the following numbers in decimal form:

84.8241

2

Find the square root of the following numbers in decimal form:

0.7225

3

Find the square root of the following numbers in decimal form:

0.813604

4

Find the square root of the following numbers in decimal form:

0.00002025

5

Find the square root of the following numbers in decimal form:

150.0625

6

Find the square root of the following numbers in decimal form:

225.6004

7

Find the square root of the following numbers in decimal form:

3600.720036

8

Find the square root of the following numbers in decimal form:

236.144489

9

Find the square root of the following numbers in decimal form:

0.00059049

10

Find the square root of the following numbers in decimal form:

176.252176

11

Find the square root of the following numbers in decimal form:

9998.0001

12

Find the square root of the following numbers in decimal form:

0.00038809

13

What is that fraction which when multiplied by itself gives 227.798649?

14

The area of a square playground is 256.6404 square meter. Find the length of one side of the playground.

15

What is the fraction which when multiplied by it self gives 0.00053361?

16

Simplify:

(i)

(ii)

17

Evaluate and hence find the value of

18

Find the value of and hence find the value of

(i)

(ii)

1

Find the square root of each of the following correct to three places of decimal.

(i) 5 (ii) 7

(iii) 17 (iv) 20

(v) 66 (vi) 427

(vii) 1.7 (viii) 23.1

(ix) 2.5 (x) 237.615

(xi) 15.3215 (xii) 0.9

(xiii) 0.1 (xiv) 0.016

(xv) 0.00064 (xvi) 0.019

(xvii) (xviii)

(xix) (xx)

2

Find the square root of 12.0068 correct to four decimal places.

3

Find the square root of 11 correct to five decimal places.

4

Give that: evaluate each of the following:

(i)

(ii)

5

Given that and find the square roots of the following:

(i)

(ii)

(iii)

(iv)