Updated By SaraNextGen

On March 11, 2024, 11:35 AM

Question 1:

Find the square root of each of the following numbers by division method.

(i) 2304 (ii) 4489

(iii) 3481 (iv) 529

(v) 3249 (vi) 1369

(vii) 5776 (viii) 7921

(ix) 576 (x) 1024

(xi) 3136 (xii) 900

Answer:

(i) The square root of 2304 can be calculated as follows.

∴ 

(ii) The square root of 4489 can be calculated as follows.

∴ 

(iii) The square root of 3481 can be calculated as follows.

Therefore, 

(iv) The square root of 529 can be calculated as follows.

∴ 

(v) The square root of 3249 can be calculated as follows.

∴ 

(vi) The square root of 1369 can be calculated as follows.

∴ 

(vii) The square root of 5776 can be calculated as follows.

∴ 

(viii) The square root of 7921 can be calculated as follows.

∴ 

(ix) The square root of 576 can be calculated as follows.

∴ 

(x) The square root of 1024 can be calculated as follows.

∴ 

(xi) The square root of 3136 can be calculated as follows.

∴ 

(xii) The square root of 900 can be calculated as follows.

∴ 

Question 2:

Find the number of digits in the square root of each of the following numbers (without any calculation).

(i) 64 (ii) 144

(iii) 4489 (iv) 27225

(v) 390625

Answer:

(i) By placing bars, we obtain

Since there is only one bar, the square root of 64 will have only one digit in it.

(ii) By placing bars, we obtain

Since there are two bars, the square root of 144 will have 2 digits in it.

(iii) By placing bars, we obtain

Since there are two bars, the square root of 4489 will have 2 digits in it.

(iv) By placing bars, we obtain

Since there are three bars, the square root of 27225 will have three digits in it.

(v) By placing the bars, we obtain

Since there are three bars, the square root of 390625 will have 3 digits in it.

Question 3:

Find the square root of the following decimal numbers.

(i) 2.56 (ii) 7.29

(iii) 51.84 (iv) 42.25

(v) 31.36

Answer:

(i) The square root of 2.56 can be calculated as follows.

∴ 

(ii) The square root of 7.29 can be calculated as follows.

∴ 

(iii) The square root of 51.84 can be calculated as follows.

∴ 

(iv) The square root of 42.25 can be calculated as follows.

∴ 

(v) The square root of 31.36 can be calculated as follows.

∴ 

Question 4:

Find the least number which must be subtracted from each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.

(i) 402 (ii) 1989

(iii) 3250 (iv) 825

(v) 4000

Answer:

(i) The square root of 402 can be calculated by long division method as follows.

The remainder is 2. It represents that the square of 20 is less than 402 by 2. Therefore, a perfect square will be obtained by subtracting 2 from the given number 402.

Therefore, required perfect square = 402 − 2 = 400

And, 

(ii) The square root of 1989 can be calculated by long division method as follows.

The remainder is 53. It represents that the square of 44 is less than 1989 by 53. Therefore, a perfect square will be obtained by subtracting 53 from the given number 1989.

Therefore, required perfect square = 1989 − 53 = 1936

And,

(iii) The square root of 3250 can be calculated by long division method as follows.

The remainder is 1. It represents that the square of 57 is less than 3250 by 1. Therefore, a perfect square can be obtained by subtracting 1 from the given number 3250.

Therefore, required perfect square = 3250 − 1 = 3249

And, 

(iv) The square root of 825 can be calculated by long division method as follows.

The remainder is 41. It represents that the square of 28 is less than 825 by 41. Therefore, a perfect square can be calculated by subtracting 41 from the given number 825.

Therefore, required perfect square = 825 − 41 = 784

And, 

(v) The square root of 4000 can be calculated by long division method as follows.

The remainder is 31. It represents that the square of 63 is less than 4000 by 31. Therefore, a perfect square can be obtained by subtracting 31 from the given number 4000.

Therefore, required perfect square = 4000 − 31 = 3969

And,

Question 5:

Find the least number which must be added to each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.

(i) 525 (ii) 1750

(iii) 252 (iv) 1825

(v) 6412

Answer:

(i) The square root of 525 can be calculated by long division method as follows.

The remainder is 41.

It represents that the square of 22 is less than 525.

Next number is 23 and 23² = 529

Hence, number to be added to 525 = 23² − 525 = 529 − 525 = 4

The required perfect square is 529 and 

(ii) The square root of 1750 can be calculated by long division method as follows.

The remainder is 69.

It represents that the square of 41 is less than 1750.

The next number is 42 and 42² = 1764

Hence, number to be added to 1750 = 42² − 1750 = 1764 − 1750 = 14

The required perfect square is 1764 and 

(iii) The square root of 252 can be calculated by long division method as follows.

The remainder is 27. It represents that the square of 15 is less than 252.

The next number is 16 and 16² = 256

Hence, number to be added to 252 = 16² − 252 = 256 − 252 = 4

The required perfect square is 256 and 

(iv) The square root of 1825 can be calculated by long division method as follows.

The remainder is 61. It represents that the square of 42 is less than 1825.

The next number is 43 and 43² = 1849

Hence, number to be added to 1825 = 43² − 1825 = 1849 − 1825 = 24

The required perfect square is 1849 and 

(v) The square root of 6412 can be calculated by long division method as follows.

The remainder is 12.

It represents that the square of 80 is less than 6412.

The next number is 81 and 81² = 6561

Hence, number to be added to 6412 = 81² − 6412 = 6561 − 6412 = 149

The required perfect square is 6561 and 

Question 6:

Find the length of the side of a square whose area is 441 m².

Answer:

Let the length of the side of the square be x m.

Area of square = (x)² = 441 m²

The square root of 441 can be calculated as follows.

∴ 

Hence, the length of the side of the square is 21 m.

Question 7:

In a right triangle ABC, ∠B = 90°.

(a) If AB = 6 cm, BC = 8 cm, find AC

(b) If AC = 13 cm, BC = 5 cm, find AB

Answer:

(a) ΔABC is right-angled at B.

Therefore, by applying Pythagoras theorem, we obtain

AC² = AB² + BC²

AC² = (6 cm)² + (8 cm)²

AC² = (36 + 64) cm² =100 cm²

AC = 

AC = 10 cm

(b) ΔABC is right-angled at B.

Therefore, by applying Pythagoras theorem, we obtain

AC² = AB² + BC²

(13 cm)² = (AB)² + (5 cm)²

AB² = (13 cm)² − (5 cm)² = (169 − 25) cm² = 144 cm²

AB = 

AB = 12 cm

Question 8:

A gardener has 1000 plants. He wants to plant these in such a way that the number of rows and the number of columns remain same. Find the minimum number of plants he needs more for this.

Answer:

It is given that the gardener has 1000 plants. The number of rows and the number of columns is the same.

We have to find the number of more plants that should be there, so that when the gardener plants them, the number of rows and columns are same.

That is, the number which should be added to 1000 to make it a perfect square has to be calculated.

The square root of 1000 can be calculated by long division method as follows.

The remainder is 39. It represents that the square of 31 is less than 1000.

The next number is 32 and 32² = 1024

Hence, number to be added to 1000 to make it a perfect square

= 32² − 1000 = 1024 − 1000 = 24

Thus, the required number of plants is 24.

Question 9:

These are 500 children in a school. For a P.T. drill they have to stand in such a manner that the number of rows is equal to number of columns. How many children would be left out in this arrangement?

Answer:

It is given that there are 500 children in the school. They have to stand for a P.T. drill such that the number of rows is equal to the number of columns.

The number of children who will be left out in this arrangement has to be calculated. That is, the number which should be subtracted from 500 to make it a perfect square has to be calculated.

The square root of 500 can be calculated by long division method as follows.

The remainder is 16.

It shows that the square of 22 is less than 500 by 16. Therefore, if we subtract 16 from 500, we will obtain a perfect square.

Required perfect square = 500 − 16 = 484

Thus, the number of children who will be left out is 16.