Class 8 Exam  >  Class 8 Notes  >  NCERT Solutions (Ex: 6.1 - 6.2) - Squares and Square Roots

NCERT Solutions (Ex: 6.1 - 6.2) - Squares and Square Roots - Class 8 PDF Download

Exercise 6.1 

Question 1: 

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

(i) 81

(ii) 272

(iii) 799

(iv) 3853

(v) 1234

(vi) 26387

(vii) 52698

[viii] 99880

(ix) 12796

[x] 55555

Answer 1: 

(i) The number 81 contains its unit’s place digit 1. So, square of 1 is 1.

Hence, unit’s digit of square of 81 is 1.

(ii) The number 272 contains its unit’s place digit 2.

So, square of 2 is 4. Hence, unit’s digit of square of 272 is 4.

(iii) The number 799 contains its unit’s place digit 9.

So, square of 9 is 81. Hence, unit’s digit of square of 799 is 1.

(iv) The number 3853 contains its unit’s place digit 3.

So, square of 3 is 9. Hence, unit’s digit of square of 3853 is 9.

(v) The number 1234 contains its unit’s place digit 4.

So, square of 4 is 16. Hence, unit’s digit of square of 1234 is 6.

(vi) The number 26387 contains its unit’s place digit 7.

So, square of 7 is 49. Hence, unit’s digit of square of 26387 is 9.

(vii) The number 52698 contains its unit’s place digit 8.

So, square of 8 is 64. Hence, unit’s digit of square of 52698 is 4.

(viii) The number 99880 contains its unit’s place digit 0.

So, square of 0 is 0. Hence, unit’s digit of square of 99880 is 0.

(ix) The number 12796 contains its unit’s place digit 6.

So, square of 6 is 36. Hence, unit’s digit of square of 12796 is 6.

(x) The number 55555 contains its unit’s place digit 5.

So, square of 5 is 25. Hence, unit’s digit of square of 55555 is 5.

Question 2: 

The following numbers are obviously not perfect squares. Give reasons. 

[i] 1057

[ii] 23453

(iii] 7928

(iv] 222222

[v] 64000

(vi) 89722

(vii) 222000

(viii) 505050

Answer 2: 

(i) Since, perfect square numbers contain their unit’s place digit 1, 4, 5, 6, 9 and even numbers of 0.

Therefore 1057 is not a perfect square because its unit’s place digit is 7.

(ii) Since, perfect square numbers contain their unit’s place digit 0, 1, 4, 5, 6, 9 and even number of 0.

Therefore 23453 is not a perfect square because its unit’s place digit is 3.

(iii) Since, perfect square numbers contain their unit’s place digit 0, 1, 4, 5, 6, 9 and even number of 0.

Therefore 7928 is not a perfect square because its unit’s place digit is 8.

(iv) Since, perfect square numbers contain their unit’s place digit 0, 1, 4, 5, 6, 9 and even number of 0.

Therefore 222222 is not a perfect square because its unit’s place digit is 2.

(v) Since, perfect square numbers contain their unit’s place digit 0, 1, 4, 5, 6, 9 and even number of 0.

Therefore 64000 is not a perfect square because its unit’s place digit is single 0.

(vi) Since, perfect square numbers contain their unit’s place digit 0, 1, 4, 5, 6, 9 and even number of 0.

Therefore 89722 is not a perfect square because its unit’s place digit is 2.

(vii) Since, perfect square numbers contain their unit’s place digit 0, 1, 4, 5, 6, 9 and even number of 0.

Therefore 222000 is not a perfect square because its unit’s place digit is triple 0.

(viii) Since, perfect square numbers contain their unit’s place digit 0, 1, 4, 5, 6, 9 and even number of 0.

Therefore 505050 is not a perfect square because its unit’s place digit is 0.

Question 3: 

The squares of which of the following would be odd number: 

(i) 431 

(ii) 2826 

(iii) 7779 

(iv) 82004 

Answer 3: (i) 431 – Unit’s digit of given number is 1 and square of 1 is 1.

Therefore, square of 431 would be an odd number.

(ii) 2826 – Unit’s digit of given number is 6 and square of 6 is 36.

Therefore, square of 2826 would not be an odd number.

(iii) 7779 – Unit’s digit of given number is 9 and square of 9 is 81.

Therefore, square of 7779 would be an odd number.

(iv) 82004 – Unit’s digit of given number is 4 and square of 4 is 16.

Therefore, square of 82004 would not be an odd number.

Question 4: 

Observe the following pattern and find the missing digits: 

NCERT Solutions (Ex: 6.1 - 6.2) - Squares and Square Roots - Class 8

Answer 4: 

NCERT Solutions (Ex: 6.1 - 6.2) - Squares and Square Roots - Class 8

 

Question 5: 

Observe the following pattern and supply the missing numbers: 

NCERT Solutions (Ex: 6.1 - 6.2) - Squares and Square Roots - Class 8

Answer 5: 

NCERT Solutions (Ex: 6.1 - 6.2) - Squares and Square Roots - Class 8

Question 6: 

Using the given pattern, find the missing numbers: 

NCERT Solutions (Ex: 6.1 - 6.2) - Squares and Square Roots - Class 8

Answer 6: 

NCERT Solutions (Ex: 6.1 - 6.2) - Squares and Square Roots - Class 8

Question 7: 

Without adding, find the sum: 

(i) 1 + 3 + 5 + 7 + 9 

(ii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 

(iii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 

Answer 7: 

[i]Here,there are five odd numbers. Therefore square of 5 is 25.

∴ 1 + 3 + 5 + 7 + 9 = 52 = 25

[ii] Here, there are ten odd numbers. Therefore square of 10 is 100.

∴ 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 102 = 100

[iii]

Here, there are twelve odd numbers. Therefore square of 12 is 144.

∴ 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 = 122 = 144

Question 8: 

(i) Express 49 as the sum of 7 odd numbers. 

(ii) Express 121 as the sum of 11 odd numbers. 

Answer 8: 

(i) 49 is the square of 7. Therefore it is the sum of 7 odd numbers.

49 = 1 + 3 + 5 + 7 + 9 + 11 + 13

(ii) 121 is the square of 11. Therefore it is the sum of 11 odd numbers

121 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21

Question 9:

How many numbers lie between squares of the following numbers: 

(i) 12 and 13 

(ii) 25 and 26 

(iii) 99 and 100 

Answer 9: 

(i)

Since, non-perfect square numbers between n2 and (n + 1)"are  2n

Here, n = 12

Therefore, non-perfect square numbers between 12 and 13 = 2 n = 2x12 = 24

[ii]

Since, non-perfect square numbers between n2 and (n + 1)2 are 2n.

Here, n - 25

Therefore, non-per feet square numbers between 25 and 26 = 2 n = 2x25 = 50

[iii]

Since, non-perfect square numbers between n and (n + 1) are 2n.

Here, n = 99

Therefore, non-perfect square numbers between 99 and 100 = 2n = 2 x 99 = 198

Exercise 6.2 

Question 1: Find the squares of the following numbers: 

(i) 32 

(ii) 35 

(iii) 86 

(iv) 93 

(v) 71 

(vi) 46 

Answer 1: 

(i) (32)= (30 + 2)2 = (30)2 + 2x 30x 2 + (2)2  [∵ (a+b)2 = a2 + b2 + 2ab]

= 900+ 120 + 4 = 1024

(ii)

(35)2 = (30 + 5)= (30)2 + 2x 30x 5 + (5)2  [∵ (a+b)2 = a2 + b2 + 2ab]

= 900 + 300 + 25 = 1225 

(iii)

(86)= (80 + 6)2 = (80)2+ 2 x 80 x 6 + (6)2     [∵ (a+b)2 = a2 + b2 + 2ab]

= 1600 + 960 + 36 = 7386

(iv)

(93)2 =(90 +3)2 =(90)2 +2x90x3 + (3)    [∵ (a+b)2 = a2 + b2 + 2ab]

= 8100 + 540 + 9 = 8649

(v)

(71)2 = (70+1)2 = 70 + 2 x 70 x 1 + 12 [∵ (a+b)2 = a2 + b2 + 2ab]

= 4900 + 140 + 1 = 5041

[vi]

(46)2 =(40 +6)2 = (40)2+2x40x6 +(6)2   [∵ (a+b)2 = a2 + b2 + 2ab]

= 1600 + 480 + 36 = 2116

Question 2: 

Write a Pythagoras triplet whose one member is: 

(i) 6 

(ii) 14 

(iii) 16 

(iv) 18 

Answer 2: 

(i)

There are three numbers 2m,m 2 -1 and m2 + 1 in a Pythagorean Triplet

Here, 2m 6  =>  m = 6/2 = 3

There fore,

Second number (m2 -1) = (3)2- 1 9 -1 8

Third number m2+1 (3)2 +1 = 9 + 1 = 10

Hence, Pythagorean triplet is (6,8,10).

(ii)

There are three numbers 2m, m-1 and nr +1 in a Pythagorean Triplet

Here, 2m = 6, m = 6/2 = 3

Therefore,

Second number (m2 - 1) = (7)2 -1 = 49 - 1 = 48

Third number m2 +1 = (7)2 + l =49 + l = 50

Hence, Pythagorean triplet is (14,48, 50).

(iii)

There are three numbers 2m,m2 -1 and m2 + 1 in a Pythagorean Triplet

Here, 2m =16  , m = 8

Therefore,

Second number (m2 - 1) = (8)2 -1 =64- 1 = 63

Third number m2 +1 = (8)2 +1 = 64 +1 = 65

Hence, Pythagorean triplet is (16, 63, 65).

(iv)

There are three numbers 2m,m2 -1 and m2 + 1 in a Pythagorean Triplet

Here, 2m =18 , m = 9

Therefore,

Second number (m2 - 1) = (9)2 -1 = 81- 1 = 80

Third number m2 +1 = (9)2 +1 = 81 +1 = 62

Hence, Pythagorean triplet is (18, 80, 82).

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FAQs on NCERT Solutions (Ex: 6.1 - 6.2) - Squares and Square Roots - Class 8

1. What is a square in mathematics?
Ans. In mathematics, a square is a number that is obtained by multiplying a number by itself. It is represented by a symbol "²". For example, 4² means 4 times 4, which equals 16. The area of a square can be calculated by multiplying the length of its sides by itself.
2. What is meant by a perfect square?
Ans. A perfect square is a number that is obtained by multiplying a whole number by itself. For example, 9 is a perfect square because it is the product of 3 and 3. Similarly, 16 is also a perfect square because it is the product of 4 and 4.
3. How to find the square root of a number?
Ans. To find the square root of a number, we need to find a number which when multiplied by itself gives the original number. For example, the square root of 25 is 5 because 5 × 5 = 25. We can use the prime factorization method or long division method to find the square root of a number.
4. What is the difference between a square and a rectangle?
Ans. A square is a special type of rectangle in which all four sides are equal in length. In other words, a square is a rectangle with all sides equal. However, a rectangle can have two sides of equal length or all four sides can be of different lengths.
5. How can we use square roots in real life?
Ans. Square roots have several applications in real life, including in engineering, architecture, and physics. For example, in construction, square roots are used to calculate the length of sides of a square or rectangular-shaped room. In physics, square roots are used to calculate the speed of an object, the energy of a particle, and the distance between two points in space.
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