8th Maths 6.1

Chapter 6

Squares and Square Roots

NCERT Class 8th solution of Exercise 6.2

NCERT Class 8th solution of Exercise 6.3

NCERT Class 8th solution of Exercise 6.4

Exercise 6.1

`text {Square you must learn.} `

`1^2=1`

`2^2=4`

`3^2=9`

`4^2=16`

`5^2=25`

`6^2=36`

`7^2=49`

`8^2=64`

`9^2=81`

`10^2=100`

`11^2=121`

`12^2=144`

`13^2=169`

`14^2=196`

`15^2=225`

`16^2=256`

`17^2=289`

`18^2=324`

`19^2=361`

`20^2=400`

`21^2=441`

`22^2=484`

`23^2=529`

`24^2=756`

`25^2=625`

`26^2=676`

`27^2=729`

`28^2=224`

`29^2=841`

`30^2=900`

`31^2=961`

`32^2=1024`

`33^2=1089`

`34^2=1156`

`35^2=1225`

`36^2=1296`

`37^2=1369`

`38^2=1444`

`39^2=1521`

`40^2=1600`

`41^2=1681`

`42^2=1764`

`43^2=1849`

`44^2=1936`

`45^2=2025`

`46^2=2116`

`47^2=2209`

`48^2=2304`

`49^2=2401`

`50^2=2500`

Q1. 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`

`text{Sol:}`

i) `text{The unit digit of 81 is 1,}``text{ so that its unit digit is 1.}`

ii) `text{The unit digit of 272 is 2,}``text{ so that its unit digit is 4.}`

iii) `text{The unit digit of 799 is 9,}``text{ so that its unit digit is 1.}`

iv) `text{The unit digit of 3853 is 3,}``text{ so that its unit digit is 9.}`

v) `text{The unit digit of 1234 is 4,}``text{ so that its unit digit is 6.}`

vi) `text{The unit digit of 26387 is 7,}``text{ so that its unit digit is 9.}`

vii) `text{The unit digit of 52698 is 8,}``text{ so that its unit digit is 4.}`

viii) `text{The unit digit of 99880 is 0,}``text{ so that its unit digit is 0.}`

ix) `text{The unit digit of 12796 is 6,}``text{ so that its unit digit is 6.}`

x) `text{The unit digit of 55555 is 5,}``text{ so that its unit digit is 5.}`

Q2. The following numbers are obviously not perfect squares. Give reason.

i) `1057`

ii) `23453`

iii) `7928`

iv) `222222`

v) `64000`

vi) `89722`

vii) `222000`

viii) `505050`

`text{Sol.}`

i) `text{The unit digit of 1057 is 7,}``text{ so that it is not perfect square.}`

ii) `text{The unit digit of 23453 is 3,}``text{ so that it is not perfect square.}`

iii) `text{The unit digit of 7928 is 8,}``text{ so that it is not perfect square.}`

iv) `text{The unit digit of 222222 is 2,}``text{ so that it is not perfect square.}`

v) `text{The number 64000 end with three 0,}``text{ so that it is not perfect square.}`

vi) `text{The unit digit of 89722 is 2,}``text{ so that it is not perfect square.}`

vii) `text{The number 222000 end with three 0,}``text{ so that it is not perfect square.}`

viii) `text{The number 505050 end with one 0,}``text{ so that it is not perfect square.}`

Q3. The squares of which of the following would be odd numbers?

i) `431`

ii) `2826`

iii) `7779`

iv) `82004`

`text{Sol.}`

i) `text{The unit digit of 431 is 1,}``text{ so that square would be an odd number.}`

ii) `text{The unit digit of 2826 is 6,}``text{ so that square would be an even number.}`

iii) `text{The unit digit of 7779 is 9,}``text{ so that square would be an odd number.}`

iv) `text{The unit digit of 82004 is 4,}``text{ so that square would be an even number.}`

Q4. Observe the following pattern and find the missing digits.

`11^2 = 121`

`101^2 = 10201`

`1001^2 = 1002001`

`100001^2 = 1underline text{     }2underline text{     }1`

`1000001^2 = underline text{     }`

`text{Sol.}`

`text{According to the given pattern}`

`100001^2 = 1underline{0000}2underline{0000}1`

`1000001^2 = underline{1000002000001}`

Q5. Observe the following pattern and supply the missing numbers.

`11^2 = 121`

`101^2 = 10201`

`10101^2 = 102030201`

`1010101^2 = underline text{               }`

`underline text{            } = 10203040504030201`

`text{Sol.}`

`text{According to given pattern}`

`1010101^2 = underline text{1020304030201}`

`underline{101010101^2} = 10203040504030201`

Q6. Using the given pattern, find the missing numbers.

`1^2 + 2^2 + 2^2 = 3^2`

`2^2 + 3^2 + 6^2 = 7^2`

`3^2 + 4^2 + 12^2 = 13^2`

`4^2 + 5^2 + underline text{     } = 21^2`

`5^2 + underline text{     } + 30^2 = 31^2`

`6^2 + 7^2 + underline text{     } =underline text{     }`

`text{Sol.}`

`text{According to given pattern}`

`4^2 + 5^2 + underline{20^2} = 21^2`

`5^2 + underline{6^2 } + 30^2 = 31^2`

`6^2 + 7^2 + underline{42^2 } = underline{43^2}`

Q7. Without adding, find the sum.

i) `1 + 3 + 5 + 7 + 9`

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

ii) `1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19`` + 21 + 23`

`text{Sol.}`

`text{We know that,}``text{The sum of first n odd numbers is n}^2`.

i) `1 + 3 + 5 + 7 + 9`

`text{5 odd numbers so, n = 5 and }``text{sum is n}^2 = 5^2 = 25`.

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

`text{10 odd numbers so, n = 10 and }``text{sum is n}^2 = 10^2 = 100`.

ii) `1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19`` + 21 + 23`

`text{12 odd numbers so, n = 12 and }``text{sum is n}^2 = 12^2 = 144`.

Q8. 

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

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

`text{Sol.}`

`text{We know that,}``text{The sum of first n odd numbers is n}^2`.

i) `text{n}^2 = 7^2 = 49`

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

ii) `text{n}^2 = 11^2 = 121`

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

Q9. How many numbers lie between squares of the following numbers?

i) `12 text{ and } 13`

ii) `25 text{ and }26`

iii) `99 text{ and }100`

`text{Sol.}`

`text{We know that,}``text{number lie between n}^2 text{ and (n + 1)}^2 = 2text{n}`

i) `12^2 text{ and } 13^2 = 2text{n} = 2times12 = 24`

ii) `25^2 text{ and }26^2 = 2text{n} = 2times25 = 50`

iii) `99^2 text{ and }100^2 = 2text{n} = 2times99 = 198`