8th Maths 6.1

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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)

$99^2 text{ and }100^2 = 2text{n} = 2times99 = 198$

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