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11.
Find whether each of the following numbers is a perfect square or not?

(i) 121 (ii) 55 (iii) 81

(iv) 49 (v) 69

(i) 121

∵           121 - 1 = 120                                       85-13 = 72
             120 - 3 = 117                                       72 - 15 = 57
             117 - 5 = 112                                       54 - 17 = 40
             112 - 7 = 105                                       40 - 19 = 21
             105 - 9 = 96                                         21 - 21 = 0
              96 - 11 = 85
i.e. 121 = 1+3+5+7+9+11+13+15+17+19+21. Thus , 121 is a perfact square.

(ii) 55

∵               55 - 1 = 54                                    30 - 11 = 19
   54 - 3 = 51                                    19 - 13 = 6
             51 - 5 = 46                                     6 - 15 = -9
             46 - 7 = 39
             39 - 9 = 30
Since, 55 cannot be expressed as the sum of successive old numbers starting from 1.

∴ 55 is not a perfact square.

(iii)    81
      Since,           81 - 1 = 80                            56 - 11 = 45
                         80 - 3 = 77                            45 - 13 = 32
                         77 - 5 = 72                            32 - 15 = 17
                         72 - 7 = 65                            17 - 17 = 0
                          65 - 9 =56

∴ 81 = 1+3+5+7+9+11+13+15+17
Thus, 81 is a perfact square.

(iv)      49
       Since,             49 - 1 = 48
                            48 - 3 = 45
                            45 - 5 = 40
                            40 - 7 = 33
                            33 - 9 = 24
                            24 - 11 = 13
                            13 - 13 = 0

∴ 49 = 1+3+5+7+9+11+13
Thus, 69 is not a perfact square.

(v) 69
Since,         69 - 1 = 68                                 44 - 11 = 33
                 68 - 3 = 65                                 33 - 13 = 20
                 65 - 5 = 60                                 20 - 15 = 5
                 60 - 7 = 53                                 5 - 17 = 12
   53 - 9 = 44

∴   69 cannot be expressed as the sum of consecutive odd num numbers starting fron 1. Thus, 69 is not a perfect square.

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12.

Express the following as the sum of two consecutive integers.

(i) 21² (ii) 13² (iii) 11² (iv) 19²

1628 Views

Short Answer Type

13.

Do you think the reverse is also true, i.e. is the sum of any two consecutive positive integers is perfect square of a number? Give example to support your answer.

1739 Views

14.

The difference between the squares of two consecutive natural numbers is equal to the sum of the two numbers.

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15.

Write the square, making use of the above pattern.

(i) 111111² (ii) 11111112

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16.

Can you find the square of the following numbers using the above pattern?

(i) 6666667² (ii) 66666667²

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Long Answer Type

17.

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

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18.

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

804 Views

Short Answer Type

19.

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

(i) 431 (ii) 2826 (iii) 7779 (iv) 82004

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20.

Observe the following pattern and find the missing digits.

11²= 121
101²= 10210
1001²= 1002001
100001²= 1 ............. 2 ..............1
10000001²= ........................

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