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Answers
Answer:
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Explanation:
Solutions:-
1)
I) 121 = 11×11 = 11²
It is a perfect square number.
ii) 69 can not be written as the product of two equal numbers .
It is not a perfect square number.
2)
i) 11²
Let the consecutive numbers are x,x+1
Sum of the two numbers = 11² = 121
=> x+x+1 = 121
=> 2x+1 = 121
=> 2x = 121-1
=> 2x = 120
=> x = 120/2
=> x = 60
So, x+1 = 60+1 = 61
So the required numbers = 60,61
60+61 = 11²
ii)19²
Let the consecutive numbers are x,x+1
Sum of the two numbers = 19² = 361
=> x+x+1 = 361
=> 2x+1 = 361
=> 2x = 361-1
=> 2x = 360
=> x = 360/2
=> x = 180
So, x+1 = 180+1 = 181
So the required numbers = 180,181
19² = 180+181
3)
I) Given numbers are 25² and 26²
Let n² = 25²
Let (n+1)² = (25+1)² = 26²
We know that
The natural numbers lie between n² and (n+1)² is 2n
The numbers lie between 25² and 26²
=> 2×25 = 50
4)
Given number = 42
The square of 42 = 42²
=> (40+2)²
This is in the form of (a+b)²
Where , a = 40 and b = 2
We know that
(a+b)² = a²+2ab+b²
=> (40+2)² = 40²+2(40)(2)+2²
=> 42² = 1600+160+4
=> 42² = 1764
5)
i) Given number = 8
Let 2m = 8
=> m = 8/2 = 4
The Pythagorean Triplets are (2m, m²-1 , m²+1)
=> m²-1 = 4²-1 = 16-1 = 15
=> m²+1 = 16+1 = 17
The Pythagorean Triplet = (8,15,17)
ii)
Given number = 12
Let 2m = 12
=> m = 12/2 = 6
The Pythagorean Triplets are (2m, m²-1 , m²+1)
=> m²-1 = 6²-1 = 36-1 = 35
=> m²+1 = 36+1 = 37
The Pythagorean Triplet = (12,35,37)
6)Given number = 81
Repeated Subtraction Method
81 -1 = 80
80-3 = 77
77-5 = 72
72-7 = 65
65-9 = 54
54-11 = 43
43-13 = 30
30-15 = 15
15-15 = 0
We get zero in 9 steps
So, √81 = 9