Question 1: Find the square of the following numbers.

(i) 32

**Answer:** 32^{2} = 32 x 32 = 1024

But above method can be tough to calculate. It is easier to calculate such values in the following way:

Since, 32 can be written as (30+2)

So, 32^{2} = (30+2)^{2} = (30+2)(30+2)

= 30(30+2)+2(30+2) = 30^{2} + 30 x 2 + 2 x 30 + 2^{2}

= 900 + 60 + 60 + 4 = 1024

Answer: 1024

(ii) 35

**Answer:** (35)^{2 }= (30+5)^{2} = (30+5)(30+5)

= 30(30+5)+5(30+5) = 30^{2} + 30 x 5 + 5 x 30 + 5^{2}

= 900 + 150 + 150 + 25 = 1225

Thus, Answer = 1225

(iii) 86

**Answer:** 86^{2} = (80 + 6)^{2} = (80 + 6)(80 + 6)

= 80^{2} + 80 x 6 + 6 x 80 + 6^{2}

= 6400 + 480 + 480 + 36 = 7396

Thus, Answer: 7396

(iv) 93

**Answer:** 93^{2 } = (90+3)^{2} = (90 + 3) (90 + 3)

= 90 (90 + 3) + 3 (90 + 3) = 90 ^{2} + 90 x 3 + 3 x 90 + 3 ^{2}

= 8100 + 270 + 270 + 9 = 8649

Thus, Answer: 8649

(v) 71

**Answer:** 71 ^{2} = (70 + 1) ^{2} = (70 + 1) (70 + 1)

= 70 (70 + 1) + 1 (70 + 1) = 70^{2} + 70 x 1 + 1 x 70 + 1 x 1

= 4900 + 70 + 70 + 1 = 4900 + 140 + 1 = 5040 + 1 = 5041

Thus, Answer: 5041

(vi) 46

**Answer:** 46^{2 }= (40+6)^{2} = (40 + 6) (40 + 6)

= 40 (40 + 6) + 6 (40 + 6) = 40 ^{2} + 40 x 6 + 6 x 40 + 6^{2}

= 1600 + 240 + 240 + 36 = 1600 + 480 + 36 = 2080 + 36 = 2116

Thus, Answer: 2116

Question 2: Write a Pythagorean triplet whose one member is:

(i) 6

**Answer:** As we know 2m, m ^{2} + 1 and m^{2} - 1 form a Pythagorean triplet for any number, m > 1.

Let us assume 2m = 6

Therefore, m = 3

And, m^{2} + 1 = 3^{ 2 } + 1= 9 + 1 = 10

And, m^{ 2 } - 1 = 3^{ 2 } - 1 = 9 - 1 = 8

Test: 6^{ 2 } + 8^{ 2 } = 36 + 64 = 100 = 10^{2}

Hence, the triplet is 6, 8, and 10 Answer

(ii) 14

**Answer:** Let us assume, 2 m = 14, therefore, m = 7

Now, m ^{ 2 } + 1 = 7 ^{ 2 } + 1 = 49 + 1 = 50

And, m ^{ 2 } - 1 = 7 ^{ 2 } - 1 = 49 - 1 = 48

Test: 14 ^{ 2 } + 48 ^{ 2 } = 196 + 1304 = 2500 = 50 ^{ 2 }

Hence, the triplet is 14, 48, and 50 Answer

(iii) 16

**Answer:** Let us assume 2 m = 16, then m = 8

Now, m ^{ 2 } + 1 = 8^{ 2 } + 1 = 64 + 1 = 65

And, m ^{ 2 } - 1 = 8^{ 2 } - 1 = 64 - 1 = 63

Test: 16^{2 } + 63^{ 2 } = 256 + 3969 = 4225 = 65 ^{ 2 }

Hence, the triplet is 16, 63, and 65 Answer

(iv) 18

**Answer:** Let us assume 2 m = 18, therefore, m = 9

Now, m ^{ 2 } + 1 = 9 ^{ 2 } + 1 = 81 + 1 = 82

And, m ^{ 2 } - 1 = 9 ^{ 2 } - 1 = 81 - 1 = 80

Test: 18 ^{ 2 } + 80 ^{ 2 } = 324 + 6400 = 6724 = 82 ^{ 2 }

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