(a) If let’s show that and

(b) If let’s show that

(c) If let’s show that

i.

ii.

(d) If let’s show that (using formula)

(e) If then let’s find out the value of

_{Formulas used:}

_{(a + b)}² = a² + 2ab + b² (1)

_{(a – b)}² = a² – 2ab + b² (2)

_{(a) Given,}

Squaring both side, we get

⇒

Taking a = x, b , in identity (1)
⇒
⇒

Again squaring both side, we get

Squaring both side, we get

Taking a = x², b , in identity (1)
⇒
⇒
⇒

(b) Given,

Squaring both sides, we get

Taking a = m, b , in identity (1)

⇒
⇒
⇒

(c) i)

Given,

Squaring both sides, we get

Taking a = p, b , in identity (2)

⇒
⇒

ii)

From Part (i), we have

Adding 2 both side,

⇒

(d) Given,

a + b = 5

Squaring both sides, we get

(a + b)² = 25

⇒ a² + 2ab + b² = 25 [1]

Also, a – b = 1

Squaring both sides, we get

(a – b)² = 1

⇒ a² – 2ab + b² = 1 [2]

Adding [1] and [2], we get

a² + 2ab + b² + a² – 2ab + b² = 25 + 1

⇒ 2a² + 2b² = 26

⇒ 2(a² + b²) = 26 [3]

Subtracting [2] from [1], we get

a² + 2ab + b² – (a²– 2ab + b²) = 25 – 1

⇒ a² + 2ab + b² – a² + 2ab – b² = 24

⇒ 4ab = 24 [4]

Multiplying [3] and [4], we get

2(a² + b)² (4ab) = 26 × 24

⇒ 8ab(a² + b²) = 624

Hence, Proved!

(e) Given,

x – y = 3

xy = 28

To find: x² + y²

We know,

(x – y)² = x² + y² – 2xy

Putting values,

⇒ 3² = x² + y² -2(28)

⇒ 9 = x² + y² – 56

⇒ x² + y² = 56 + 9

⇒ x² + y² = 65