x +

Hence, option B is the correct answer.

a² + b² + 1 = 2a

or, a² – 2a + 1+ b² = 0

or, (a – 1)² + b² = 0

or, a – 1 = 0

so, a = 1

and b = 0

a⁴ + b⁷ = 1⁴ + 0⁷ = 1 + 0 = 1

Hence, option A is the correct answer.

Let x/8 = p

We know that, if p + 1/p = 1, then p³ = -1

∴ x³/8³ = -1     [From (i)]

⇒ x³/512 = -1

⇒ x³ = -512

Hence, option D is the correct answer.

= (100 + 99)(100 – 99) + (98 + 97)(98 – 97) + …………+ (22 + 21)(22 – 21)

[since a² – b² = (a + b)(a – b)]

= (100 + 99)(1) + (98 + 97)(1) + …………+ (22 + 21)(1)

= 100 + 99 + 98 + 97 + …………+ 22 + 21

It is an A.P series, where n = 80, first term(a) = 21 and last term(l) = 100

We know that,

Sum of n terms(n) = [n/2][a + l]

= [80/2][21 + 100]

= [40][121] = 4840

Hence, option D is the correct answer.

= (1 – x)(1 + x) (1 + x²) (1 + x⁴) (1 + x⁸)

= (1 - x²)(1 + x²) (1 + x⁴)(1 + x⁸)

= (1 – x⁴)(1 + x⁴)(1 + x⁸)

= (1 – x⁸)(1 + x⁸)

= 1 – x¹⁶

Hence, option C is the correct answer.

p –  = 6

p² + = (p – )² + 2 = 6² + 2 = 38

p⁴ + = (p² + )² – 2 = 38² – 2 = 1442

"Hence, option C is the correct answer."

x⁴ – 15x³ + 15x² – 15x + 40

= 14⁴ – (14+1)14³ + (14+1)14² – (14 + 1)14 + 40

= 14⁴ - 14⁴ – 14³ + 14³ + 14² – 14² – 14 + 40

= 40 – 14 = 26

"Hence, option B is the correct answer."

(a + b)² – (a – b)²

= (a² + 2ab + b²) - (a² - 2ab + b²)

= a² + 2ab + b² - a² + 2ab - b²

= 4ab

"Hence, option D is the correct answer."

We know that:

If x + 1/x = n, then x³ + 1/x³ = (n)³ – 3(n)

x + 1/x = 2

∴ x³ + 1/x³ = (2)³ – 3(2)

= 8 – 6

= 2

Hence, option A is the correct answer.

⇒

⇒

⇒

⇒

⇒

⇒

∴

= 64 – 12

= 52

Hence, option A is the correct answer.