Question

If (1 + x)¹⁰ = a₀ + a₁ x + a₂ x² + a₃x³ +.......+ a₁₀x¹⁰, then (a₀ – a₂ + a₄ – a₆ + a₈ – a₁₀₎² + (a₁ – a₃ + a₅ – a₇ + a₉)² is equal to

(a)  3¹⁰ 

(b)   2¹⁰ 

(c)   2⁹ 

(d)  none of these

Answer 1

Correct option  (b)  2¹⁰ 

Explanation:

 (1 + i)¹⁰ = (a₀ – a₂ + a₄ – a₆ + a₈ – a₁₀) + i(a₁ – a₃ + a₅ + a₅ –a₇ + a₉ ^{)  ....(1)}

^Also, (1– i)¹⁰ = (a₀ – a₂ + a₄ –a₆ + a₈ – a₁₀) – i(a₁ – a₃ + a₅ – a₇ + a₉ )  .....(2)

Multiply (1) and (2)

((1 + i)(1 – i)¹⁰ = (a₀ – a₂ + a₄ – a₆ + a₈ – a₁₀)² + (a₁ – a₃ + a₅ – a₇ + a₉)²

2¹⁰ = (a₀ – a₂ + a₄ – a₆ + a₈ – a₁₀)² + (a₁ – a₃ + a₅ – a₇ + a₉)