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Maths Binomial Theorem & Mathematical Induction Mix Subjective Type

Published on: August 13, 2026

Prove that (x 2 + y 2 ) 5 = (x 5 – 10x 3 y 2 + 5xy 4 ) 2 + (5x 4 y – 10x 2 y 3 + y 5 ) 2 .

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Sol. (x 2 + y 2 ) 5 = (x + iy) 5 (x – iy) 5

= [x 5 + 5 C 1 x 4 (iy) + 5 C 2 x 3 (iy) 2 + 5 C 3 x 2 (iy) 3 + 5 C 4 x(iy) 4 + (iy) 5 ]

[x 5 – 5 C 1 x 4 (iy) + 5 C 2 x 3 (iy) 2 – 5 C 3 x 2 (iy) 3 + 5 C 4 x (iy) 4 – (iy) 5 ]

= [(x 5 –10x 3 y 2 + 5xy 4 ) + i(5x 4 y – 10x 2 y 3 + y 5 ) ×[(x 5 –10x 3 y 2 + 5xy 4 ) –i (5x 4 y –10x 2 y 3 + y 5 )]

= (x 5 –10x 3 y 2 + 5xy 4 ) 2 + (5x 4 y –10x 2 y 3 + y 5 ) 2

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