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Most Important Selected Qs for JEE AdvancedMathematicsBinomial Theorem

Consider two polynomial f(x) and g(x) given by f(x)= _ r=0 ²⁰⁰ _r x^r and g(x)= _ r=0 ²⁰⁰ _r x^r such that _r=1 100 r 200 and f(x+1)=g(x) . Let A= _ r=100 ²⁰⁰ _r , then the remainder when A is divided by 15 is equal to

Correct answer

1

Step-by-step solution

aligned & _ r=0 ²⁰⁰ _r x^r= _ r=0 ²⁰⁰ _r(1+x)^r & ₀+ ₁ x+ .+ ₂₀₀ x²⁰⁰= ₀+ ₁(1+x)+ .+ ₂₀₀(1+x)²⁰⁰ & = ₀+ ₁(1+x)+ .+ ₉₉(1+x)⁹⁹+ (1+x)²⁰¹-(1+x)¹⁰⁰ x aligned Equating coefficient of x¹⁰⁰, x¹⁰¹, ., x²⁰⁰ respectively we get aligned & ₁₀₀= 201 101 , ₁₀₁= 201 102 , ., ₂₀₀= 201 201 & A= _ r=100 ²⁰⁰ _r= ₁₀₀+ ₁₀₁+ . .+ ₂₀₀=2²⁰⁰ aligned When 2²⁰⁰ is divided by 15 we get remainder 1

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