Let ( x 3 + α x 2 + 2 x − 5 ) 19 ( x 2 + β x − 41 ) 8 ( x 4 − x 3 + x − 7 ) 6 = x 97 + 391 x 96 + a 95 x 95 + a 94 x 94 + … + a 1 x + a 0 be an identity, where α , β , a 95 , a 94 , … , a

Question

Let ( x 3 + α x 2 + 2 x − 5 ) 19 ( x 2 + β x − 41 ) 8 ( x 4 − x 3 + x − 7 ) 6 = x 97 + 391 x 96 + a 95 ​ x 95 + a 94 ​ x 94 + … + a 1 ​ x + a 0 ​ be an identity, where α , β , a 95 ​ , a 94 ​ , … , a 1 ​ , a 0 ​ are integers. If α + β < 10 , then the smallest possible value of α is

TestHub · Accepted Answer

It will be an identity even if we replace x by y 1 ​ and considering numerator alone. Differentiating on both sides with respect to y, at y =0 we get 19 α + 8 β = 397 , α + β = 10 − k where k is positive integer.Put β = 10 − α − k in first equation we get 11 α − 8 k = 317 ∴ α = 31

Mathematics - Binomial Theorem Question with Solution | TestHub

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