Misc 2 - Class 11

Transcript

Misc 2 Find a if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal. We know that General term of expansion (a + b)n is Tr+1 = nCr (a)n–r . br For (3 + ax)9 , Putting a = 3 , b = 9x & n = 9 General term of (3 + ax )9 is Tr+1 = 9Cr (3)9–r . (ax)r = 9Cr 39–r . ar . xr We need Coefficient of x2 & x3 Given that Coefficient of x2 = coefficient of x3 9C2 (3)7 . a2 = 9C2 (3)6 . a3 ﷐9!﷮2!﷐9 −2﷯!﷯ 37 . a2 = ﷐9!﷮3!﷐9 −3﷯!﷯ 36 . a3 ﷐9!﷮2!﷐7﷯!﷯ 37 . a2 = ﷐9!﷮3!﷐6﷯!﷯ 36 . a3 ﷐﷐9!﷮2! 7!﷯ . 37﷮﷐9!﷮3!﷐9 −3﷯!﷯ . 36﷯ = ﷐𝑎3﷮𝑎2﷯ ﷐9!﷮2! 7!﷯ . 37 × ﷐3! 6!﷮9! . 36﷯ = a ﷐9! × 3! × 6!﷮9! × 2! × 7!﷯ × ﷐37﷮36﷯ = a ﷐3 ×2! ×6!﷮2! ×7 ×6!﷯ . 3 = a ﷐3﷮7﷯ × 3 = a ﷐9﷮7﷯ = a Hence, a = ﷐9﷮7﷯