If \( A \) and \( B \) are binomial coefficients of the 30\(^\text{th}\) and 12\(^\text{th}\) terms of the binomial expansion \( (1 + x)^{2n-1} \), and \( 2A = 5B \), then the value of \( n \) is

Question

Let \(y=y(x)\) be the solution of the differential equation \[ x\frac{dy}{dx}=y-x^2\cot x,\quad x\in(0,\pi) \] If \(y\!\left(\frac{\pi}{2}\right)=\frac{\pi^2}{2}\), then \[ 6y\!\left(\frac{\pi}{6}\right)-8y\!\left(\frac{\pi}{4}\right) \] is equal to:

Answer 1

The general term for the binomial expansion of \( (1 + x)^{2n-1} \) is \( T_{r+1} = \binom{2n-1}{r} x^r \). The 30th term is \( T_{30} = \binom{2n-1}{29} = A \), and the 12th term is \( T_{12} = \binom{2n-1}{11} = B \). Given that \( 2A = 5B \), we substitute the expressions for \( A \) and \( B \) to get \( 2 \binom{2n-1}{29} = 5 \binom{2n-1}{11} \). Solving this equation yields \( n = 21 \).

If \( A \) and \( B \) are binomial coefficients of the 30\(^\text{th}\) and 12\(^\text{th}\) terms of the binomial expansion \( (1 + x)^{2n-1} \), and \( 2A = 5B \), then the value of \( n \) is

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The Correct Option is B

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