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99 Percentile Qs Bank for JEE MainMathematicsBinomial Theorem

For n ∈ N , in the expansion of x - 3 4 + a x 5 4 n , the sum of all binomial coefficients lies between 200 and 400 and the term independent of x is 448 . Then the value of a is

Options

  1. A1
  2. B2
  3. C1 2
  4. D0

Correct answer

B. 2

Step-by-step solution

Given that, x - 3 4 + a x 5 4 n General Term T r + 1 =   n C r ( 1 x 3 4 ) n - r a r ( x 5 4 ) r = n C r a r x - 3 4 n + 3 4 r + 5 4 r For independent of x - r = n 3 4 3 4 + 5 4 = 3 n 4 8 4 ∴   r = 3 n 8 The term independent of x is 448 . ⇒   C 3 n 8 n a 3 n 8 = 448 Sum of coefficient between 200 and 400 is given as 200 < 1 + a n < 400 According to the question, n C r a r = 448 ⇒ 8 C 3 a 3 = 448 ⇒ 56 a 3 = 448 ⇒ a 3 = 8 ⇒ a   =   2 ⇒ n = 8

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