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JEE Main202427 Jan 2024Morning ShiftMathematicsBinomial TheoremActual

C r n - 1 = k 2 - 8 C r + 1 n if and only if :

Options

  1. A2 2 < k ≤ 3
  2. B2 3 < k ≤ 3 2
  3. C2 3 < k < 3 3
  4. D2 2 < k < 2 3

Correct answer

A. 2 2 < k ≤ 3

Step-by-step solution

Given: C r n - 1 = k 2 - 8 C r + 1 n . . . i ⇒ r + 1 ≥ 0 , r ≥ 0 ⏟ r ≥ 0 ⇒ C r n - 1 C r + 1 n = k 2 - 8 ⇒ n - 1 ! r ! n - r - 1 ! n ! r + 1 ! n - r - 1 ! = k 2 - 8 ⇒ r + 1 n = k 2 - 8 ⇒ k 2 - 8 > 0 ⇒ ( k - 2 2 ) ( k + 2 2 ) > 0 ⇒ k ∈ ( - ∞ , - 2 2 ) ∪ ( 2 2 , ∞ ) . . . ( i i ) Using equation i , ⇒ n ≥ r + 1 ⇒ r + 1 n ≤ 1 ⇒ k 2 - 8 ≤ 1 ⇒ k 2 - 9 ≤ 0 ⇒ - 3 ≤ k ≤ 3 . . . ( i i ) From equation i i and i i i we get, k ∈ [ - 3 , - 2 2 ) ∪ ( 2 2 , 3 ]

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