If 1 + x n = C 0 + C 1 x + C 2 x 2 + . . . . + C n x n , ∑ r = 0 n r + 1 2 C r = 2 n - 2 f n and if the roots of the equation f x = 0 are α and β , then the value of α 2 + β 2 is equal to (where C r denotes C r n )

Question

If 1 + x n = C 0 + C 1 x + C 2 x 2 + . . . . + C n x n , ∑ r = 0 n r + 1 2 C r = 2 n - 2 f n and if the roots of the equation f x = 0 are α and β , then the value of α 2 + β 2 is equal to (where C r denotes C r n ), 13, 10, 17, 20

MARKS App · Accepted Answer

1 + x n = ∑ r = 0 n C r x r ⇒ x 1 + x n = ∑ r = 0 n C r x r + 1 Differentiating w.r.t. x we get x n 1 + x n - 1 + 1 + x n = ∑ r = 0 n r + 1 C r x r Again multiplying both sides by x 1 + x n - 1 n x + 1 + x x = ∑ r = 0 n r + 1 C r x r + 1 Again differentiating w.r.t. x we get, d d x 1 + x n - 1 n x 2 + x 2 + x = ∑ r = 0 n r + 1 2 C r x r ∑ r = 0 n r + 1 2 C r x r = 1 + x n - 1 2 n x + 2 x + 1 + n x 2 + x 2 + x n - 1 1 + x n - 2 Putting x = 1 on both sides, we get, ∑ r = 0 n r + 1 2 C r = 2 n - 1 2 n + 3 + n + 2 ⋅ n - 1 2 n - 2 = 2 n - 2 n 2 + 5 n + 4 Now, f x = x 2 + 5 x + 4 = x + 1 x + 4 ⇒ α = - 1 , β = - 4 Hence, α 2 + β 2 = 17

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