The roots of (x – 41)49 + (x – 49)41 + (x – 2009)2009 = 0 are -�

1.	The roots of (x – 41)49 + (x – 49)41 + (x – 2009)2009 = 0 are - (A) all necessarily real (B) non-real except one positive real root (C) non-real except three positive real roots (D) non-real except for three real roots of which exactly one is positive
Answer» Correct Option :- (B) non-real except one positive real root Explanation :- (x – 41)⁴⁹ + (x – 49)⁴¹ + (x – 2009)²⁰⁰⁹ = 0 f(x) = (x – 41)⁴⁹ + (x – 49)⁴¹ + (x – 2009)²⁰⁰⁹ f ′ (x) = 49(x – 41)⁴⁸ + 41(x – 49)⁴⁰ + 2009 (x – 2009)⁴⁸> 0 hence f(x) will cut x-axis only once. 1 real root .

The roots of (x – 41)49 + (x – 49)41 + (x – 2009)2009 = 0 are - (A) all necessarily real (B) non-real except one positive real root (C) non-real except three positive real roots (D) non-real except for three real roots of which exactly one is positive

Answer»

Correct Option :- (B) non-real except one positive real root

Explanation :-

(x – 41)⁴⁹ + (x – 49)⁴¹ + (x – 2009)²⁰⁰⁹ = 0

f(x) = (x – 41)⁴⁹ + (x – 49)⁴¹ + (x – 2009)²⁰⁰⁹

f ′ (x) = 49(x – 41)⁴⁸ + 41(x – 49)⁴⁰ + 2009 (x – 2009)⁴⁸> 0

hence f(x) will cut x-axis only once.

1 real root .

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The roots of (x – 41)49 + (x – 49)41 + (x – 2009)2009 = 0 are - (A) all necessarily real (B) non-real except one positive real root (C) non-real except three positive real roots (D) non-real except for three real roots of which exactly one is positive

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