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Nature of Roots

Assertion :Th...

Question

Assertion :The number of irrational roots of $2 x^{100} + 5 x^{99} + 2 x^{98} + . . . . + 2 x + 5 = 0$ is $98$ Reason: If $a x^{2} + b x + c = 0$ has irrational roots then $D = b^{2} - 4 a c$ is not a perfect square.

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion is incorrect but Reason is correct

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Solution

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
Both Statement-1 and Statement-2 are individually true but statement-2 is not the correct explanation of Statement-1
$f (x) = 2 x^{100} + 5 x^{99} + 2 x^{98} + . . . . . + 2 x + 5$
$= x^{99} (2 x + 5) + x^{97} (2 x + 5) + . . . . . (2 x + 5)$
$= (2 x + 5) (x^{99} + x^{97} + . . . . . + x^{2} + 1)$
Also, $x^{98} + x^{96} + . . . . . + x^{2} + 1 > 1$
$∴ f (x) = 0$ has two rational roots.
Hence, number of irrational roots $= 100 - 2 = 98$

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