Assertion :Let $f (x) = 6 x^{4} + 5 x^{3} - 38 x^{2} + 5 x + 6$ then all four roots of $f (x) = 0$ are real & distinct out of which two are positive & two are negative. Reason: $f (x)$ has two changes in sign in given order as well as when $x$ is replaced by $- x$ .

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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Both Assertion and Reason are incorrect

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Solution

The correct option is C Assertion is correct but Reason is incorrect
$f (x) = 6 x^{4} + 5 x^{3} - 38 x^{2} + 5 x + 6 = 0$
Hence
$(x - 2) (2 x - 1) (x + 3) (3 x + 1) = 0$
Hence
$x = 2, \frac{1}{2}$ and $x = - 3, \frac{- 1}{3}$
Hence
Both assertion is correct, and reason is incorrect.

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Assertion :Let f(x)=6x4+5x3−38x2+5x+6 then all four roots of f(x)=0 are real & distinct out of which two are positive & two are negative. Reason: f(x) has two changes in sign in given order as well as when x is replaced by −x.

The correct option is C Assertion is correct but Reason is incorrectf(x)=6x4+5x3−38x2+5x+6=0Hence(x−2)(2x−1)(x+3)(3x+1)=0Hencex=2,12 and x=−3,−13HenceBoth assertion is correct, and reason is incorrect.

Assertion :Let $f (x) = 6 x^{4} + 5 x^{3} - 38 x^{2} + 5 x + 6$ then all four roots of $f (x) = 0$ are real & distinct out of which two are positive & two are negative. Reason: $f (x)$ has two changes in sign in given order as well as when $x$ is replaced by $- x$ .

The correct option is C Assertion is correct but Reason is incorrect
$f (x) = 6 x^{4} + 5 x^{3} - 38 x^{2} + 5 x + 6 = 0$
Hence
$(x - 2) (2 x - 1) (x + 3) (3 x + 1) = 0$
Hence
$x = 2, \frac{1}{2}$ and $x = - 3, \frac{- 1}{3}$
Hence
Both assertion is correct, and reason is incorrect.