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Assertion :If roots of the equation $$ x^{2}-b x+c=0 $$ are two consecutive integers, then $$ b^{2}-4 c=1 $$ Reason: If $$ a, b, c $$ are odd integer, then the roots of the equation $$4 abc
x^{2}+\left(b^{2}-4 a c\right) x-b=0 $$ are real and distinct.


A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Both Assertion and Reason are incorrect
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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
 According to statement $$ 1, $$ given equation is

$$x^{2}-b x+c=0$$

Let $$ \alpha, \beta $$ be two roots such that

$$|\alpha-\beta|=1$$

$$\Rightarrow \quad(\alpha+\beta)^{2}-4 \alpha \beta=1$$

$$ \Rightarrow \quad b^{2}-4 c=1 $$

According to statement $$ 2, $$ given equation is $$ 4 a b c x^{2}+\left(b^{2}-4 a c\right) x-b=0 . $$ 

Hence

$$D =\left(b^{2}-4 a c\right)^{2}+16 a b^{2} c$$

$$=\left(b^{2}+4 a c\right)^{2}>0$$

Hence, roots are real and unequal.

Mathematics

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