  Question

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 $$4 abcx^{2}+\left(b^{2}-4 a c\right) x-b=0$$ are real and distinct.

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion  B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion  C
Assertion is correct but Reason is incorrect  D
Both Assertion and Reason are incorrect  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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